There are 15 small square blocks placed in a tray. Each block has a letter on it and they are placed as in the diagram:

Because of the empty space in the bottom corner, it is possible to slide the blocks about within the tray’s boundary to get other arrangements, for example:

Which of the following arrangements is it possible to obtain by sliding the blocks about?

[enigma673]

]]>Look at any arrangement of the digits 1 to 9 equally spaced around a circle. For example:

In this example, reading both ways around the circle, pairs of adjacent digits make the following two-figure numbers:

56, 68, 81, 19, 93, 32, 24, 47, 75, 74, 42, 23, 39, 91, 86, 65, 57

just three of which are prime. If I reverse the position of, say, the 3 and the 7 in the circle, then I get an arrangement which produces five primes.

Your job today is to find an arrangement of the digits 1 to 9 around a circle so that the 18 two-figure numbers read around the circle in both directions do not include any primes and such that there are two different digits whose sum is 10 with the property that if you interchange those two digits then the new arrangement still produces no primes.

What are those two interchangeable digits?

[enigma670]

]]>It is the year 2100 and the Mars pioneers have built an agri-bubble in which they will be able to cultivate their own food. The crop is a form of grass that grows at a steady rate and can be harvested and turned into nutritious protein snacks (yum!). Now it is time to populate the planet.

Scientists have figured out that if there are 40 adults living in the bubble, the crop will only feed them for 20 days. However, with only 20 adults, the crop will keep them going for 60 days — so half as many adults can survive for three times as long! Why? Because without overharvesting, the crop is able to replenish itself.

Of course, the pioneers want a food supply that keeps the population sustained indefinitely. Based on the numbers above, how many people should be in the first Mars cohort?

[puzzle#179]

]]>My niece Melinda likes to solve mazes, so I gave her a problem involving an odd maze of numbers:

I told her that from any odd number

ngreater than one, she could move to another odd number as follows: take an odd divisor ofn – 1and multiply it by an odd divisor ofn + 1. From 11, for example, she could move to 5 × 3 = 15, because 5 divides 10 and 3 divides 12. 1 is allowed as a divisor, so she could also move to 1 × 3 = 3, 5 × 1 = 5, or to 1 × 1 = 1. 1 is a dead end, of course, and certain numbers, like 31, inexorably lead to 1. I asked Melinda to find the shortest path from 13 to 19 and back to 13 again (without falling into the Black Hole at 1!).Can you help Melinda? Find a path from 13 to 19 and back again in the least number of steps.

[enigma672]

]]>As Christmas approaches and the winter weather takes hold, so two nuns bring food and clothing to the refugee camp that has been established near to their hospital. There are 27 huts in the refugee camp, ranged in a circle and numbered 1, 2, 3, …, 26, 27 as we go round the circle, with 27 also being next to 1. In order to brighten the camp up for Christmas, each hut has a coloured flag in front of it, red or blue or yellow. The rule for the camp is that if two huts are next to each other in the circle then their flags must be of different colours. Each morning the flags are put out as follows:

RBY RBY RBY RBY RBY BRY RBY RBY RBY.

That is to say, 1=R, 2=B, 3=Y, …, 27=Y.

The children in the camp have a supply of flags of all three colours. They choose a hut and go and change its flag, ensuring that they keep the camp rule. They then choose another hut and go and change its flag, still keeping the camp rule. They carry on in this way, choosing one hut after another.

It was late on Christmas Eve when the nuns completed their task and set off for the hospital. When they looked back at the refugee camp they saw it was bathed in moonlight shining through a gap in the clouds, while the surrounding area was in darkness. The younger nun remarked on the contrast and the older agreed and continued, “but it does help us see where our priorities lie”.

On various days the children try to achieve certain patterns of flags. Which of the following are they able to produce?

(i) RBY RBY RBY RBY RBY RBY RBY RBY RBY.

(ii) RYR BYR BYR BYR BYR BRY RBY RBY RBY.

(iii) RBY RBY RBY RBY RBY BRY BRY RBY RBY.

I have made every effort to transcribe this puzzle correctly. (I believe it is incorrect on the *New Scientist* web site). To make it easier to read I have added spaces in the sequences of colours, so they are in groups of three.

[enigma854]

]]>A story:

A hero enters a cave. Inside is a monster with three numbered heads. It attacks! Our hero chops off a head, but two more heads grow in its place. One of the new heads attacks and the hero greets it with a quick chop, too. Standing back, he realises that every chop produces at least one prime-numbered head that, when multiplied by the number on the other new head, gives the number from the head that was chopped. He also notices that all the prime-numbered heads are friendly.

The story’s illustrator squints at the author’s scribbled notes. She can make out the numbers on two of the original three heads (418 and 651) but not the third. Reading ahead, she notices that when the hero collapses triumphant at the feet of the now fully friendly monster, all the heads show different numbers and their sum is 113.

“Aha!”, she declares, and draws the original monster with its three snarling, numbered heads.

What was the third number?

[puzzle#178]

]]>What is the big fuss over age? Some people prefer not to disclose their age, while others attempt the futile task of claiming the same age every year.

I suspect the honesty of the average person falls somewhere between, as show in the addition below. In this addition, different letters stand for different digits, but the same letter stands for the same digit whenever it appears. And

Yis twiceR.Find the value of

AGES.

[enigma671]

]]>I have 26 boxes numbered 1 to 26 and 26 cards labelled A to Z. I start with A in box 1, B in box 2 … Z in box 26 and I want to end with A in box 2, B in box 3 … Y in box 26, Z in box 1.

I am allowed to make a series of swaps, where a swap consists of choosing any two boxes and swapping over the two cards in them.

So, can I achieve my aim with a series of:

(1) 27 swaps;

(2) 26 swaps;

(3) 25 swaps;

(4) 23 swaps?

[enigma850]

]]>Scientists have been studying two rare monkey species in a forest.

In one part of the forest live the

Equalismonkeys, which are split 50-50 between males and females. In another part of the forest are theFraternismonkeys, of which exactly two-thirds are male — the evolutionary aspect of this isn’t yet known.Both species are monogamous, with families coming in all shapes and sizes. Some parents stop at one offspring, but there are others with 10 or more, so some monkeys have lots of brothers, while others have none at all. The sex of any infant is independent of others in the family.

Among

Equalismonkeys, which should expect to have more brothers, the males or the females? And how about theFraternismonkeys?

[puzzle#177]

]]>There are 12 males in the Utopian Royal family: King Nosmo and his sons, grandsons and greatgrandsons, the Princes Airey, Barry, Carey, Derek, Eric, Fred, Gary, Harry, Igor, John and Kevin. They all have truly palindromic birthdates, that is, not only do the figures read the same each way but there is also no need to rearrange the dashes, so that 15/6/51 would be truly palindromic while 1/5/51 would not. They were all born in different years and have different zodiacal signs.

