Take a red, a blue and a green bucket. Ask a friend to put into them three red, three blue and three green balls in such a way that:

1. There are three balls in each bucket.

2. There is a blue ball in each bucket.

3. There are no green balls in the green bucket.Now point to a bucket (without looking inside it) and have him throw you one ball from it at random. Repeat the process until you have collected one red, one blue and one green ball.

What is the smallest number of balls collected in this way which cannot fail to include the selection required?

A variation of this puzzle appears in the book **Tantalizers** (1970) under the title “Billiard Balls”.

[tantalizer13]

]]>When I went to bed last night I glanced at my digital bedside clock just before dozing off. When I stirred briefly between one and two hours later, I glanced at the clock again and confirmed that I had been asleep for that long. And then I had another sleep and when I stirred again, an exact number of hours after first dozing off, I again confirmed this on the clock.

When I can’t sleep I often ponder on the numbers displayed on the clock (like noting at bedtime one evening that the 1056 on the clock, as illustrated above, was the product of my age and my house number). On those three occasions when I looked at the clock last night, one of the numbers which I saw was palindromic, another was a perfect square and the other consisted of consecutive digits in increasing order.

In fact what I’ve just told you in the previous paragraph is rather misleading: it isn’t true about the three actual times because on the middle occasion I was very drowsy and I didn’t look at the clock but at its reflection in the bedside table top. I didn’t realise my mistake immediately because the number I saw did seem like a reasonable time.

At what time did I first doze off?

[enigma838]

]]>I collect Russian dolls, the type where each doll can be opened to reveal a smaller one inside. I am particularly fond of my simple, single-coloured ones, which come in sets of five (and, unusually, have a hollow smallest doll). I have five lovely sets of them, each a different colour.

Alas, while I was out, my daughter Kira rearranged them so that each large doll now contains one each of the four other colours. She proudly tells me that no blue doll contains a doll that has a yellow doll anywhere within it. There is no doll that contains a pink doll with a red doll anywhere within it. And no yellow doll contains a green doll with a pink doll anywhere within it.

“By the way, have you seen my wedding ring?” I ask her.

“Ah, I put that inside the smallest blue doll” replies Kira.

Which coloured doll should l open first if I want to find the ring as quickly as possible?

[puzzle#224] [headscratcher224]

]]>“Be a dear”, said Aunt Agatha, “and write to that nice man about the things”.

“Which nice man about what things?” I asked.

“The man who collects beetles about the hymn-books”, replied my aunt without hesitation. “Here’s his address:

Ernest Baggins,

Mallards,

Appleton,

Kent”.“That doesn’t sound right, dear”, interrupted Aunt Maud. “I’m sure he’s not Baggins. Where’s my book? Yes, here we are:

Ernest Boggins,

Halyards,

Bladon,

Surrey”.“That’s not what I’ve got”, put in Aunt Jobiska. “I’ve got:

Edward Biggins,

Haystacks,

Cuxham,

Surrey”.“Not Biggins, Jobiska, Boggins”, this from Aunt Kate.

“Edward Boggins,

Pollards,

Appleton,

Sussex”.“Stuff!”, said Aunt Tabitha rudely, “He is called Ernest Buggins, poor man, and his address is:

Willows,

Bladon,

Surrey”.As no Aunt was willing to give way, I had to ring the vicar and he, it turned out, was both deaf and loquacious. However, I got the name and address in the end and found that each aunt had been right in exactly two out of her five particulars.

What is his name and address?

A version of this puzzle appears in the book **Tantalizers** (1970) under the title “Aunt Maud”.

[tantalizer12]

]]>The teams in the Midshires league play each other once during their season, which runs from mid-September to March. They each play up to three times a week, getting three points for a win and one point for a draw.

At the end of the past season, the teams’ total points were calculated and the teams were placed in decreasing order. The top team had one more point that the second which had one more point than the third, which had one more point than the fourth and so on until the penultimate team, which had one more point than the bottom team.

More than a third of the teams played in no draws at all, more than a third of the teams played in one draw, a quarter of the teams played in two draws, and the rest played in more than two.

How many teams are there in the league?

[enigma937]

]]>“Not one of your best, is it?”, smirked Michael, peering over Leo’s shoulder at the portrait he was painting. “The colours are so drab. Who is she?”

“The name’s Lisa”, said the model, smiling enigmatically from the other side of the easel.

“I’m trying to mix a glaze to perfect the tone of her face”, sighed Leo. “But I seem to have run out of paint”.

“Yes, about that”, said Michael. “I might have borrowed some for a ceiling. In any case, it looks like you’ve got two brownish dollops there”.

“One of them is equal parts yellow, red and blue. The other is five parts yellow, three parts red and four parts blue. But anyone can see her cheeks require 10 parts yellow, eight parts red and nine parts blue!” said Leo.

Lisa sat and listened quietly, with a knowing look in her eye. Or maybe sad, or bored; it is hard to say. But if Leo is to finish his portrait, in what proportions should the two dollops be mixed to produce the right tone?

This puzzle sees the name of the series switch from *Puzzle* to *Headscratcher*. Probably related to the book *Headscratchers* due to be published in October 2023 [link].

[puzzle#223] [headscratcher223]

]]>Old Charlie is night-watchman at the Kite Company. His parish consists of 12 buildings and a gatehouse. laid out thus:

His orders are to inspect all 12 buildings during the night. He is to inspect each building the same number of times, beginning with No 1, keeping to the paths shown and ending up finally at the gatehouse. Having inspected a building, he must inspect at least one other (or the gatehouse, which he may inspect as often as he likes) before inspecting that building again. Each stretch of path is 100 yards long, except 1-6 and 1-11, which are 200 yards.

Old Charlie has a conscience and rheumatism so he carries out his orders faithfully but walks not one yard further than he need.

How far must he walk in the night?

This puzzle appears in the book **Tantalizers** (1970).

[tantalizer11]

]]>We arrange the 26 letters of the alphabet in a row as follows:

THEQUICKBROWNFXJMPDVLAZYGS(*)Now take any letter, say

P, and find the longest chains in (*) in alphabetical order ending withP. We find some of length 5, for exampleEIKMP, but none any longer. Next we find the longest chains in (*) in reverse alphabetical order starting withP. We find some of length 3, for examplePLG, but none any longer. We sayPhas alphabetical length, α=5, and reverse alphabetical length, ω=3.

Question 1:I have chosen a letter which comes beforePin the alphabet and to the left ofPin (*). Can you say for certain my letter has α less than 5?

Question 2:I have arranged the 26 letters in a new row (**). Can you say for certain that if you choose a letter in (**) and I choose a different letter in (**) then they will have different α’s or different ω’s?Now I want you to write on a piece of paper, a list of 25 possibilities for α and ω so [α=1 ω=1], [α=1 ω=2], …, [α=1 ω=5], [α=2 ω=1], …, [α=2 ω=5], [α=3 ω=1], …, [α=5 ω=5].

Next I want you to take each letter in (*) and work out α and ω for it and mark it on the list, for example you will write

Pagainst [α=5 ω=3]. Unfortunately, some letters will have a combination of α and ω that is not on the list, for instanceXhas α=6 and ω=4.

Question 3:Can you arrange the 26 letters of the alphabet in a row so that every letter has a combination of α and ω that is in the list? If your answer is “yes” then give such a row.

Question 4:Can you be certain that you will be able to find in my row (**) a chain of 6 letters that are in either alphabetical or reverse alphabetical order?

[enigma837]

]]>Being sentimental, Patty likes to use her grandmother’s beam scales when weighing out ingredients to make a birthday cake for her own granddaughter. The only problem is that the scales aren’t accurate as the two arms are of slightly different lengths.

To overcome this, she uses both pans and measures half the required quantity in each. For example, to weigh 2 kilograms of flour, she will put a 1-kilogram weight in the right-hand pan and weigh the flour on the left-hand pan, then place the weight in the left-hand pan and weigh a second batch of flour on the right-hand pan. The combined portions of flour will, she thinks, weigh exactly 2 kilograms.

Is she right or will she have more or less than 2 kilograms?

[puzzle#222]

]]>“An inelegant crime”, observed Holmes, surveying the wrecked library. The mangled body of Sir Plutus Gnome sprawled on the rug and beside it lay the hammer and sickle which had produced its present unpleasing condition. Pocketing his magnifying glass, Holmes turned to the five police inspectors.

“Well, gentlemen, what does the evidence tell us of the culprit?”

The first inspector cleared his throat: “We are looking”, he said, “for an unmarried right-handed woman who is shorter than the deceased.”

“No”, said the second, “for an ambidextrous married communist the same height as the deceased.”

“No”, declared the third, “for a married man who is an anti-communist and taller than the deceased.”

“No”, put in the fourth, “for an unmarried, left-handed, female communist.”

“I disagree”, remarked the fifth, “we want a right-handed man, taller than the deceased, who has no feelings about communism either for or against.”

Holmes gave them a glance both penetrating and scornful. “You have done very well, gentlemen” he pronounced. “Each of you is right in exactly two particulars. Now, it is clear that there are only four possible suspects; George Crabtree, the victim’s bachelor nephew, Miss Pringle, his secretary, Henry Hetherington, his accountant, or Henry’s wife, Mary. You will no doubt be able to determine which of the four perpetrated the deed.”

Who mangled the millionaire?

This puzzle appears in the book **Tantalizers** (1970) under the title “The Mangled Millionaire”.

[tantalizer10]

]]>A group of friends were in the Rose and Crown, debating who should pay for the round. By chance they were seated round the table in the sequence Alan, Brian, Charlie, David … with alphabetically consecutive first initials up to Mr Smith’s.