Furthermore:

(1) Although Kevin was just over eleven months old on King Nosmo’s 80th birthday, he has still not reached his first birthday.

(2) Airey, Nosmo’s eldest son, was born in the month of his 18th birthday.

(3) Barry is a year and 10 days older than Carey; Derek is a year and 10 days older than Eric; Gary is a year and 10 days older than Harry; and Igor is a year and 10 days older than John.

(4) Igor was born in June.

(5) Neither Barry nor Gary is an Arien.

Who is the Taurean?

[enigma856]

]]>A stairway consists of a number of steps and each step contains two of the digits 0, 1, 2, …, 9, for example you might have:

If you have a step [

AB] then the step that goes on that step is [CD] so:where

C= the units digit ofA×B, andD= the units digit ofB+C.So if we want to add a step on top of [85] above we find 8 × 5 = 40, so

C= 0, and 5 + 0 = 5, soD= 5, so we put [05] onto [85].What is the largest number of steps that you can have in a stairway without repeating a step?

[enigma669]

]]>When the giant town-hall clock chimed 2 o’clock, an ant resting by the number 2 woke from its nap. Spotting that the minute hand was edging towards it, the ant started walking clockwise round the rim of the clock face. Thinking it had escaped the minute hand, it was shocked when it caught up with the hand again. At that point, the ant turned round and walked anticlockwise back round the rim, at the same constant speed as before, reaching the minute hand for a second time after a further 15 minutes. At this point, it decided to give up and take another nap.

At what time did the ant stop walking?

[puzzle#176]

]]>When Pembish first moved into Pembish Hall he found the grounds littered with cannon balls, so he ordered his elderly retainer, Punnish, to collect them all and stack them into a single triangular pyramid. Each layer of such a pyramid consists of an equilateral triangle of balls having one ball fewer along its edge than the layer immediately below it, except for the very bottom layer, of course. So the top “layer” contains just one ball, the one below it 3 balls, the one below that 6, the one below that 10 and so on.

Punnish counted the balls and found that it was possible to construct such a pyramid. In fact, he was just putting the last, crowning, ball in place when, to his horror, he heard the monotonous tones of the Reverend Neverend from over the hedge: “Can you tell me the time?” Now Punnish had once slept through the first 17 hours of one of the cleric’s impromptu sermons and he didn’t want to have to hear again the story of how the Reverend had once escaped the Boers disguised as an (extremely boring) chicken.

It was imperative to distract him and so to cut a (very) long story short, Punnish called out: “No, I can’t. But I can tell you the number of cannonballs in this triangular pyramid is the 6-digit number CANCAN, in which each letter stands for a digit. And even if I can’t can cans, I can cancan!”. So saying, Punnish cancanned into the conservatory. The boring cleric thought at first it could be 179 layers, but that would require 971970 balls. So that wasn’t quite right.

How many balls are there in the pyramid?

[enigma668]

]]>You will need a box of matches. Divide the matches into a number of piles, not necessarily the same number of matches in each pile. For example, you might lay out:

7,5,4,7,6,19,3.You are allowed to pick any of two piles. If the piles are equal, put them together to make one pile; if the piles are not equal, take from the larger the number of matches that are in the smaller and add them to the smaller.

Using the example above, if you select the

5and3, you get:

7,2,4,7,6,19,6.If you then select the two piles of

6you get:

7,2,4,7,12,19.If you then select the second

7and the19, you get:

7,2,4,14,12,12.You carry on in this way, repeatedly acting on the piles you got from your previous action. Your target is to collect all the matches into one large pile. Sometimes that is possible, sometimes it is not.

For which of the following layouts is it possible to collect all the matches into one pile?

A:17,4,5,5,1,2,3,15,12.

B:17,4,5,5,9.

C:51,72,57,78,78,48.

D:1,1,1,2,2,2,2,3,3,3,3,3,23.

There are now **1600** *Enigma* puzzles available on the site. And there are 192 puzzles remaining to post. Which means about 89.3% of all *Enigma* puzzles are now available.

And between **Enigmatic Code** and **S2T2** there are now **2750** puzzles available.

[enigma648]

]]>The students at Yellow Brick High School for Girls couldn’t decide if their maths and drama teacher, Ms Gale, was a genius or just overworked when she announced the new school play, a “mathemusical” called

The Wizard of Odds. But the real challenge, as usual, was in the casting, and the parents, students and faculty had various demands, summed up as follows:1. If Megan doesn’t get the lead role, Dorothree, then Kasey will play either the Square Crow or the Ten Man.

2. If neither Leah nor Nicki are the Cowardly Line, Jane will will be Dorothree.

3. If Leah doesn’t get the Square Crow, Jane or Kasey will get Dorothree.

4. If Nicki isn’t the Ten Man and if Leah doesn’t get Dorothree, then Kasey will play the Wicked Witch of the Word Problems.

5. If Leah isn’t the Cowardly Line and if Nicki isn’t the Wicked Witch of the Word Problems, then Jane will be cast as the Square Crow.

Remembering that “if

x, theny” doesn’t imply “if notx, then noty“, can you help Ms Gale accommodate this tornado of requests by assigning the roles?

[puzzle#175]

]]>In the multiplication below, the digits have been replaced by letters and asterisks. The same letter stands for the same digit wherever it appears, while the asterisks, of course, can be any digit:

The only zero in the multiplication is denoted by an asterisk and I can tell you that it is not in the second partial product (fourth line down). But why worry, you will find it soon enough.

What is

WORRY?

[enigma667]

]]>We hope to have a few magic moments this Christmas. I’m giving my parents an unusual calendar. Three rulers (of the same thickness, depth and material) slide through a frame.

The 6-inch ruler has marks for Saturday-Friday at one-inch intervals, the 11-inch ruler has marks for January-December at one-inch intervals, and the 15-inch ruler has marks for 1-31 at half-inch intervals. Each day the details of the date are lined up in the frame. The illustration shows how the calendar should have looked on Tuesday 14 March.

The calendar should be anchored to the wall by two central nails in the frame. If the upper nail alone were used the calendar could swing and hang at some peculiar angles.

In fact it would only have hung horizontally on one day in 1995. Which day?

[enigma855]

]]>I have a 12-hour digital number display alarm clock. As is normal on digital clocks, each of its four digits is constructed using seven possible segments.

I go to bed when the display is at its dimmest and awake when it is at its brightest.

How long am I in bed?

[puzzle#174]

]]>An “absolute difference triangle” is a triangle of numbers such that each number below the top row os equal to the absolute value of the difference of the two numbers above it. For example:

In the following one, letters have been substituted for digits in the top row, with different letters being consistently used throughout for different digits. Stars may have any value:

If I own up and tell you that

OWNis a perfect cube andONE(as you might expect) is a perfect square, then what isENIGMATIC?