They took part of a pack of cards, shuffled it, and placed it face down on the table. They agreed to draw one card each, Alan, then Brian, and so on round and round the table, until someone drew a black card. That man would buy the drinks. If they had studied the cards first, they would have discovered that they had more than half the pack and that they had equal chances of drawing the first black.

In the event, the drawing process lasted as long as it possibly could with that selection of cards. What was the initial of the man who bought the drinks?

[enigma836]

]]>“Drat”, said Ron the reporter. “Now the Logical World Cup is over, the editor wants to know how many games each team won, drew and lost, but all I have are the points totals”.

“Maybe I can help”, said Martha the mathematician. “Show me what you’ve got”.

Ron passed her the sheet of paper he had been glaring at:

“Hm. I presume it was a round robin with three points for a win and one for a draw?”

“But of course”, said Ron.

“Then I can tell you the other columns” said Martha.

Can you?

[puzzle#221]

]]>Now that our school has introduced the personalised time-table, it is no fun being a housemaster. Take Digweed, for instance.

He has two years to go before A-level and must take exactly three subjects in each year. He is to do Geography for at least one year and English for exactly one year. If he does English in the second he must take French in the second. If he does English in the first, he must take Geography in the first. If he does not do French in the second, he may not do Geography in the first.

If he does not do History in the first, he may not do Geography in the second. If he does not do French in the first, he is to do History in the second. His first year may not include both French and Geography; nor his second both French and History.

Digweed tells me with a smirk on his face that his time-table is impossible. But he has forgotten that I can force him to do Latin, if need be.

What subjects will he be doing in each year?

This puzzle does not appear in the book **Tantalizers** (1970), but a similar puzzle appeared as **Enigma 260**.

[tantalizer9]

]]>Whenever I play darts I keep track of my score by writing down how many points I scored for each go (consisting of three darts) followed by the number of that go. For example, if I scored 27, 154 and 84 on my first three visits I would have written 2711542843.

After a recent game I noticed that if the digits were consistently replaced by letters, with different letters for different digits, then it read:

TREBLETOPSATDARTSMy total score was a prime number.

Please find the value of

PLEASE.

[enigma835]

]]>There was controversy at this year’s Artificial Intelli-Vision song contest, in which each of the competing countries used Al to generate their entries.

Every nation had a judging panel that gave a score to each of the others. The “songwriters” all tried to engineer a higher score for their country by letting an Al generate their ditty as a danceable blend of one other country’s all-time favourite tunes.

This led to a strange outcome. Each judging panel awarded 10 points to the song tailored to its national preferences and the same lower number of points to all of the others. For example, the Transylvania panel gave a perfect 10 to Ruritania’s artificially intelligent effort “Everybody Let’s Dance Last Night Tonight”, while giving only a 7 to all the rest.

The song contest’s board decided to restore artistic integrity to this prestigious event by deducting the inflated 10 from each country’s set of scores. After this, the grand total of all scores was 222, with no two nations tied for any position.

Can you figure out how many countries took part and how many points the winning song scored?

[puzzle#220]

]]>When the old destroyer,

Fantastic, was turned into a floating missionary chapel, instead of being scrapped, the following worthies were present at her re-launching:The Bishops of:

Armuth

Bangor

ConeAdmirals:

Sir Desmond Drake

Sir Evelyn Easy

Sir Francis Fish

Sir Gregory Grogg

Sir Harry Hornpipe.Afterwards they repaired to the wardroom to drink the toast of “The Church Afloat”. It was no thimble-sized toast and presently the eight dignitaries were arm in arm on the deck and singing lustily. Everyone, I regret to report, was wearing someone else’s hat. This is what the bishops sang, each topped with a cockaded nautical tile:

Armuth:My hat is on the head of a man,

whose hat is on Sir Desmond.

Bangor:My hat is on the head of a man,

whose hat is on the head of a man,

whose hat is on Sir Evelyn.

Cone:My hat is on the head of man,

whose hat is on the head of a man,

whose hat is on the head of a man,

whose hat is on Sir Francis.Sir Harry Hornpipe, resplendent in episcopal mitre, then opened fire:

Hornpipe:My hat is on the head of a man,

whose hat is on the head of a man,

whose hat is on the head of a man,

whose hat is on the head of a man,

whose hat I’m proudly wearing.Luckily it started to rain at this point, before anything worse befell. It thus only remained to restore the hats to their rightful owners.

Who was wearing whose?

A variation on this puzzle appears in the book **Tantalizers** (1970) under the title “The Sailor’s Puzzle”.

[tantalizer8]

]]>This week I have replaced every letter of the alphabet by a digit. Of course, that means that some digits may represent several different letters, but any particular letter is replaced by the same digit throughout.

With my particular use of the digits you will find that each of the following seven:

TWO

THREE

FOUR

FIVE

SIX

SEVEN

EIGHTis a number and that although one of the three-figure numbers is not a square all the remaining six numbers are perfect squares.

Find the value of

ROOTS.

[enigma834]

]]>“You know that debt you owe me?”, says Svengali, rather menacingly. “I am prepared to write it off — but only if you have a bit of luck”.

He takes out a regular pack of playing cards and gives them a thorough shuffle. “Now”, he declares, “I am about to turn over all the cards one at a time, but first I want you to predict the position of the second red queen that I will turn over. If your guess is correct, I will write off your debt”.

My odds aren’t high. There are 52 cards in a pack. Only two of them are red queens the queen of hearts and the queen of diamonds. And only one of those will be the second red queen to be turned over.

“I will give you one bit of advice”, says Svengali, noting my pensiveness. “Don’t choose the top card, because by definition that cannot be the second red queen — though it might be the first”.

Which card position in the range 2 to 52 should I nominate for the second red queen? Or should I just pick a random number?

[puzzle#219]

]]>When George, Harris and I go on the river, we take it in turns to row, steer and cook. With George rowing and Harris steering the boat goes slower than with Harris rowing and me steering but faster than with me rowing and George steering.

We have found that the boat’s speed is a simple sum of a rowing element (in knots) and a steering element (in knots) and that each of us makes a measurable contribution at the oars or the helm. George is the best helm or the worst oar or both.

Each of us is the best of us in one department — I mean rowing, steering or cooking — and the worst in another.

Which of us is the best cook?

Both this puzzle and the one published the previous week were **Tantalizer 7**.

This puzzle appears in the book **Tantalizers** (1970).

[tantalizer7b] [tantalizer7]

]]>Anna and Wesley each have seven book tokens and each token is for a whole number of pounds. The total value of Anna’s tokens is no more than £126 and the total of Wesley’s is no more than £127.

Anna sat down at the table and worked through all the possible combinations of her seven tokens, finding the total value for each combination. She did this by first taking each token by itself, then each pair of tokens, then each combination of three tokens and so on. She was looking for two combinations which had the same total. Wesley did the same thing with his tokens.

Anna had two cousins and she wanted to give each of them some tokens so that each received the same total value.

Wesley wanted to the same for his two cousins.

On the basis of the information we have, answer each of the following questions “Yes” or “No”:

(a) can we say for certain that Anna was able to find two combinations which had the same total?

(b) can we say for certain that Wesley was able to find two combinations which had the same total?

(c) Can we say for certain that Anna was able to give some tokens to her two cousins so that they each received the same total value?

(d) Can we say for certain that Wesley was able to give some tokens to his two cousins so that they each received the same total value?

[enigma833]

]]>I was eating a sandwich on a park bench when a horde of children from the local school descended on the playground in front of me and divided into two groups.

I couldn’t help overhearing the kids by the swings shout across to the kids near the climbing frame: “If two of you run over here, our group will be double yours!”.

The kids by the climbing frame replied: “But if two of you come over here, the groups will be the same size!”.

The mathematician in me couldn’t resist figuring out how many children there must be in each group for this to be true. But l also discovered something else. If a number of children going one way makes one group double the other, and that same number going the other way makes the two groups the same, the total number of children must always be a multiple of… what?

[puzzle#218]

]]>The courtyard at Jude’s College is paved with large regular hexagons (as shown).

Professor Xerxes is at present stationed on No. 1 and his old enemy, Professor Youthful, on No. 7. To settle their latest scholars’ feud, they have agreed to a contest under these terms:

1. They will take it in turns to move.

2. Each will move at any turn only to a hexagon he “commands”.

3. Each commands at any time all and only those hexagons which lie in a straight line across any side of the hexagon he is then standing on. (Thus X now commands 2, 3, 4, 12, 11, 10; but not 13, 15, 7).

4. Neither may move to or across any hexagon commanded by the other.

5. Whoever is first unable to move will be deemed the loser.Xerxes has won the toss. Should he move first? If so, what is his winning strategy? If not, what is his winning strategy after Youthful’s move?

Both this puzzle and the one published the following week were **Tantalizer 7**.

In the book **Tantalizers** (1970) a modified version of this puzzle appears as “Battle of Minds”.

[tantalizer7a] [tantalizer7]

]]>The country in which Uncle Fibo at present resides has introduced a peculiar new tax system. It is based on the following principles:

1. All taxable income is taxed at a “basic tax rate”.

2. On all taxes payed, a rebate is given, calculated at the basic tax rate.

3. All tax rebates are treated as taxable income.Uncle Fibo has calculated that, at the basic tax rate applicable to him, the effective tax which he has to pay would be the same if he paid his entire taxable income for a given period over to the taxman and then received back from him a once-off non-taxable rebate on this amount, calculated at the basic tax rate.

What is Uncle Fibo’s basic tax rate, given to the nearest tenth of a percentage point?