[enigma666]

]]>Alan, Brian and Charles have surnames Adams, Brown and Collins (not necessarily respectively) and occupations of architect, builder and carpenter (again not necessarily respectively). Each of them is either thoroughly honest or thoroughly dishonest. Below are some statements (not necessarily by more than one person) which involve these three people’s names and jobs: (each blank space originally contained one of the surnames — Mr Adams deleted the names after seeing the first few statements):

Alan says:

(i) I am not the architect.

(ii) Brian is a carpenter.

(iii) Charles’s surname is [……….].Mr Adams says:

(i) The architect’s surname is not Brown.

(ii) The builder’s surname is [………].

(iii) The two spaces contain the same surname.The architect says:

(i) The builder isn’t called Charles.

(ii) It’s now possible to work out all our names and jobs.Please state (in order) the [deleted] surnames.

If I’ve counted correctly there are now “only” 200 *Enigma* puzzles remaining to post. (Actually I think there are 196).

[enigma665]

]]>My son is obsessed with chess, and has been acting out the game’s moves everywhere we go, running like a bishop and jumping like a knight on tiled floors. He was tickled to see that on the number pad of my keyboard he could type 27 using a knight’s move, because the move from 2 to 7 is an L-shape, like a knight moves on a chessboard.

Alas, he can’t make 27 using a bishop move. Bishops move diagonally any number of spaces, so a bishop, using multiple moves, could make a number like 484 or 9157. In a similar fashion, a knight could make numbers such as 167 or 8349.

Yesterday, he made a happy discovery: a three-digit knight number that is exactly 27 more than a three-digit bishop number. (Actually, I found I could put another digit, call it “X”, at the front of my son’s numbers, and still have a knight number that is exactly 27 more than a bishop number).

What numbers did my son find?

[puzzle#173]

]]>Harry, Tom and I were challenged to find four perfect cubes, one consisting of one digit, one of two digits, one of three digits and one of four digits, such that the ten digits used included nine different digits. We were allowed to regard 0 as a valid solution for the one-digit cube.

Our solutions were all different. You might say that we each found two solutions, though each of us simply found an alternative for one of our numbers, the other three numbers being common to both solutions. The three numbers common to both my solutions did not appear in any of Harry’s or Tom’s solutions.

Please list those three numbers in ascending order.

[enigma857]

]]>I have a lucky number. When the numbers of a date add up to the square of my lucky number I call that a “lucky day”. (For example, if my lucky number were 11 then 24/3/94 would be a lucky day since 24 + 3 + 94 = 121). I especially like the lucky days which fall on Sundays.

I once noted the occurrences of my lucky days as they happened over a period of three consecutive calendar years. I had one day each month. Furthermore, there were more lucky Sundays in the first year than in the second, and more lucky Sundays in the second than the third.

What was the date of the first lucky day in that three-year period?

[enigma859]

]]>Emma Neesha is a forgetful sort, and she has just locked herself out of her house with her keys still inside.

Not to worry, she installed keypads on both her house and car for such occasions. Unfortunately, she has also forgotten the four-digit code to her house.

This shouldn’t be a problem, because she keeps a code clue in her wallet. It says: “Four times the car’s four-digit code”.

Well, that is fine, but she has forgotten that one too. Fortunately, Emma is prepared: she has a clue written down for the car’s code as well. Pulling that one out, it reads: “The reverse of the house code”. Oh dear.

Can you help?

[puzzle#172]

]]>Within the Earth’s atmospheric range, it is not possible to convert any whole number of degrees Fahrenheit into Celsius by reversing the digits. There is a good approximation in the case of 82 °F, which is nearly 28 °C the same may be said for 61 °F and 16 °C, but in neither case is the conversion exact.

It is possible to construct another scale to Celsius, sharing zero as freezing point, but with a different boiling point, which is a whole number berween Celsius’s 100° and Fahrenheit’s 212°, so that there is a positive point on the Fahrenheit scale which may be converted exactly into the equivalent temperature on the alternative scale simply by reversing the two digits.

What is boiling point on the alternative scale, and what is the convertible Fahrenheit temperature?

[enigma858]

]]>I used to run a village shop and I had a pair of scales and a set of 10 weights. Each weight was a whole number of ounces and the total weight was less than 1024 ounces. I found that I could [

not] put one or more weights on each of the two pans of my scales so that they balanced.One day I lost a weight. By chance I was able to replace it with a spare; unfortunately it was not the same weight as the one I lost. However, now I could put one or more weights on each of the two pans of my scales so that they balanced; in fact, I could arrange for each side to weigh 6 ounces. Unfortunately I could not weigh 13 ounces, although I could weigh 3 ounces.

Q1.What was my set of weights after the replacement?

Q2.Suppose I was back with my original set of weights. Could I have selected one and replaced it by some other whole-number-of-ounces weight, of my own choosing, so that the total weight was still less than 1024 ounces and I still could not put one or more weights on each of the two pans of my scales so that they balanced?

Q3.AsQ2but now I can replace six weights instead of just one. The new set of weights must not be the same as my original set.

This statement takes into account the correction to the puzzle published with **Enigma 866**.

I also added the “**not**” in square brackets into the text, as without it the puzzle doesn’t make sense to me.

[enigma860]

]]>“Here’s your 21st birthday present”, said Amy.

“A bracelet?”, frowned Sam.

“Not just any bracelet, it is a magic number bracelet because I know you love numbers. See how it has got five beads, each with a different positive number on it. You can find all the numbers from 1 to 21. But to find most of them, you have to add together adjacent beads.”

“For example, to make the number 17 you add together these three beads”, she said, pointing to the beads in positions A, B and C on the diagram. “Other numbers are found by adding two, three or four adjacent beads. And, of course, to get 21, you add up all five”.

What are the five numbers on Sam’s bracelet?

[puzzle#171]

]]>In Churchester there are nine churches of the

Angel, of theBell, ofCharity, ofDestiny, ofEndurance, ofFriendship, ofGod, ofHelp and ofInspiration. The map of Chuchester shows the nine square parishes and each has its church at its centre. I can only remember where the church of theAngel and the church ofCharity are:There are nine doves in Churchester and each one visits three churches, flying around in a triangle in a clockwise direction. For example, one dove flies from

Help toBell, toCharity, and then back toHelp again. The routes of the doves areHBC(the one just mentioned),DAF,BGI,GEA,HID,CEF,CAI,BEDandFHG.Complete the map of Churchester.

[enigma664]

]]>With eight handsome bachelors and eight ravishing spinsters in the office, prospects look bright for matrimony. The snag is, however, all are fanatical Catholics or Protestants and the chances against a random bride and groom being of the same religion are nine to seven. In a random mixed marriage, the groom would probably not be a Catholic, even though there are more Catholic men than Protestant women.

How many of the 16 are Protestants?