[enigma832]

]]>It made sense at the time. The enemy was coming and Neville the Mighty But Not That Bright had created a circular moat in which he placed a fast-swimming, flesh-eating moat monster. He stood at the centre of the island, sword in hand, guarding the only bridge, and ready to fight whoever dared to cross. The plan was foolproof… until the enemy burned down the bridge.

Now, Neville is stranded on an island encircled by a moat with a monster in it. He can just barely run and jump the moat, but the monster swims four times as fast as Neville can run and sense where he is at all times. If Neville tries to jump the moat while the monster is directly beneath him, he will be snatched out of the air like a sausage being caught by a dog.

Is Neville doomed by his own plan? Or is there a chance of escape?

[puzzle#217]

]]>Five old bachelors live together at Nag’s End, where they are thoroughly overrun by their pets. Each pet is a perfectly normal specimen of its kind. The pets have between them 10 heads and 30 legs. The only quadrupeds are cats or dogs and there are more dogs than cats. The two dog owners keep at least one cat each.

Arthur keeps a budgie but no dogs. Brian owns six legs (including his own two). Charlie and Dan keep the same number of pets and their pets have, collectively, the same number of legs. Edward has had some trouble in training his pets not to follow him upstairs at night.

Who keeps what?

In the book **Tantalizers** (1970) a reworded version of this puzzle appears.

[tantalizer6]

]]>In the snooker match between Davis and Whyte, the winner was the first player to win a certain number of frames, not greater than 16; the match ended as soon as one player had won the required number of frames. It is impossible for a frame to be halved.

Davies won all the frames played (including the first frame), whose numbers were perfect squares. Whyte won all the frames played whose numbers were primes; the sum of the numbers of the frames won by Davies was exactly the same as the sum of the numbers of the frames won by Whyte. Davies won the match.

How many frames did each player win?

Who won the penultimate frame?

[enigma831]

]]>Let’s play a game, dear reader. I have placed three stones on a number line: one on 0, one on 20 and one on 40. You and I shall take the game in turns; on a turn, a player picks up one of the two outer stones and places it on a whole number between the other two stones. The game continues until the stones occupy three consecutive whole numbers, whereupon no further moves can be made, and the game ends. The player who made the final move is the winner.

As the game is played by my rules, I will let you have the first move. I feel compelled to warn you, however, that only perfect play will result in my defeat. So here is your challenge: to play this game of stones and claim victory over its creator.

[puzzle#216]

]]>George and Mabel, who live at one of the 24 towns shown on this map of the island of Angula, have been planning their summer motoring holiday.

George proposed that they should motor along every road shown once and once only. Mabel agreed this was a feasible plan but thought the town they would finish at was a horrible dump. She proposed instead that they should seek the afternoon sun, never motoring north, east or north-east. George agreed and they have also agreed upon their destination. They are now arguing which of the eight possible routes they will take to get there.

Where do they live and where will they be spending their holiday?

In the book **Tantalizers** (1970) a reworded version of this puzzle appears under the title: “The Beggerman’s puzzle”.

[tantalizer5]

]]>On my calculator each digit is formed by some of the seven liquid crystal strips being illuminated. For example the “8” uses all seven strips but the “1” only uses two.

Recently the calculator developed a fault. I displayed a single digit. Then at least one of the illuminated strips went out leaving a different digit displayed … Then at least one of the illuminated strips went out leaving a different digit displayed … I forget how many times this happened but eventually I read off the digits which I saw, and used them in that order to form one number. This number was not divisible by any of the individual digits used in it (in fact it was a prime).

What was the number?

[enigma821]

]]>The team members were assembled for their regular meeting with their eccentric manager.

“Before we get to the main agenda, we’ll observe our weekly ‘Trivia Tuesday’. Today’s trivia titbit is that the month and day are the same number. It is 4 April, which is the fourth day of the fourth month. How about that?”

The rest of the office rolled their eyes. “Seems like the boss forgot until the last second again”, whispered Grace. “I guess he just got lucky with the date”.

Clancy thought for a moment. “Sadly, it isn’t the only time this year when he’ll be able to use the same kind of ‘trivia’ for Trivia Tuesday”.

Without looking at a calendar, can you work out how many more Trivia Tuesdays after 4 April will fall on the

nth day of thenth month?

[puzzle#215]

]]>One of the best loved sights of the Roman Arena was a duel to the death between a Gladiator and a Retiarius. The Gladiator, being in armour and carrying a sword, was slow in movement but lethal at close quarters. The Retiarius, having no armour but carrying a net and trident, was most deadly at a distance.

Those who wish to test where the odds lay for themselves will, in these softer days, have to make do with a diagram.

Let us suppose that Gladiator starts at 32 and Retiarius at 1 and that they move in turn. Gladiator moves three circles at each turn and Retiarius four. Both must always move along the lines but can change direction or double back during their move. The duel is won by whoever first lands on top of his opponent at the end of a turn.

Can either player be sure of winning? If so, who?

In the book **Tantalizers** (1970) a reworded version of this puzzle appears under the title: “The Lion and The Unicorn”.

[tantalizer4]

]]>In an inebriated state I tried to address two letters to each of five friends. I have listed my efforts below. Each envelope has four items:

– first name

– surname

– house name

– townAll the correct items occurred somewhere on the ten envelopes. But on each envelope the four items which I wrote all turned out to be from different people.

1: David Davis, Rose Cottage, Ely

2: Brian Clark, The Meadow, Carlisle

3: Ed. Andrews, Riverside, Bradford

4: Alan Eyres, Waters’ Edge, Altrincham

5: Ed. Clark, Belle View, Carlisle

6: David Andrews, Waters’ Edge, Ely

7: Clive Brown, Belle View, Doncaster

8: Brian Eyres, Riverside, Doncaster

9: Clive Clark, Riverside, Ely

10: Ed. Davis, Rose Cottage, Carlisle.What is Brian’s name and address?

[enigma830]

]]>My burglar alarm won’t stop beeping. I need to enter the four-digit code, but I just can’t remember what it is.

I do remember that I chose a number that is divisible by 7. I also know that I chose a number in the thousands. I recall that the numbers formed by the first three digits and the last three are divisible by 7 as well. I also deliberately chose four different digits. Finally, I know that if I add all the digits, their sum isn’t divisible by 7, but if I add the digits of

thatsum, the resultisdivisible by 7.Can you help me work out what my code is to stop the incessant beeping?

[puzzle#214]

]]>My friend Jones has food poisoning, which serves him right. Last night he thought he would impress two girls by taking them to dinner at the newly opened Café de Gaulle. Since the girls have survived the ordeal, we may assume that the trouble lay in a dish which Jones ate and they did not.

The menu was this:

Potage Tiede … 1s

Haricots sur Toast … 2s

Coctaile de Crevettes … 3sBulle et Couic … 4s

Crapeau dans le Trou … 5s

Trotteurs de Cochon … 6sFromage Souriciere … 2s

Morceau Singulier Gallois … 3s

Becasse Ecossaise … 4sJones tells me that each of them ate one item from each course and that, not counting tips etc., he paid eight shillings for himself, nine shillings for Polly and 10s for Gladys. No dish eaten by Polly was also eaten by Gladys.

He was in fact able to recall what the girls had eaten and so even though he had forgotten what he ate himself, we could deduce what had poisoned him. The offending dish can, however, be deduced merely from the information given so far plus the information that someone had Becasse Ecossaise.

Which is it?

In the book **Tantalizers** (1970) a reworded version of this puzzle appears under the title: “Café des Gourmets”.

[tantalizer3]

]]>My word processor has developed a fault. There’s one particular digit which it will not type: if you press the appropriate key absolutely nothing happens.

I can, for example, type the cube of 4 correctly, but not the square. Whereas for the larger number 78 (whose number of digits is less than the one-figure number I cannot type) I can correctly type both its cube and square.

Which digit do I keep missing out?

Between the **Enigmatic Code** and **S2T2** sites there are now 3000 puzzles available.

On **Enigmatic Code** there are now 1658 **Enigma** puzzles available (which leaves 134 remaining to post). All 90 puzzles from the **Puzzle** series are available, as well as 215 from the **Tantalizer **series (and about 283 that are not yet posted). And we have all puzzles from the current **Puzzle #** series (which is ongoing, and most recently reached **Puzzle #213**).

And on the **S2T2** site there are currently 840 **Teaser** puzzles available (these are also ongoing, and has just reached **Teaser 3156**, so there are quite a lot of those remaining. But I have been working through the published books of puzzles and newspaper archives that are available).

Along with a few additional puzzles that brings the total to the magical 3000.

If you have been playing along with me and have solved all the puzzles posted so far, then well done! It has been quite a journey.

As long as I have the time I will keep posting puzzles to the sites. Thanks to those who have contributed to the site, either by sourcing puzzles or sharing their solutions.

Happy Puzzling,

— Jim

[enigma829]

]]>Older age does bring some benefits. My daughters Kate and Laura have offered to help me by taking on the maintenance of my garden, which is rectangular with a small, rectangular vegetable plot in one corner. The remainder is lawn.

To make it fair for them, I have agreed that my last job in the garden will be to partition it into two with a straight fence, with each daughter getting the same area.

Kate suggested that we forget about the vegetable plot, and only divide the lawn. She sketched a line on the diagram that would give them each exactly half the lawn (with no awkward pinch point to get the mower through). Laura, meanwhile, drew a fence that would divide the lawn and the vegetable patch into halves. To make their lines, neither daughter needed to measure anything, they just needed a straight edge.

Can you draw the lines on which Kate and Laura propose to build fences?