[tantalizer270]

]]>As you can see from my friend’s latest letter, there’s been a general election in Utopia too:

“You will recall that the state of the four parties after the 1989 election was Sinistrals 289, Dextrous 243, Others 36, Indeterminates 32. This remained the case right up to the breakup of the ruling coalition last month, when new elections to the Scitting were held. This time the following occurred:

1. The Sinistrals finished with more seats, the other three parties with less.

2. The Sinistrals lost an equal number of seats to each of the other three parties and gained some seats.

3. The Dextrous and Indeterminate parties both won seats from the Sinistrals only.

4. The relative positions of the parties are unchanged.

5. Each party gained

andlost either a perfect square or a perfect cube of seats to finish with either a perfect square or perfect cube number of seats.6. The total number of seats is unchanged and no other party won a seat.

So now you can work out the new state of the parties.”

Can you?

[enigma662]

]]>I visited my niece Melinda again, and found that she had acquired another new puzzle (where does she get them?). This one is called Clock Hop, and is a kind of peg solitaire. The pegs are arranged in a circle of five holes, with one hole left empty. The pegs are numbered consecutively, from 1 to 4. The puzzle is to rearrange them in the order 4, 3, 2, 1, with the empty hole in the original position, by shifting pegs — but one is only allowed to shift a peg as many spaces as its number indicates (and, of course, there has to be an empty hole waiting to receive it!). Pegs can move in either direction around the circle — clockwise of counterclockwise. With so few pegs, it seems like a very easy puzzle.

What is the minimum number of moves required to solve my niece’s Clock Hop puzzle?

[enigma663]

]]>“Have you written your thank-you letters, Kayleigh?”

“Not yet mum, just doing it now. Do I thank Amelia for the nail varnish, Beth for the book, Clara for the lip balm, Diaz for the pencils and Elinor for the sunglasses?”

“At least four of those are wrong.”

“Ah, then was it Amelia for the book, Beth for the lip balm, Clara for the pencils, Diaz for the nail varnish and Elinor for the sunglasses?”

“That’s better, but you’ve still made mistakes.”

“How about Amelia for the lip balm, Beth for the sunglasses, Clara for the pencils, Diaz for the book and Elinor for the nail varnish?”

“Even better, but still not full marks. Clara didn’t give you the pencils or the sunglasses!”

“I give up.”

Can you help?

[puzzle#170]

]]>Arthur Amble, the town clerk of Footle, has just put his signature to the final minute in a file dealing with Environmental Recycling (Re-use of Paperclips). The file has many fascinating features, not least the fact that each of Footie’s 12 senior officers has now signed it twice.

The town hall has a rigid chain of command. Amble presides with the help of his deputy (Bumble) and Bumble’s deputy (Crumble). Crumble has three deputies, the Planning Officer (Dimwit), the Engineer (Eggwit), and the Treasurer (Frogwit). These last three are of equal and independent status and each has his own deputy (Gumling, Halfling, and Inkling respectively). These too are of equal and independent status and they share a deputy between them, who rejoices in the name of Junket. Junket’s deputy is Krumpet, and Krumpet’s is Limpet. Limpet is this the deputy-deputy-deputy-deputy-deputy-deputy-deputy-town clerk, which sounds better in German; he has no deputy.

It is a strict rule at Footle that memoranda and minutes may be passed only between officer and immediate deputy. Thus Dimwit, for instance, can deal only with Crumble and Gumling – not even with Eggwit or Frogwit. Each minute must be signed.

Given that Eggwit signed the paperclip file before Halfling and that Frogwit signed it before Gumling, can you list the 24 signatures in order?

[tantalizer321]

]]>Tom, Dick, Harry, Anne and Belinda are on holiday together on the island of Alvbe (Ars Longa Vita Brevis Est), where it is the custom to celebrate diversaries (rather than anniversaries) of everything – and particularly of one’s birth.

Today Tom is celebrating a palindromic diversary of his birth. “Happy prime diversary”, says Anne. “And tomorrow you’ll be twice a prime,” says Dick. “And the day after tomorrow you’ll be three times a prime,” says Belinda. “And the day after that you’ll be four times a prime,” says Harry.

So, how old is Tom today (in, of course, days – the number of days since his birth)?

[enigma861]

]]>Some town squares are designed as giant chessboards, but urban planner Dominica has paved her town’s new piazza with giant dominoes instead.

Picking different dominoes at random from a set, she laid them down flat to form a 7×7 square of numbers (pictured, below), leaving one space in the centre for a fountain.

Using the numbers on the diagram, can you draw the outlines of the dominoes that Dominica used, and figure out which dominoes she left out?

(Remember that a full set of dominoes contains every pair of numbers from 0-0 to 6-6. There were no duplicates).

[puzzle#169]

]]>The great dictator Fidelio T. Castersugar was forced to flee his banana republic after a counter-counter-coup and take up residence on a deserted island with his chauffeur Miguel. Now there was nothing for Miguel to drive, but that did not mean there was not plenty for him to do. While Fidelio planned his counter-counter-counter-coup, Miguel made fires, gathered coconuts, swept the beach and did the laundry. His main task was to make a temporary humidor for the cigars Fidelio had brought with him.

On the first day Fidelio let Miguel smoke one cigar and then smoked a thousandth of the number remaining. On the second day he gave Miguel two cigars and then smoked a thousandth of the number remaining. On the third day he gave Miguel three cigars before smoking a thousandth of the number remaining. He carries on like this, each day letting Miguel smoke one more cigar than he had on the day before, before smoking a thousandth of the number of cigars remaining, until, one fine day, there are no more cigars left.

Given that neither Fidelio, nor indeed Miguel, would even dream of smoking a fractional cigar, or relighting a cigar which has gone out or (horror of horrors!) making a new cigar from stale cigar butts:

How many cigars had Fidelio smoked up? How many had Miguel smoked up?

[enigma661]

]]>One perpetual novelty, at least a century old but always good for a fresh marketing under a new name, is the set of four coloured cubes best known as devils dice. If you opened them up and laid them flat, they would look like the diagram, A, B, C and D being the four colours.

According to legend, the devil, always a sportsman, once offered them to a dying sinner as a last chance to save his soul. If the poor lad could stack them in a column each of whose four sides showed four different colours, Old Nick would stay his hand. If, within the obvious time limit, not, then not.

Perhaps you would like to try. Please do not complain at having a diagram instead of the real thing. The diagram makes it easier, at any rate for those wanting to reason it out. It should take about 15 minutes, if you start by asking what each die consists of for purposes of the puzzle.

There are now **2700** puzzles available between the **Enigmatic Code** and **S2T2** sites. If you have been playing along and have solved them all, Congratulations!

In total there are 1586 *Enigma* puzzles available, with 208 puzzles remaining to post. So, about 88.4% of all *Enigma* puzzles are now available.