[puzzle#213]

]]>Frank French, our local chess secretary, was writing out the results of our all-against-all tournament, when in popped Barbara Bocardo, the well-known logician.

“What gives?”, she asked, pouncing on the score sheet, which looked like this:

“The half points are draws”, he explained. “I’ve filled in all there were and next I shall record the 1’s (wins) and 0’s (losses)”.

“Don’t do that. Let me guess. How did you do yourself?”

“Well, I drew in four of the five rounds, as you see. But alas, I finally shared bottom place with Alapin, my opponent in the first round.”

“A lot of draws, surely?”

“Yes. At least one in every round. Each of us drew in at least two consecutive rounds.”

“Did the winners (I see there must have been two of them), meet in the second round?”

“No”.

“Good. It is now possible to deduce whom they played in the last round.”

Can you perform this logical feat?

[tantalizer2]

]]>I had a day at the health club recently. I planned to have one full session of squash, one full session of badminton and one full session in the sauna (but not necessarily in that order) with at least a one-hour break between each of the sessions. The club’s timetable of sessions is shown above.

By coincidence my colleagues Mark and Jenny also spent the day there with the same idea in mind (although none of us necessarily did any of the activities together).

Our boss (who knew all the above facts) tried to telephone me at one stage but was told I was busy. From this he was able to work out my exact day’s schedule.

An hour later he tried to telephone Mark but was told he was busy and he was told the activity which Mark was engaged in. From this my boss was able to work out Mark’s exact schedule.

Another hour later he tried to telephone Jenny but was told she was busy and he was told the activity which Jenny was engaged in. From this my boss was able to work out Jenny’s exact schedule.

Which is the correct order of the men’s, women’s, and mixed sessions in the sauna?

[enigma814]

]]>“Why isn’t the sound working?”, Mum muttered as she hit the mute key on the remote control.

“You’ve probably got the wrong remote, Mum”, Sam said. “Remember, it’s the long, thin one for the television, the wide one for the set-top box and the little one for the speakers. Which one did you mute?”

“I can’t remember”, said Mum, as she got her thinking cap on to try to fix things.

To get sound, all three remotes have to be unmuted. Mum came up with the most efficient system for cycling through the possible combinations of muting and unmuting, and got to work.

What is the maximum number of presses needed if she wanted to be sure of getting the sound back?

[puzzle#212]

]]>Kappa, Lambda, Mu and Omicron are at present uneasily seated in the Warden’s study at Jude’s College, awaiting summonses from the committee which will appoint one of them to the vacant Fellowship in Greek Literature. Each is hugging his only published work and each suspects that the post will go to the author of the longest, irrespective of all possible merit.

From their stilted but cunning conversation, the following facts have so far emerged:

Each book has a whole number of pages over 100.

Only Lambda’s book and Mu’s book have the same number of pages.

The total number of pages in all four books is 500.

Mu then asked Omicron whether the number of his (Omicron’s) pages was a perfect square. From Omicron’s answer Mu and Kappa made silent and independent deductions with impeccable logic. Mu deduced that Omicron’s book was the longest. And Kappa, who was not a perfect square, deduced that Omicron’s answer was not the truth.

How many pages are there in each man’s book?

This was the first **Tantalizer** puzzle published in *New Scientist*. It was accompanied by the following introduction:

This is the first of a series of logical puzzles compiled by Martin Hollis. No mathematical knowledge is required for their solution. A new puzzle will appear each week, and the answer will be printed in the following week’s issue.

[tantalizer1]

]]>The schedule of matches has been drawn up for the next tournament between Albion, Borough, City, Rangers and United, in which each team will play each of the other teams once. Two matches will take place on each of five successive Saturdays, each of the five teams having one Saturday without a match.

Two of the five teams will be meeting their four opponents in alphabetical order. Given this information you could deduce the complete schedule of matches if I told you either one of the matches scheduled for the first Saturday.

1. Which teams will be meeting their opponents in alphabetical order?

2. Which matches are scheduled for the first Saturday?

[enigma813]

]]>Debbie and Hoi are playing a game where they take turns to cross out numbers written on a piece of paper.

Each player must cross out a divisor or a multiple of the number most recently crossed out. The first player who is unable to cross out a number loses.

Hoi goes first and crosses out 11. Debbie smiles, knowing she can now win in three moves. What number does she cross out?

[puzzle#211]

]]>Seven couples rented a villa for the week in Majorca. They arrived on a Monday and dived beneath the sheets. They didn’t see much of the ocean but devoted a fabulous week to musical beds instead.

Partners were changed daily. Angela, for instance, spent Monday with Tommy, Tuesday with Upton, Wednesday with Vaughan, and Thursday with William.

Barbara kicked off with Xerxes followed by Yvan, Upton, and Zeno in that order. Cutie partnered with Tommy on Wednesday, Xerxes on Thursday, and William on Friday.

Delia spent Monday with Vaughan, Wednesday with William, Friday with Xerxes, and Saturday with Yvan. Yvan devoted Thursday to Esther and Monday to Fiona, whose partner on Thursday was Upton. Esther and Xerxes spent Saturday together, as did Fiona and Tommy. Esther and William made their whoopee on Monday.

These facts are kindly supplied by Gillian, who adds demurely that Sunday’s pairs were all decently married to one another. This was not only proper but also necessary for all possible mixed pairs to have had a day’s delight by the end of the holiday.

Who is married to whom?

[tantalizer313]

]]>I once had a job in a department store, working in the House Name Department. For example, I had to make “Dunromin” (taking 8 letters) and “Four hundred and twenty-one” (taking 23 letters).

On one occasion a customer ordered his three-figure house number spelt out in this way. I prepared the invoice and wrote on it (in figures) the number of letters used. But the invoice clerk thought this referred to the house number so he replaced it (in figures) with the number of letters that house number would take.

His superior again thought the number referred to the house number so he replaced it (in figures) with the number of letters that house number would take.

The auditor again thought the number referred to the house number so he replaced it (in figures) with the number of letters that house number would take. He then prepared the bill accordingly. The customer turned out to be very lucky: at each stage in this long process the number had been reduced and the bill was for less than half what the Es alone would have cost.

How many letters should the customer have been charged for?

[enigma812]

]]>I have a train to catch! I was planning to drive or cycle to the station, but, to my dismay, I realise that my car has a flat battery and my bike has a puncture.

I look at the clock and realise that, based on past experience, if I were to set out right now and walk to the station, I would miss my train by 10 minutes. However, if I were to run at my top speed, I would be 15 minutes early. It is freezing outside and I don’t fancy waiting on the platform for any longer than I have to.

It is too late to do any more calculations as I need to leave right now. I decide to run for half the distance and walk the rest of the way. But where are my running shoes? After a frantic search, I find them.

I leave the house 2 minutes later. Do I make it in time to catch the train?

[puzzle#210]

]]>Uncle Delroy had £3 for each of his nephews Sam and Tom and his niece Anna. For each child Uncle Delroy changed the money into 300 pence. He gave them 300 tests and they got a penny for each test they succeeded at.

Each test involved three boxes labelled 1, 2 and 3. While the child was not looking Uncle put a coin in one of the boxes. The child then chose one of the boxes. Uncle then chose one of the two other boxes; he always chose an empty box. He opened it and showed it was empty to the child. He then asked the child if he or she wished to change their choice to the other unopened box. After the child has either changed boxes or stayed with their original choice, they opened their final choice of box; if it contained the coin then they kept it otherwise the coin went to Uncle Delroy’s favourite charity.

Sam reckoned that Uncle showing him an empty box was no help so he never changed his choice. On the other hand Anna reckoned that uncle showing her an empty box did tell her something so she always changed her choice. Tom did not change in his first test but, for every other test, he changed if and only if he had lost in the previous test.

After the tests the following facts emerged. They each made a correct initial choice in the same number of tests. Tom lost in his first test, won in his last test and never won in two consecutive tests. Uncle paid out a total of £4.

How much did each child get?

[enigma811]

]]>There was a knock at Professor Numero’s door. It was a postal worker.

“Excuse me disturbing you, but can you help? This letter has a cryptic address. I can’t make head or tail of it:”

To the Resident,

The house with a number whose digits when multiplied together give five times what they sum to, Long Road.The professor pondered. “Well, you’ve come to the right road. And, luckily, there’s only one house number in the road with this mathematical property — the one at the end, where Colonel Crypto lives”.

As you would expect, the houses in Long Road are numbered consecutively from 1 upwards, with no missing numbers.

How many houses are there in the road?

[puzzle#209]

]]>In rugby union a try is worth 7 points if converted or 5 points if unconverted; a penalty goal or drop goal is worth 3 points. There are no other forms of scoring.

1. If in a match the winning side’s score is such that they cannot have scored any unconverted tries and the losing side’s score is such that they must have scored a converted try, what is each side’s score?

2. If in a match the winning side’s score is such that you know the number of tries that they scored and the losing side’s score is such that you cannot be sure of the number of tries that they scored, what is each side’s score?

3. If in a match the winning side’s score is such that they must have scored at least one penalty goal or drop goal and the losing side’s score is such that even if I tell you their score and the number of tries that they scored it is possible that you still cannot be sure how many of those tries were converted, what is each side’s score?

[enigma809]

]]>Ivor Plant is the head gardener of Lady Bird’s estate. Her large chrysanthemum garden needs to be weeded and pruned, so he assigns his two apprentices, Lupin and Heath, to the rather tedious task. The garden consists of a central 4-metre-sided square (pink) inscribed in a circle, and four outer areas (blue) enclosed in semicircles that are connected to the square’s corners.

“If you give me two of your chocolate biscuits, I will let you pick whichever area you want to weed: the outside or the inside”, says Heath to Lupin. Always eager to get out of extra work whenever possible, Lupin agrees.

If he is looking to weed the smaller of the two areas, should he choose the blue or the pink section?

[puzzle#208]

]]>Old George was recalling the long defunct West Wessex Railway in the snug at the Railway Arms the other night. There were nine stations – Axle, Bundle, Cordwain, Dawdle, Egdon, Foxfair, Gudgeon, Hangover and Inkwell – all exactly one mile apart in alphabetical order in a straight line.

George was the ticket inspector and took his duties seriously. He used to board the early train at Hangover and thereafter alight at a station every so often and wait for another train from there. It was a long, grimy day, since, of each pair of stations more than two miles apart, he boarded a train at one and got off it at the other. Each of these pairs, however, were only so used once during the day and only in one direction. Nevertheless he was glad to step out of the final station and into the snug of the Railway Arms.

Where is the Railway Arms?

[tantalizer312]

]]>Having a few moments to spare one Thursday, I decided to measure the dining table. This is a fairly conventional piece of furniture in dark oak, rectangular in shape, the longer side less than double the width.

It emerged that whether the surface area is expressed in square yards, square decimetres, square feet, hectares or square light-years, the first significant digit is the same. Furthermore, whether the length of the table is expressed in yards, miles, millimetres or light-hours, the first significant digit is again the same one, and this applies also to the length of the diagonal.

The length of the perimeter is an exact whole number of half-inches, the area in square centimetres is an integral number which is a perfect cube, and the speed of light in my dining room is 0.3 kilometres per microsecond.

Please ascertain the width of the table in feet, to four significant figures.

[enigma810]

]]>I recently had the incredible opportunity of talking to an intergalactic traveller about her encounter with four extraterrestrial beings.

“The aliens have purple skin, long, floppy legs and large orange eyes”, she told me. “And they revealed to me the true nature of dark matter”.

I leaned in close with excitement.

“Unfortunately, I don’t remember anything about that. I do remember a great puzzle they told me, though. I asked them their ages, and they said that if you sum up the ages of only three of them, the possible totals are 24, 53, 54 and 61. From this, they told me that I could work out all four of their ages”.

Well, it isn’t exactly the secrets of the universe, but it will have to do. Can you work out the ages of the four aliens?

[puzzle#207]

]]>To keep fit the monks and novices of the monastery of St Tantalus play a vicious game in the cloisters called Guide the Missal. The rules are simple – mainly a matter of each side trying to stuff a missal by fair means or foul through a porticulum at the opponent’s end. But it is exercise almost as tough as the spiritual kind, as the winners are the first to score 21 goals.

By tradition it is monks versus novices. The novices play from the Refectory end and the monks from the Miserere end. But it has recently been proved that it is three times as easy to score from the Refectory end and a custom has grown up of changing ends at the halfway stage. To be precise the change over occurs when the novices have scored 11 goals. Eleven being more than half 21, the monks are wont to regard this as an indulgence to the still soft novices. And it is certainly true that the monks usually win.

You may well wonder why. Is it that extra prayer has toughened the monks? Or does the new custom need changing and, if so, by how much?

[tantalizer311]

]]>Susan is on a 2-day holiday, staying near one of the stations on the East-West railway line. There are stations every mile along the line. Tickets are sold in packs. The Go East pack contains a random selection of 9 tickets and each one is for travel of 1 or 2 or 3 or 4 or 5 miles in an easterly direction. The Go West pack contains a random selection of 5 tickets and each one is for travel of 1 or 2 or 3 or 4 or … or 9 miles in a westerly direction.

Susan’s plan for the first day of her holiday starts at her local station, where she will buy one pack of each kind. From the Go East pack she will choose a ticket at random and catch a train going east. When the ticket is used up she will alight; she will then take the pack for travel towards her home station, select a ticket at random, catch a suitable train and alight when the ticket is used up. She hopes to carry on in this way until she alights at a station she has been to before; she will then walk home.

The second day of her holiday again starts at her local station with her buying one pack of each kind. She will sort through the two packs and try to find a selection of East tickets and a selection of West tickets so that the total distances for the two selections are the same; she will then use all the selected tickets to make a trip that ends at her home station.

Q1. On day one, is there a possibility Susan might find herself on a station, wanting a ticket in a particular direction and having none of those left?

Q2. On day one, is she certain eventually to reach a station she has been to before?

Q3. On day two, is she certain to be able to find selections of tickets that meet her requirements?

[enigma807]

]]>“What ho!” boomed Aunt Nicola. I could tell she was about to talk cricket at me. “Have you been following the test match between Pythagorea and Lagrangia?”

“Auntie, you know I prefer Navier-Stokes to Ben Stokes”. “Well”, she said, “you might be interested — there’s maths involved! In their first innings, Lagrangia’s total score was a square number”.

“Innings?” I asked. “It’s the word for a team’s turn to bat. They each have two. In their first, the Pythagoreans also got a square number, but they were more than 300 behind!”

“That sounds insurmountable”. “You might think so”, she said. “Then, when Lagrangia batted again, they added a different square number — less than 50 — so that their lead and overall total were also square numbers”.

“Goodness”. “But the Pythagoreans battled back in their second innings”, she continued, “and the game ended dramatically in a tie”.

I then knew enough to work out the totals of the four innings in order. What were they?

[puzzle#206]

]]>Miss Prism, being duly sworn, was asked for her age but remained mulishly silent.

“Come, come, my good woman”, snapped the judge, “you must answer the court’s questions”.

Counsel intervened smoothly. “I understand, M’Lud, that she is certainly over 40 and under 50, if it pleases you, M’Lud”.

“It does not please me, Sir Ambrose; I am thinking of having her committed”.

This threat overcame some of Miss Prisms reticence and she hastily deposed as follows: “When I stop speaking, I shall have made exactly two true statements. My age is divisible by 2. It is a perfect square. It is not 42. It is a prime number. It is divisible by three”. At this point she stopped speaking.

“Perjury, I fancy”, sniffed the judge.

“With respect, M’Lud, I fancy she has answered the question, if Your Lordship will attend carefully to the various possibilities”.

How old is the lady?

[tantalizer310]

]]>In a recent maths test five statements were labelled

AtoE. StatementAwas “it is an odd two-figure number”, and statementCwas “it is a two-figure number which is an odd perfect square”.The other three statements (whose order I forget) were:

“it is a two-figure prime”,

“it is a two-figure number which is the sum of two consecutive numbers”, and

“it is a two-figure number with one odd digit and one even”.

Candidates who chose an easy question were given a particular two-figure number and asked to tick which applied:

Ais true

Bis true

Cis true

Dis true

Eis trueFor those whose chose a hard question the boxes had the following different headings and the candidates were asked to tick the box or boxes which were always true:

BimpliesC

CimpliesD

DimpliesE

EimpliesA

AimpliesBThe box or boxes which should be ticked were the same for both questions.

What was statement

D? And which of statementsAtoEwere true in the easy question?

There are now 1650 *Enigma* puzzles available on the site, and 142 remaining to post. So there is currently around 92% coverage.

[enigma808]

]]>The burial chamber of Queen Count-M-Up of the Mat e’Matic people has been discovered! Archaeologists know that her people greatly appreciated diversity in life, and they believe she was buried with a set of painted shields that illustrate this by displaying all possible combinations of three colours (gold, silver and blue) in four sections.

Opening the tomb, they quickly find seven complete shields and one broken one. What colour is the missing section? And when they discover the ninth shield, what will it look like?

[puzzle#205]

]]>If you ever visit Ratselheim, you will no doubt want to see the Stephankirche and the Volksmuseum. Moreover you may be tempted to travel from one to the other by subway. Don’t!

The map shows all lines and stations, each stretch of clear line being 1 km. But not every train stops at every station it passes through. So, by a triumph of absurdity, the distance between Stephanskirche and Volksmuseum by subway is a preposterous 16 km, involving several changes of train but never taking you over the same stretch of line twice.

Whereas if you walk it, the direct distance is … well, how far?

[tantalizer309]

]]>I recently entered a local darts tournament where, to make things more difficult, most of the standard board had been covered up, leaving only six of the segments and the outer bull on which to score. No two adjacent segments were uncovered. In my seven visits to the board each of my darts scored (with three darts for each of the first six visits, but I did not need all three on my last visit). I kept a note of the remaining total required after each dart (starting at 301 and ending at 0, not necessarily finishing with a double).

In this list of decreasing numbers I noticed that after each visit to the board the remaining total required was a perfect cube. I also noticed that all the other remaining totals required were “triangular numbers” (that is, the sum of some consecutive integers from 1 upwards). The sum of the numbers on the six uncovered segments was also a triangular number.

Which segments were uncovered?

(A standard dartboard has twenty segments which are, in clockwise order from the top, 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5. Each segment has a single section, a double section and a treble section. The outer bull scores 25).

[enigma806]

]]>My son’s book of mathematical magic tricks includes this one:

1. Think of a whole number from 1 to 50.

2. Add 9 and double the result.

3. Multiply by 6 and add 12.

4. Subtract 60.

5. Divide by 12 and subtract 4.

6. Subtract your original number.

7. And the answer is (drumroll…) 1!Unfortunately, when he is reading, my son sometimes repeats a line and sometimes he skips one. That happened with this trick. He did one of the lines twice and then missed the final instruction to subtract his original number. Yet, by an incredible fluke, he still managed to end up on the number 1.

What number did my son think of?

[puzzle#204]

]]>Wanting a sure thing for the 2:30, I nipped around to the paddock to ask the only three horses running. “Psst!” I hissed, “what gives?”