[tantalizer320]

]]>The mileage chart shows the distances between various secret government establishments, each of which is designated by a three-digit code. The distance between two locations can be calculated by adding the differences between the three pairs of corresponding digits, ignoring the signs of the differences. Thus the distance between 689 and 773 is (7 − 6) + (8 − 7) + (9 − 3) = 8.

For reasons of security the government wants these locations to be as far apart as possible, and is concerned that two of them are only four miles apart. Locations 000 and 999 must be retained, but the other four can be moved to locations represented by any three-digit codes, to make the closest pair as far apart as possible.

What is the greatest possible distance between the closest pair?

[enigma862]

]]>University student Rick Sloth has spent his life avoiding work, and even though it is exam season he has no intention of mending his lazy ways.

He is studying palaeontology, which he thought might be an easy option when he signed up for it, as he loves dinosaurs, but he has now discovered that it requires rather more study than he was expecting.

It turns out there are 18 topics in the syllabus and his end-of-year exam will feature 11 essay questions, each on a different topic. Fortunately for Rick, candidates are only required to answer four questions in total.

Rick wants to keep his exam preparation to a bare minimum, while still giving himself a chance of getting full marks.

How many topics does he need to revise if he is to be certain that he will have at least four questions that he can tackle?

And can you come up with a general formula for the minimum number of topics you need to study based on the number of exam questions and topics in the syllabus?

[puzzle#168]

]]>I have ten volumes of

together in order on my shelves. Each volume has 100 pages and the page numbering continues through the volumes from 1 to 1000.Applications of Abstract AlgebraA bookworm started to nibble through the even-numbered side of a leaf (whose number was a perfect square): it kept nibbling in a straight line — through covers as well — and when it stopped it had just emerged through a leaf onto its odd-numbered side (whose number was also a perfect square). Ignoring the covers, it had eaten through a number of leaves which was also a perfect square.

What were those three squares?

[enigma660]

]]>There is a rumour going around that in choosing your six lottery numbers it is better to choose higher numbers because those balls weigh more and are therefore more likely to fall to the bottom of the selection machine!

In fact there’s an element of truth in this rumour. This is due to the fact that, although the original blank balls are identical and a particular digit weighs the same whichever ball it is on, it is true that some digits use more paint than others. Therefore some of the balls from 01 to 49 do weigh a little more than others.

In fact, no ball numbered more than 25 is lighter than any ball numbered less than 25. But there are fewer balls which are actually heavier than ball 25 than there are lighter than it.

How many of the balls are heavier than ball 25?

The following clarification was published along with **Enigma 866**:

The penultimate sentence was meant to imply that fewer than half the balls were heavier than 25.

[enigma863]

]]>At the shopping mall, while their mother’s attention is conveniently fixed on a window display, Jill furtively approaches her brother.

“Hey, what do you say to a race up to the next floor and back? Winner does the other one’s chores”.

“You’re on”, says Jack.

There are two escalators that move at the same speed, one going up and one going down, and also a regular set of stairs.

The children decide that each will choose on which of the three they want to race, but whatever they pick, they will have to run both up up and down on that choice. Jack knows he can run upwards and downwards at about the same speed (and much faster than the escalator).

Which of the three methods should he choose for the race: the up escalator, the down escalator or the stairs?

[puzzle#167]

]]>I was taken to and from the secret base on two occasions but it was only on the two return trips that I was able to learn anything about the location of the base.

On my first return trip we left the base on a straight road which was headed for Arcville. After a time, one of my captors said we had reached a crossroads which was halfway between Arcville and the base. We turned off at the crossroads and continued on a straight road at right angles to our original road. After another interval of time the car stopped, I was put onto the road and the car drove off. I waited there until the police found me.

On my second return trip the same sort of thing happened except that we started from the base on a straight road headed for Cluechester and turned off at a crossroads halfway between Cluechester and the base.

The inspector in charge of the case drew a map so:

The inspector drew two circles on the map and put his finger on two points — “The base is at one of these two points. Let’s go!”

On a copy of the map, draw the two circles that the inspector drew and mark the two points he put his finger on.

[enigma658]

]]>“If Albert Tatlock enters the beer-drinking contest again this year, I shall cancel it”, said the vicar firmly. (Last year Albert drank all the profits meant for the church roof).

With this proviso all was made ready for the Loose Chippings sports day. Even so, however, all did not go for the best, as the hated Cocksure brothers carried off all the medals in all the events. Each brother won just two medals, being the only one to take medals in just those events and having just one brother who took a medal in neither.

Scoring 3 for a gold, 2 for silver, and 1 for a bronze, all except George finished with a different total. George, with medals for sausage-stuffing and pie-throwing, scored the same total as Henry, who got a medal for the sack race and a better one for the egg and spoon race.

Precisely which medals did George and Henry win? And did Albert enter for the beer drinking?

[tantalizer319]

]]>Three local teams form a league in which they play each other several times, earning 2 points for a win and 1 for a draw. Part way through the season the table is such that, with letters consistently replacing digits and different letters used for different digits, part of its reads as above.

Although Attack is ahead of Brave which is ahead of Calamity (if teams have equal points then goal difference determines the order), Calamity has played more games than Brave which has played more games than Attack.

What were the scores in the games between Attack and Brave?

[enigma866]

]]>In the Zordik language, the seven days of the week start with Ardik (Monday). The other days are Bordik, Curdik, Deldik, Endik, Fordik and Gandik, but not exactly in that order.

In her Zordik class, teacher Miss Taik asked the students to recite the days in order, starting with Ardik. Sven called them out in alphabetical order.

“You got three in the right place”, said the teacher. “Kim, you have a go”.

“Deldik, Bordik, Ardik, Curdik, Endik, Gandik and Fordik”.

“Three right again. Help them, Raki”.

“Gandik, Deldik, Curdik, Bordik, Ardik, Fordik and Endik”.

“Unbelievable — you only got three right as well. Still, you now have all the information you need to work out the days of the week in order”.

What are they?

[puzzle#166]

]]>This is the “working” of a long division sum. Each digit has been consistently replaced with a letter.

What was the problem (dividend and divisor) to be solved?

[enigma659]

]]>Tom, Dick and Harry settled down to a serious game of dominoes in the snug at the Puzzler’s Head last night. They shuffled a standard set of 28 tiles, running from 0-0 to 6-6 and drew nine each. By chance each drew the same number of pips.

Tom played first and thereafter each added a domino to one end of the growing line on the usual principle of pairing like with like. (There was no extra turn for playing a double). After three rounds, Tom and Dick had played dominoes totalling eight pips each, whereas Harry’s totalled seven.

Tom then played a double, Dick a domino with nine pips and Harry a double; Tom a domino with five pips, Dick one with six pips, and Harry a double; Tom a domino with nine pips, Dick a double, and Harry a domino with eight pips. This left Tom unable to play.

One domino remained undrawn at the start of the game. Which was it?

[tantalizer318]

]]>Somewhere and sometime, in a parallel and alternative world, Nugel Lawson is Chancellor of the Earth Exchequer and has just received from Ursa Major a minicalc (price: 7 bongs from the Yellow moon) able to display in one go numbers of up to 17 digits. “At last”, he sighed, “a calculator on which it will be possible to display our interplantetary trade deficit in Earth pounds”. Now, this deficit, by the the way, had quite an amusing property. Dividing it by 4 did nothing at all to the digits apart from making the digit at the left-hand end of the number zoom to the right-hand end of the number, and all other digits move one place to the left. “Back of the queue!” gleegled Nugel as he worked this trick. In the course of playing with the minicalc he also discovered that the interplanetary trade deficit this Earth month was the largest number having this property that would fit onto the display of the minicalc. “Not to worry”, he said to himself as he settled back in his space hammock for a stellar snooze, “if things get any worse, there are plenty of other alternative universes…”.

What was the interplanetary trade deficit?

[enigma653]

]]>Penny Wise, chief financial officer of the Hartree Power Company, has decided to rent out surplus land beside its cooling towers, in an effort to boost her company’s annual income. The land, outlined in red on the plan above, was to have been for a fourth tower, which is now no longer required, plus the land between the three existing towers.

Her legal officer has been tasked with preparing the rental agreement, but is having difficulty in calculating the area of land for rent.

Can you help?

[puzzle#165]

]]>The country road runs east-west. Mary’s farm is 13/24 miles due south of the only tree on the road. The morning bus passes the tree at 8am and travels at 26 miles per hour along the road.

Each morning Mary picks some point on the road and runs from the farm, in a straight line, at 10 miles per hour, until she reaches that point on the road.

The bus will stop and pick Mary up provided she reaches the point on the road by the time the bus reaches it. By varying the point she chooses, Mary can vary the time she has to leave home.

What is the latest time that Mary can leave home and still catch the bus?

[enigma654]

]]>To find the relationship between a man and an ape imagine four different squares with the characteristics shown below. In these the digits have been replaced by capital letters and question marks. Different letters represent different digits, and a letter always stands for the same digit, while a question mark can be any digit:

Square I has an area =

MONKEY.Square II has a perimeter =

?MY?, and length of side =PET.Square III has length of side =

M?Yand an area =P?E?T(and its perimeter is twice that of one of the other squares).Square IV has a perimeter =

APE.That should enable you to discover the missing link and to find the difference between

APEandMAN.

[enigma656]

]]>One relationship between the year 1997 and Hong Kong is shown in the multiplication below, where the missing digits have been replaced by letters and asterisks. Each letter stands for the same digit whenever that letter appears, while question marks indicate where the remaining digits should be.

I don’t really need to tell you whether there is any relationship between the letters and digits shown, but I can say that the only zero in the multiplication is denoted by a letter.

So what comes after 1997?

[enigma652]

]]>Amira presented her homework to her teacher:

“Wrong, Amira, please check your working.”

“I promise you it is right. It is just that I have used a code. Every digit used represents a different digit, and the same digit is always represented by the same ‘wrong’ digit. For example, maybe I replaced all the 6s with 4s. Or maybe I did something else…”

“You are giving me a headache, Amira.”

What is the correct sum?

[puzzle#164]

]]>I have just bought a new electronic clock and I have to put in the fraction of a second which elapses between updates of the clock’s display. For example if I put in 8/13 then the clock’s display will update at the following times after I start the clock:

8/13, 1 + 3/13, 1 + 11/13, 2 + 6/13, 3 + 1/13, 3 + 9/13, 4 + 4/13, 4 + 12/13, 5 + 7/13, …

The clock’s display actually gives the time to the nearest second and so in this case it shows, in turn:

1, 1, 2, 2, 3, 4, 4, 5, 5, …

I can put in any fraction I choose, provided that the denominator is odd.

For each of the following sequences, say whether or not I can put in a fraction so that the clock’s sequence of displays begins with the given sequence:

(a) 1, 2, 3, 4, 5, 5, 6, 6.

(b) 1, 2, 3, 4, 4, 5, 6, 7, 8, 8.

(c) 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13.

(d) 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

[enigma864]

]]>Barbara bought 5 lbs of carrots, 7 lbs of apples, 6 lbs of bananas and 2 lbs of potatoes. John bought 6 lbs of carrots, 2lbs of apples, 7 lbs of bananas and 15 lbs of potatoes. Mark bought 10 lbs of carrots, 5 lbs of apples, 1 lb of bananas and 4 lbs of potatoes. Susan bought 1 lb of carrots, 6 lbs of apples, 9 lbs of bananas and 4 lbs of potatoes.

The four bills came to £6.70, £7.70, £7.80 and £8.60, but I forget which bill belonged to which. However, I do remember that the price per pound of each vegetable and each fruit was a multiple of 10p.

What were the prices per pound of carrots, apples, bananas and potatoes?

When this puzzle was originally published the apples were missing from John’s shopping list, which meant there wasn’t a sensible solution. A correction was published with **Enigma 656**, and is included in the above text.

[enigma651]

]]>In the TV number quiz show

Down for the Count, four contestants are challenged to combine each of five number cards exactly once to achieve a target number. They are allowed to use standard arithmetical operations +, –, × and ÷ (as well as brackets, if required).“I’ll take five cards from the top row”, requested one of the contestants, and the presenter Anne-Marie obligingly revealed these five numbers:

10

100

1,000

10,000

100,000.The studio computer then generated today’s challenge:

“Produce a whole number without any zeroes.”

No zeroes? Wow. Each of the four contestants thought hard, and after the timer ran out, each announced that they had used the five cards to produce a positive number smaller than 10. All four were different.

Which numbers did they get, and how?

[puzzle#163]

]]>Old MacDonald had a farm – and on his farm he had some sheep pens. Five rectangular pens, in fact, one being completely surrounded by the other four, as shown in the sketch, which is not to scale. The pens are formed by 12 straight lengths of fence, all different lengths and each a whole number of yards long (ignore the size of the corner posts).

What is the smallest total length of fencing which can form such an arrangement?

**Note:** Although it is not explicitly stated it seems that, in order to arrive at the (eventually) published solution, the fence panels are only allowed to be joined at the marked posts.

A correction to the original solution was published with **Enigma 866**.

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

[enigma852]

]]>While window-shopping at a toy store, my partner and I came across a set of dice that was partially obscured at the back of the shop’s display.

My partner took one look and bet me that I couldn’t tell them what the sum of all the touching faces between the dice would be. I accepted and gave the correct answer.

What was it?

[puzzle#162]

]]>In what follows, the digits from 0 to 9 have been replaced consistently by letters, different letters being used for different digits:

ODDis odd;

SQUARESis a perfect square;

CUBEDis a perfect cube.Find the value of

CARD.