Lightning laughed. “Merryman is an ant”, he said. “Ninepin is a bee”.

Merryman snorted. “The ant will not win”, he said. “It is going to rain”.

Ninepins snickered. “Lightning is a crow”, he said. “Lightning will beat Merryman”.

You will find this all a bit foxing, unless you know that “ant” is horse-talk for an invariable liar, “bee” for a horse invariably truthful and “crow” for one who alternates between true and false (or false and true). The three horses included one of each.

How did they finish?

[tantalizer308]

]]>I was recently filling in a passport application for my family and in writing down a particular year of birth for one of them I included an extra digit in error. The passport officer rejected the form but the quick-witted clerk noted that the five-figure “year” which I had written was a palindromic prime. Interestingly, the same was true if the number was read upside down.

What year had I intended to write?

[enigma805]

]]>This magic square has been known since ancient times. Each row, column and diagonal of three numbers adds to the same magic constant, 15.

There is, however, a different type of 3×3 magic square, in which every row, column and diagonal can be multiplied together to produce the same number. There are plenty of ways to do this, but if every number in the square has to be a different, positive whole number, then the smallest example has a magic constant of 4096.

Which numbers go where in the square?

**Note:** Although the puzzle states that 4096 is the *smallest* magic constant for a 3×3 multiplicative square, this is not the case. But it is possible to construct a 3×3 magic square with a magic constant of 4096, so the puzzle can still be solved.

[puzzle#203]

]]>How much do George and Mabel earn, the neighbours are asking, and how old is Charles? Well, George and Mabel live at 3 Down Tantalizer Avenue (a cube), with their sons Arthur (age 6 Across – a square), Bob (age the sum of the digits of George’s age or, if you prefer, the sum of the digits of Mabel’s age) and Charles (age 1 Across – twice as old as Bob). George is older than Mabel. His age is the last two digits of 1 Down and hers the first two digits of 2 Down. Their phone number (4 Down) is a cube. George’s income is a number of four digits each two greater than the previous one (3 Across or 3 Across reversed – I shall not tell you which); hers (5 Across) consists of four even digits summing to 12.

Can you satisfy the neighbours’ curiosity?

[tantalizer307]

]]>Last week’s

Noughts and Crosses Buglehad the following problem:“A game in which O began has reached the following position:

X is now to go and win.”

There were some O’s and X’s in the diagram but I have forgotten where they were. This week’s

Buglesays the solution is as follows:“X goes in

f. After O has gone, at least one ofeandiwill be empty and X can go in either that is empty to complete a row.”What was the diagram in the original problem?

[enigma798]

]]>Every day I play a little game with the date, to keep my mind sharp. The rules are very simple: you take the three numbers that express the date in its simplest form and combine them mathematically in various ways to try to make the numbers 1 to 10.

You are allowed to use the four operations of addition, subtraction, multiplication and division, as well as brackets, square roots and powers. You may use any of these as many times as you want, but you can only use each number in the date once and you have to use all three numbers each time.

My favourite date of 2022 was 16 September (expressed as 16/9/22) because I found a way to make every number from 1 to 10 following those rules, though it took me a while and I used square roots a lot. For example, I found “1” this way: √(22 + √9) − √16 = 1.

Can you find a way to get the numbers 2 to 10 with that date?

Happy New Year from *Enigmatic Code*!

[puzzle#202]

]]>Alan has told Bill and Charlie that he had chosen two different integers, each in the range 1-15, and has noted their product and their sum. He shows Bill the product, and Bill remarks that he cannot deduce the pair of numbers.

Alan then shows Charlie the sum, and Charlie, who has heard Bill’s remark, but not seen the product, remarks that he cannot deduce the pair of numbers.

Bill then declares that he can now deduce the two numbers, but does not name them.

Charlie then names the two numbers.

What are they?

[enigma804]

]]>The four senior clerks at Messrs Peabody and Quant each has a hat peg. Each has his name printed in faded ink over his peg and each guards his privilege with a fierce territorial imperative. None of this has escaped the notice of Jack the office boy.

Yesterday Amble arrived first, hung his hat and went into his cubicle, followed shortly by Bumble, who acted likewise. Jack promptly rose and swapped their hats over. Then Crumble arrived, hung his hat and went into his cubicle. Jack Rose again and re-pegged each of the three hats, leaving Dimwit’s peg empty. Then, on the stroke of eight and cutting it a little fine even for a senior clerk, Dimwit arrived, hung his hat and went into his cubicle. Jack rose again and re-pegged each of the four hats.

The day’s labour done, each departed wearing the hat on his peg. Just two of them are wearing their own hats. Who, Jack asks, was certainly one of them?

[tantalizer306]

]]>Two nuns were visiting the villages of the valley and helping the villagers to share their vegetables with one another. The nuns had started out with 10 bags of each of potatoes, carrots, beans, sprouts and turnips on their cart. At each village they made one of four possible exchanges of bags. They were (a) 5 of potatoes, 4 of beans and 9 of sprouts for 8 of carrots and 11 of turnips, (b) 4 of potatoes, 5 of carrots and 5 of turnips for 3 of beans and 13 of sprouts, (c) 3 of carrots and 8 of turnips for 3 of potatoes, 5 of beans and 4 of sprouts, and (d) 4 of beans and 6 of sprouts for 1 of potatoes, 2 of carrots and 5 of turnips.

The nuns noticed a curious thing. Each time they approached a village they counted up the bags on their cart, then they multiplied the number of bags of potatoes by 3, the number of bags of carrots by 1, the number of bags of beans by 2, the number of bags of sprouts by 2 and the number of bags of turnips by 3, and finally they found the total of the five products. They did a similar calculation as they left the village after making an exchange. They found that the two totals were the same. The nuns soon realised why this was so; for example, if they made exchange (a) then the change to the special total was (5 × 3) – (8 × 1) + (4 × 2) + (9 × 2) – (11 × 3) = 0, and similarly for the other exchanges.

As the nuns made their way home on Christmas Eve the younger nun remarked that one or two villagers had been reluctant to give their surplus to their needy neighbours in case they needed it themselves later in the winter. “I can understand their worry”, said the older nun, “but the actual need of today must come before the possible need of tomorrow”. “Yes”, said the younger nun, as the carols of the villagers filled the valley, “and that is one of the most important messages of Christmas”.

Which of the following combinations of bags might possibly have occurred as the contents of the nuns’ cart at some point on their journey between the villages?

(i) 11 of potatoes, 6 of carrots, 13 of beans, 12 of sprouts and 8 of turnips.

(ii) 15 of potatoes, 8 of carrots, 10 of beans, 8 of sprouts and 7 of turnips.

(iii) 8 of potatoes, 9 of carrots, 11 of beans, 8 of sprouts and 13 of turnips.

[enigma802]

]]>→ [ **2021** | **2020** | **2019** | **2018** | **2017** | **2016** | **2015** | **2014** | **2013** | **2012** ]

There are now 1646 *Enigma* puzzles on the site, along with 206 from the *Tantalizer* series, all 90 from the *Puzzle* series, and 196 from the *Puzzle #* series (and a few other puzzles that have caught my eye). There is a (mostly) complete archive of *Enigma* puzzles published between January 1979 to December 1992, and from November 1995 up to the final *Enigma* puzzle in December 2013, which make up almost 92% of all the *Enigma* puzzles published. There are 146 *Enigma* puzzles remaining to post (mostly from 1993 – 1995).

In 2022, 100 *Enigma* puzzles were added to the site (and 29 *Tantalizers*, 55 *Puzzle #s*, so 183 puzzles in total).

Here is my selection of the puzzles that I found most interesting over the year:

**Enigma 648: Piles of fun****Enigma 657: East is east and west is west****Enigma 662: State of the parties II****Enigma 664: The way of the dove****Enigma 678: It’s in the bag****Enigma 684: Take to the hills****Enigma 685: Food and fast****Enigma 691: Excuse me, please!****Enigma 696: A kinky tale****Enigma 794: Squares on cubes****Enigma 801: Forty-eight squares****Enigma 852: Sheep pens****Enigma 854: Colourful Christmas****Enigma 862: Distances****Enigma 866: Own goals galore****Enigma 869: Solitaire****Enigma 915: Double domino****Enigma 922: One or two routes**

::

::

::

I have also been collecting *Teaser* puzzles originally published in *The Sunday Times* on the **S2T2** site. There are currently 804 *Teaser* puzzles available on the **S2T2** site, 196 were added in 2022.

Here is my selection of the more interesting puzzles posted over the year:

**Teaser 1949-12-25: Four fours****Teaser 118****Teaser 119: Weights****Teaser 150: Paving the way****Teaser 560: Ribbon counter****Teaser 2534: Fiftieth anniversary****Teaser 2538: Octahedra****Teaser 2690: Leaving present****Teaser 2695: Powers behind the throne****Teaser 2720: Better tetra****Teaser 2721: Milliner’s hat-trick****Teaser 2743: Line-up****Teaser 2846: Bingo!****Teaser 2868: Prime logic****Teaser 3098: Wonky dice****Teaser 3110: Many a slip****Teaser 3116: Poll positions****Teaser 3117: Stop me if you’ve heard this one****Teaser 3119: Hidden powers****Teaser 3124: Lawn order****Teaser 3130: Making squares****Teaser 3136: Fill cells mentally, my dear Watson**

I now have physical copies of all 8 published collections of *Teaser* puzzles, and have been working through them. Currently about 64% of the puzzles contained in these books are available on the **S2T2** site, and I will continue to add puzzles as time allows.

::

Between both sites I have posted 381 puzzles in total this year, bringing the total number of puzzles available to 2934.