[enigma650]

]]>A group of friends stand in line, each holding a number written on a card.

Alan’s number is the square root of one more than the number that Brian holds.

Brian’s number is the square root of four more than Colin’s number.

Colin’s number is the square root of 16 more than Dave’s number.

Dave’s number is the square root of 64 more than Eric’s number.

This is virtual reality, where even the impossible is easy. The line is infinitely long, and the constant 1, 4, 16, 64 … increases by a factor of four at each step.

What is Alan’s number?

[enigma865]

]]>“When exactly is a person supposed to celebrate their half birthday?”, wondered Lionel. “I mean, you could celebrate your 182/365th birthday, or your 183/365th birthday, but unless it is a leap year, the 1/2 fraction never arrives”.

“The world’s greatest mystery”, said Jill.

“But the good news is that if you write every date as a fraction, so 1 January is 1/365 and 2 January is 2/365 and so on, then my birthday falls on a day where that fraction is reducible”, says Lionel.

“Reducible?”

“You know, can be simplified, like 12/18 can be reduced to 2/3. My birthday fraction can be reduced, but the next day can be reduced even more, to numbers on the top and bottom that are even smaller”.

“I’ll be sure to celebrate by reducing the number of presents I get you”, says Jill.

When is Lionel’s birthday?

[puzzle#160]

]]>I asked Susan to choose a number and she chose 40. I went over to a large pile of bricks and built the following wall which contained precisely 40 bricks.

A wall can be any width and any height greater than 1, but each row, apart from the bottom, must contain one less brick than the row below it.

I then asked Susan to choose any year from the past 1000 years, and told her that I would try to build a wall with that number of bricks in it.

Give a list of all the years she could choose for which I could not build a wall.

[enigma649]

]]>The class kept pestering the teacher to tell them her age. So she gave them a numerical exercise leading to it. She wrote two whole numbers on the board which, as she told the class, added together gave the square of her age.

Most of the class then correctly worked out her age, but one boy used his calculator and made a couple of errors. First, between entering the two numbers he forgot to press the “add” button. Then, when taking the square root, he pushed that button twice instead of once. This too gave him a whole-number answer, but it turned out to be the age of the teacher’s husband, who was older than her.

How old were the teacher and her husband?

[enigma851]

]]>Click! The camera shutter opened and closed just as the creature’s head ducked back beneath the surface of the lake, creating a large ripple.

“I got it! I finally got a picture of the Loch Ness monster!”, exclaimed Lily. She looked at the result on her digital camera. “It’s blurry!”. She hung her head in defeat.

The boat passed directly over the spot where they saw the creature, but it was nowhere to be found. “How far away do you think it was when I took the picture?” Lily asked Amelia later that day.

“Well, we were travelling in a straight line at 2 metres per second towards where we saw Nessie, and it took us 5 seconds to reach that large ripple it created, and another 10 seconds to get to the other side of the ripple. So that means it was 5 seconds plus an additional 5 to the middle of the ripple, which would make it 20 metres!”.

“Sadly, I think your maths is as fuzzy as my photo,” said Lily.

How far away was the monster at the time the picture was taken?

[puzzle#159]

]]>In this problem each letter stands for a different digit and the same letter represents the same digit wherever it appears. Note that the second line of the sum is

UNE, as in French, notONE.If you hope to have la fortune of winning a fortune, you need to find the value of

FORTUNE.

[enigma853]

]]>Make an enlarged copy of the triangle above with the horizontal and vertical sides each 10 centimetres in length.

Take four coloured pens — red, blue, green and yellow — and colour in the triangle in any way you like so that each point gets one of the four colours.

For each of the following statement say whether it is “true” or “false” that however we colour the triangle we are certain to be able to find two points of the same colour which are at a distance apart of:

1. at least 2 centimetres.

2. at least 5 centimetres.

3. at least 8 centimetres.

4. at most 0.001 centimetres.

5. at most 0.000001 centimetres.

6. at most 0.000000001 centimetres.

[enigma868]

]]>The streets of New Addleton are set out in a rectangular grid. Seven coffee vendors (the circles in the diagram above) have stalls at metro stations and want to set up a central depot to collect supplies from each morning. They want to minimise their combined cycling distance from stall to depot. Pat has four candidates for the depot location: A, B, C and D.

“Are you sure one of those four is optimal?” asks Shahin. “I suppose we could work out the total vendor-depot distance for every point on the grid.”

“No need, I can tell you the best place just by looking at the diagram,” announces Kim.

Which location does Kim recommend and why is she so confident?

[puzzle#159]

]]>In a tennis doubles match, one player serves in the first game and his partner will serve in the third game; of their opponents, one will serve in the second game and the other in the fourth game; thereafter each player will serve in every fourth game, except that at the start of a new set either player of the pair whose turn it is to serve may serve in the first game, with his partner serving in the third game, and either of the opponents may serve in the second game, with his partner serving in the fourth game; thereafter each player will serve in every fourth game for the rest of the set.

A set is won by the first pair to win six games, except that if it goes to 5-5 it is won 7-5 or 7-6. In the match that we are considering, which went to five sets, if all the games are numbered from 1 to

n(wherenis the number of games played) Wood only ever served in prime numbered games (remember 1 is not a prime). So if I told you the score of the third set you would be able to identify all the games in which Wood served.(1) What was the score in the third set?

(2) In which prime-numbered games did Wood’s partner serve?

[enigma935]

]]>The TV sporting highlight of my childhood was always the Five Nations rugby championship, which involved a series of matches between England, Scotland, Wales, Ireland and France.

Every fortnight, on Saturday afternoon, there would be two matches, with the fifth country having the day off.

The fixture list had an elegant symmetry to it. Each country played every other country once, with two matches at home and two away, and each country alternated between playing at home and away.

I recall that in one year, the fixtures on the opening Saturday were Ireland vs England and France vs Wales, and that, on the third Saturday on which matches were played, Wales competed at home.

If those memories are correct, what were the final two matches of the competition?

[puzzle#158]

]]>To enter the lottery I chose six lucky numbers: I called this “selection A”. I have added 1 to each of those six numbers to form a new “selection B”, I’ve added 1 again to each to form “selection C”, and repeated this twice more to form “selection D” and “selection E”. So, for example, the lowest number in selection E is four more than the lowest number in selection A. Of course, as always, each selection is in the range 1-49.

In one of the selections (but not selection A) the six numbers are all two-digit numbers with the same first digit.

In one of the selections (but not selection A) there are no prime numbers.

Each week I decide to have two goes at the lottery and I choose at random two of my five selections. I know that whichever pair I choose, it would be possible for one of the selections to contain four of the six winning numbers and the other selection to contain none. On the other hand, whichever pair I choose, it would also be possible for both selections to contain four of the six winning numbers.