Thanks to everyone who has contributed to the sites in 2022, either by adding their own solutions (programmatic or analytical), insights or questions, or by helping me source puzzles from back-issues of *New Scientist*.

::

As a bonus New Year puzzle you might like to try inserting mathematical symbols into the following countdown, to make the resulting expression equal to **2023**:

**10 9 8 7 6 5 4 3 2 1** = ** 2023**

Here are a couple of solutions:

**10** × (**9** × **8** + **7** + **6** × **5** × **4** + **3**) + **2** + **1** = **2023
10** +

but there are many others.

]]>Faith, Hope and Charity were rivals for the post of mistress of the Glebe this year, a key office in the Federation of Whortleford Valley Women’s Clubs. Each canvassed ruthlessly and each amassed (as she supposed) 79 promises. As the number of ladies entitled to vote was just 100, the final count was very enjoyable.

The Valley Federation has eight branches, all of different sizes and none smaller than three members. Each branch pledged itself and voted as a block. But not all voted as they had pledged. To be exact, three branches voted for the one candidate they had promised to support, this being a different candidate in each case. Three branches pledged themselves to two candidates and then sneakily voted for the third. Of these the largest voted for Faith, the next for Hope and the smallest for Charity. One branch made no promises and cast no votes at all. One branch weakly pledged itself to all three candidates, got cold feet and also failed to vote.

What were the results of this poll?

[tantalizer305]

]]>Your challenge is to fill each hexagon in the Christmas tree with a different prime number. Each side of the triangle the tree makes must add to the same sum and each line’s total should be as small as possible.

Note that, by convention, 1 isn’t a prime number.

This is one of 4 puzzles published in the same issue of *New Scientist*.

[puzzle#201]

]]>‘Twas just a few days before Christmas. Santa was out for a revitalising walk on the hills around his workshop with his sleep-deprived elf companion Bubby, when he spied a group of other elves moving around on the next hill. “Ah”, he surmised. “They’re using their special secret elf communication system that only elves can understand, elf-maphore”. The system involves holding two trees in specified positions to represent a letter of the alphabet.

As they watched, three separate groups of elves ran up to the top of the hill and each group spelled out a different word. “What does the message say?”, Santa asked. Bubby flipped her elf hat over to the right, as elves always do when using the language, and squinted her tired eyes at the elves on the far away hill, watching them repeat the message. “Well, it appears to say

PSTAHERHLLTNORL. But that can’t be right!”.What was the message?

This is one of 4 puzzles published in the same issue of *New Scientist*.

[puzzle#200]

]]>Dear Santa,

I can explain. We weren’t planning to peek at the gifts Aunt Emily brought us, but the receipt was sticking out of one of the bags and Joe accidentally saw it and accidentally told us, so then we knew she got us all lightsabers, two green and three blue. Edmund wants a blue one and I want green. Timothy said it wouldn’t really be cheating if we only peeked at the gift of the sibling younger than us, but I said that isn’t fair because I’m youngest, so we agreed to look at the gifts of both the siblings younger and older than us, which means I only looked at one and it was only a tiny peek.

After that, Georgia asked if anyone knew what they themselves got, and we all said no, except then she smiled and said that if no one else knew, then she did know. I guess I’Il just have to wait to know what colour mine is. I hope you can still bring me presents and please take the lightsabers off the list I sent you.

Love,

SophieWhat colour was Sophie’s lightsaber?

This is one of 4 puzzles published in the same issue of *New Scientist*.

[puzzle#199]

]]>Your turkey needs exactly 6 hours to slow roast. Alas, you have no timer, just three candles: one that burns in 3 hours, one that burns in 4 hours and another that burns in 5 hours.

How can you measure 6 hours using these candles, putting the roast in the oven at the same time as you light the first candle?

This is one of 4 puzzles published in the same issue of *New Scientist*.

[puzzle#198]

]]>One Christmas Goldilocks called at the bears’ cottage with three pretty boxes of honey cakes. “Happy Christmas, bears!” she called, “Here is a little present for Baby Bear, a bigger one for Mother Bear, and an even bigger one for Father Bear. Honey cakes! I expect you’d like to play a little game with them. So each open your box, without letting the others see”.

They did as bid and Goldilocks announced the total number of cakes, adding that not even Father Bear had more than 10.

“Now I wonder who can work out how many each of the others has?” she asked brightly.

“Not I” said Baby Bear. Pause. “Nor I”, said Mother Bear. Pause.

“Nor I”, said Father Bear in his deep growly voice. Pause. “Nor I”, said Baby Bear. Pause. “Nor I”, said Mother Bear. Pause. “Nor I”, said Father Bear in his deep growly voice. “I knew Mummy Bear couldn’t”, squeaked Baby Bear, “even before she spoke last time. But I can now.”

Since the bears made all their deductions right, you too can work out how many cakes each had.

[tantalizer304]

]]>My friend and I go swimming at the local 25-metre pool. We both swim at our own steady speed and when she has completed one length I have swum a certain whole number of metres.

We start at one end of the pool together and she gradually gets ahead. After completing a length she turns, and, when we meet, I turn round and swim with her again. Of course she soon gets ahead again and we repeat this procedure each time we meet after she has turned around at either end of the pool. When she has completed four lengths and we meet when she is on her fifth length, I calculate that, to the nearest metre, I have completed three lengths.

How many metres do I swim when she swims a length?

[enigma799]

]]>We bought some marshmallows to toast on the fire, but my daughter had other ideas. She started building shapes by joining marshmallows together with cocktail sticks.

In one effort, she put four marshmallows on the table and, keeping them on the surface, joined them with six cocktail sticks, as shown here. To do this, she had to break three of the sticks to the same shorter length (shown by the dotted lines).

“Very neat”, I thought, “though she wouldn’t have had to shorten any sticks if she’d arranged them as a pyramid”.

A few minutes later, she came up with a different way of joining four marshmallows to each other using a fresh batch of six sticks. Again, the marshmallows were flat on the table and three of the sticks were full length, while the other three had been shortened. As before, the three shortened sticks were the same length as each other.

Can you find her other arrangement?

[puzzle#197]

]]>It’s strange to think of Australians playing cricket while we shiver at Christmas. One of my Australian friends was noting the scores at each fall of a wicket in a recent match:

12 for 1; 38 for 2; 112 for 3; 250 for 4 (declared), which he wrote as 12138211232504 and then, after a few beers, he was completely unable to interpret what he’d written.

On another occasion the team again declared at less than a thousand runs and he wrote down the scores in a similar fashion. Then he consistently replaced each digit by a letter, using different letters for different digits, and it gave him the list:

CHRISTMASCRICKETERSHow many runs had they scored “for 5” and how many when they declared?

[enigma803]

]]>Tuff and Rumble are two of the snappiest journalists in the business. Both write faster than they think. Indeed each writes n times faster than he thinks and n is the same number for both. But Tuff is starting to crack up a bit these days and writes only as fast as Rumble thinks. Rumble is still in his prime and writes four times as fast as Tuff thinks.

They polished off 60 feature pieces last month between them, each devoting the same thinking effort to each of his pieces and the same writing effort too. Both of them spent the same total time on doing his pieces. How many of the 60 did Tuff do?

[tantalizer302]

]]>I have three wooden cubes. On each face of each cube one of the digits from 0 to 8 is written, with each of those digits on more than one of the cubes. The sum of the digits on each cube is the same.

By moving the cubes around I can make the top faces form many numbers (with the 6 being also used for a 9);

for example:

In this way I can make all the three figure perfect squares.

How many of the three-figure perfect cubes cannot be made in this way?

There are now 1642 *Enigma* puzzles available on the site, and there are 150 remaining to post. So 91.6% of all *Enigma* puzzles are available.

[enigma794]

]]>Three top racehorse trainers, Kate Winsalot, Britney Spurs and Harry Trotter, were invited to compete in the prestigious Champion Trainers Stakes.

Each had to enter three horses in the race, with points being awarded to the trainer according to the finishing position of their mounts. The trainer of the winning animal would be awarded 9 points, the second placed horse 8 points, and so on, down to 1 point for the trainer of the last horse. The person receiving the most points is the Champion Trainer.

The results of the race were delayed waiting for Bored Bronco, one of Harry’s horses, to finally finish and also because two of Kate’s runners had to have their positions determined by photograph, with her third horse finishing further back.

When the scores were calculated, it turned out that the three trainers had tied on points.

Which trainer’s horse came first?

[puzzle#196]

]]>I carried out a long division calculation and then replaced all the digits with

Xs as follows:The problem, of course, is to work out what the original calculation was, though you need some more information for that. It would be sufficient if I told you a certain fact involving just one of the digits in the five-figure quotient, but I won’t.

What was the eight-figure dividend?

[enigma705]

]]>Five candidates stood for the Chair in Cognitive Pneumatics at Wessex recently, and the selectors went carefully. They ranked the five under each of three headings, giving 5 for 1st, 4 for 2nd, etc. in each case. As the criteria were Weight of Publications, Sonority of Lecturing, and Persistence in Administration, the overall winner was bound to be the best man. No one gained the same place under two or more headings and each candidate finished with a different total.

Dr Airwick was top in nothing, scored 3 for Lecturing and finished third. Dr Borehead was top in Publishing and bottom in something else. Dr Crumthink scored 2 for Publication. Dr Drinkwell out-pointed Dr Empty in Lecturing. The new Professor is not the top lecturer nor was the top publisher the runner-up.

Can you produce the selectors’ complete table?

Today marks the 11th anniversary of the founding of the **Enigmatic Code** site.

There are now 2900 puzzles available between the **Enigmatic Code** and **S2T2** sites.

On the **Enigmatic Code** site there are 1640 *Enigma* puzzles, 202 *Tantalizer* puzzles and 90 puzzles from the original *Puzzle* series, as well as 189 from the new *Puzzle* series. And on the **S2T2** site there are 787 *Teaser* puzzles available.

[tantalizer303]

]]>The Pebblies are a range of 11 hills which have one-letter names, A to K, so:

Pussicatto has been commissioned to paint the following 13 views: D-G (that is to say the hills D, E, F, G), E-J, B-J, D-I, B-F, A-I, B-I, A-J, C-K, B-K, D-K, A-H, C-J. First he went to the Pebblies to take a number of photos to paint from. He wanted a set of photos such that when he came to paint any of the 13 views he could select some photos and stand them up, possibly overlapping, so that they showed precisely the hills he had to paint. He could paint B-I if his photos included, say, B-D, E-H, F-I. He wanted to take as few photos as possible.

When Pussicatto came to paint he decided that for certain of the paintings he would paint his wife Tabitha climbing one of the hills in each of those paintings. For example, he could paint Tabitha on hill F in painting A-H. In order that she did not dominate the paintings he decided that: (a) she must not be on the same hill in two different paintings, and (b) there must not be two paintings which both contain her and which both contain the two hills she is on in those two paintings, for example he did not allow B-J with Tabitha on D and C-K with Tabitha on G. Pussicatto wanted to include Tabitha in as many paintings as possible.

How many photographs did Pussicatto take? How many paintings included Tabitha?

[enigma684]

]]>“Go on without me. Just go!”, exclaimed Jeff.

Bart shook his head dramatically. “We never leave a man behind”.

In reality, though, Bart’s car only held three passengers, so the five friends had no clear way for all of them to make it to the pizzeria together. Hunger was closing in fast, so they thought up the following solution: Jeff would start walking as Bart drove his other three friends towards the pizzeria. Bart would drop them off somewhere along the route, go back and pick up Jeff, then drive to the pizzeria, while his other three friends walked the rest of the way.

They wanted to time it just right so that all five friends arrived at the pizzeria together. As it happens, the car drove seven times faster than Bart’s friends walked.

How far down the 5-kilometre route to the pizzeria did Bart need to drop off his first group of passengers?

[puzzle#195]

]]>Perhaps it was the effect of a recent whitewater rafting trip. I awoke the other night from the strangest dream. I was seated in the extreme rear of a 4-metre-long rubber raft, travelling down an immense river with huge rolling waves. As the dream began, we were just about to ascend one mammoth wave. As we rose up each wave, I found myself struggling forward in the boat, exactly 1 metre with each wave. But as we slid down from the crest of each wave, the entire raft stretched out, elongating by exactly 4 metres each time. The portions of the raft fore and aft of me stretched in proportion, so I maintained my relative position after each such stretch. (I myself didn’t seem to get stretched!)

Can you tell me during which wave (10th?, 25th?) I finally reached the bow of the raft and awoke?

[enigma700]

]]>Thirteen candidates came through the recent civil service exam with equal points. This being too many, the selectors set a final test of five questions, each with a yes-or-no answer. The 13 candidates produced 13 different answers, six of which were judged equal best. As the much-sought-after Ministry of War and Fish was six men short, that was all very convenient.

Among the disappointed were Ponsonby (who answered No, No, Yes, No, No), Quidnunc (who answered Yes, No, No, Yes, Yes) and Ribbontape (who answered Yes, Yes, Yes, No, Yes). All are now with the Department of Surplus Services.

Never mind the questions. What were the right answers?

[tantalizer301]

]]>The aim of this

Enigmais to place the numbers 1 to 48 into the 48 squares of this 4 × 12 rectangle so that consecutive numbers touch each other either by being above and below each other or by being to the left and the right of each other. The two numbers already inserted make the solution unique.

[enigma801]

]]>It is time for the men’s Football World Cup. Countries in eight groups of four will play each other in a round robin, with the top two in each group qualifying for the last 16.

The final tables of the group stage provide ready-made puzzles because it is often possible to figure out the results of a group’s matches just from the information in a table. For example, the following is the final table from one of the groups in the 2018 World Cup.

Can you figure out the results of all the matches played by teams A, B, C and D?

[puzzle#194]

]]>Simple Simon is not as zany as some people imagine, especially when it concerns wherewithal.

To illustrate, consider the following two additions where, as usual, the digits have been replaced by letters — the same letter standing for the same digit wherever it appears and different letters standing for different digits:

YES + LETS + SMILE + SIMON + I + SEE = MONEY

SEE + SIMPLE + SIMON + SMILES = MOSTLYSimple Simon tells me his middle name,

IS, also happens to be a two-digit odd number.How much is

PLENTY?

[enigma702]

]]>This year has been declared the 3000th anniversary of Homer’s birth. But one minor detail is worrying the organisers – they do not know where he was born. Legend insists that it is one of seven places which form an old hexameter: Smyrna, Chios, Colophon, Salamis, Rhodos, Argos, Athenai. Some are more plausible than others and, of course, some offer more commercial advantages. So the committee finally marked each place out of 25 for plausibility and out of 25 for prospective profit and then added the two figures for a total. Just one copy of the secret results has leaked. It is pretty fragmentary but it is also known no two places tied on any list and that the figure scored by Colophon in the first column also occurs somewhere in the second.

Can you fill in the whole table?

There are now 200 *Tantalizer* puzzles available on the site.

[tantalizer299]

]]>Ms Curricula chalked up the following equations on the blackboard:

116 = N × 116

236 = N × 236

356 = N × 356

476 = N × 476

596 = N × 596“The numbers on the right of the equations are all written in base 10”, she told the class. “Can anybody give me the value of the integer N?” she continued, “… and I don’t want ‘N equals 1′”, she added quickly, before the hands had time to shoot up.

Can you tell her?

[enigma800]

]]>Bonnie is refitting her bathroom and her plumber has brought round some very peculiar tiles to cover the floor. Her bathroom is a 5×5 square and each tile is an L shape made up of three squares (see the diagram above).

Bonnie has realised it will be impossible to tile the whole bathroom without cutting tiles, as each tile covers three squares and 25 isn’t a multiple of 3. When she points this out, however, her plumber says it is OK, as they will need to leave one square of floor space free for the toilet.

“Well, clearly I don’t want the toilet in the middle of the room!” she says.

In which squares could the toilet be placed?

[puzzle#193]

]]>I started with this star-like framework, threw a normal die, and wrote the uppermost number in one of the triangles. I then did this a further 11 times until there was a digit in each of the 12 triangles of the framework.

The sum of the five digits in each of the arrowed lines was the same. Also the five-figured numbers read in each of those arrowed directions was a perfect square.

I can tell you that the six digits in the outermost “points” of the star alternated odd/even around the start.

Fill in the star showing my layout of digits.

[enigma697]

]]>Matthew, Mark, Luke and John decided to test their powers of deduction. So they picked four aces and two kings out of a pack [of cards], shuffled the face down and took one card each. Each then showed his card to the others but did not look at it himself (the two undrawn cards remained unseen).

“Right”, said John heartily, “who knows whether he has an ace or a king?”

“Not I”, replied Matthew.

“But I do”, remarked Mark, after a pause.

Which did Mark hold?

[tantalizer298]

]]>Each Christmas two of the nuns from the abbey visit the six villages in their valley with a cart full of food. This year they began their journey with two boxes of apples, a box of carrots, three boxes of jars of damson jam and three boxes of fennel. The reason for the visit was to allow the villagers to obtain supplies of those foods where their own harvest had failed. The nuns visited the villages of Angel, Birth, Crisp, Deep, Even and Flock once each, although not necessarily in that order. At each village the nuns gave some food in exchange for some other food as described in the table.

The nuns gave these boxes:The villagers gave these boxes:Angel3 beetroot, 1 elderberry 2 apples, 1 carrot Birth2 apples, 1 damson, 2 elderberries 3 beetroot, 1 carrot, 1 fennel Crisp2 apples, 1 carrot, 2 fennel 2 beetroot Deep2 carrots, 1 elderberry 2 apples, 3 damsons Even2 beetroot, 1 fennel 1 damson, 3 elderberries Flock2 damsons 1 carrot, 2 elderberries, 1 fennel [→ image] It was Christmas Eve as the two nuns approached the abbey at the end of their journey. One said to the other. “The winter had been making me feel quite depressed, but helping the villagers has given me new life”. Her companion smiled and, as the sound of carols from the villages mingled with those from the abbey, replied, “Isn’t that the true gift of Christmas?”

In what order did the nuns visit the villages?

[enigma698]

]]>It was decided that Friendly Co. Ltd and Amicable Co. Ltd would merge and form the Friendly Amicable Co. Ltd. Alas, things have proved less than friendly in the boardroom.

First, there were some angry resignations from the original dozen or so directors at the two companies.

And then, the first board meeting of the new company didn’t begin well, with old enmities preventing a group of the board members from shaking hands among themselves. This meant that the number of handshakes was exactly one-third down on the number there would have been if everyone had shaken hands with everyone else.

Let’s just hope the share price doesn’t go the same way.

How many members are on the new board?

[puzzle#192]

]]>If you add together consecutive pairs of the digits of an

ENIGMA(E+N,N+I,I+G, and so on), the results will beWRONG(if they are right!).In the words and letters in capitals above, different letters as usual represent different digits. The same letter always represents the same digit.

What is the value of

NOW?

I don’t have a source image for the puzzle as originally published, but I believe the above text is correct.

[enigma796]

]]>