Which six numbers form “selection A”?

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

[enigma936]

]]>The season had to come to an end with each of the six teams, A, B, C, D, E, and F, playing each other once and no game ending in a draw. The sports editor allocated one point for each win and totalled the points for each team. She intended to list the teams in the order of their total points, but then she had an idea. There had been 15 results during the season and she would like her list to reflect as many of those results as possible, in the sense that if X had beaten Y, then X should be above Y in her list. For example, F had beaten C, and so she would like her list to reflect that result by F being above C.

At first she only looked at lists that were in order for points totals, that is if X got more points than Y, then X was above Y on the list; if X and Y had the same number of points then they could be in either order. Doing that, she found the list, [L] = C D E F A B (C at the top, B at the bottom) reflected the largest number of results. However, when she turned to the other lists she found that the list C D E F B A (C at the top, A at the bottom) reflected more results than the list [L].

1. How many results did [L] reflect and what were the results it did not reflect?

2. What is the largest number of results in that season that could be reflected by a list?

3. What is the smallest number of results that season that could be reflected by a list?

[enigma934]

]]>At Shady Hills retirement home, Ike and Eileen are arguing over weather predictions.

“I’m telling you, Eileen, it’s going to rain tomorrow and the next day, and then it won’t rain on the third day”.

“Balderdash, Ike. The way my knee is acting up, I’m sure that it won’t rain tomorrow, but it will rain for the following two days. I’lI bet my Monday night cheesecake on it”.

“I’ll take that bet. But what if we’re both wrong?”

“Then we’ll just keep waiting until there are three consecutive days that match one of our predictions”.

Two of the staff overhear the conversation. “This time of year, there’s a 50-50 chance that it will rain on any given day!”, says one, “so I guess their chances are even”.

“You’re right about the rain”, says the other. “But even so, I think one of them has a much better chance of winning”.

Which one, and why?

[puzzle#157]

]]>In golf matchplay the two players keep a count of how many holes they have won. As soon as one player’s lead exceeds the number of holes left to play (for example, 7 up and 6 to play; or 5 and 3; or 1 and 0, etc.) then the round ends and the score is noted. I noted my winning scores on several rounds recently and wrote them down in one line. So, for example, had they been the scores quoted above I would have written 765310. Then I noticed that if I replaced different digits everywhere by different letters, with the same letter being used for the same digit wherever it occurred, then I could make it:

MATCHPLAY.Of course, each round was over 18 holes. Indeed both

TEEandHOLEare divisible by 18.What is your

TOTAL?

[enigma933]

]]>What happens when a sadist meets a masochist? Does the former torture the latter and, if so, to whose satisfaction? Logicians have a long worried about such questions, while psychologists have worried about the logicians.

Useful data may well be forthcoming, if the rumoured American and Russian two-man space probe goes ahead as planned. Each nation is to supply one astronaut, to be chosen in each case randomly from a short list of eight. The American eight comprise six sadists and two masochists.

Each member of the Russian eight is also a sadist or a masochist. (Logically minded readers will appreciate that no member can be both). The chances of a probe manned by a sadist and a masochist are 11 to 5 in favour.

By the way, how many of the Russians are masochists?

[tantalizer317]

]]>There are eight hockey teams in my area and they have quaint one-letter names:

A,B,C,D,E,F,G, andH. In each of the years ’94, ’95, ’96, ’97 they have organised a knockout competition. Curiously, over the four years no two teams have played each other more than once.I watched

BF(that is to sayBversusF),CGandEGin ’94.AH,CH,DGandDHin ’95.AG,EH,CF,DFandFGin ’96. AndAC,BHandFHin ’97. I had planned to watchAin the ’97 final but they were knocked out in an earlier round. I remember thatADandCEwere not in the same year. AlsoEFwas not in ’95.Name the years which:

(i)

AplayedB;

(ii)BplayedC;

(iii)CplayedD;

(iv)DplayedE.

[enigma926]

]]>I have a solid wooden tetrahedron, each of whose four faces consists of an equilateral triangle of 20 cm sides.

A woodworm started at the midpoint of one edge and burrowed in a straight line through the tetrahedron and emerged at the midpoint of an opposite side.

Meanwhile a spider started to walk from the same point where the woodworm started and it took a shortest route on the surface of the tetrahedron to the point where the woodworm had emerged.

(1) How far did the spider walk?

(2) What was the initial angle between the two different routes?

[enigma927]

]]>At first glance, you wouldn’t think it was possible to put these three L-shaped tetrominoes together to make a flat, symmetrical shape.

And yet, it turns out that there are two different ways to make a shape with mirror symmetry using all three Ls. Can you find them?

Hint: one solution looks a bit fishy, but you might love the other one.

[puzzle#156]

]]>Here is a map of Enigma Island, showing 16 villages and all the roads. For a delightful walking holiday, I suggest you start at A and continue: M N D E F E D C B A L K J O P I H G F N O G H I J K L P M C B A. As each village starts with a different letter, there can be no confusion and you can return finally to A, having been over each stretch of road on the island just once. Don’t worry that the villages aren’t labelled – a mere oversight and, anyway, all routes are equally pleasant.

What’s that? The suggested tour is impossible? Why, so it is! I see I have inadvertently transposed two consecutive villages at one point in the route. Never mind, I’m sure you will quickly work out which two they are.

Well, which are they?

[tantalizer316]

]]>Come with us now to those 10 faraway tropical islands with quaint one-letter names,

A,B,C,D,E,F,G,H,IandJ. Transport between the islands is by boat. For certain pairs of islands, boats go back on the route between the two islands. I do not remember any routes that have a boat, but I do remember that none of the four routesAG,BD,CD, andGI, have a boat.It is possible to make the journey from any island to any other island. However, the journey may involve more than one route, in which case you would have to change boats at intermediate islands. You always use the minimum number of routes to make your journey. You are charged £1 for each route you use on your journey, except that if you use only one route then you are charged £2 for your journey. The cost of the journeys are as follows:

£5:

AE;

£4:AH,EG,EI,EJ;

£3:AB,AF,AI,CE,DE,FG,GH,HI,HJ.

All other journeys cost £2.How many routes have boats on them?

[enigma922]

]]>Ada Hubbard loves creating riddles and takes every opportunity to pose them. She occasionally includes her daughter Betty in her brain-teasers.

When someone recently asked for Ada and Betty’s ages, Ada gleefully supplied a riddle instead:

“I am five times as old as Betty was when I was as old as Betty is now. When Betty is as old as I am now, then I will be 77”

How old are Ada and Betty?

[puzzle#155]

]]>Jim is a keen amateur horticulturalist who has succeeding in breeding a new type of plant with most unusual properties. For the past three years he has had a batch of these plants under test in his greenhouse. It appears that for the plants with red flowers one year, a fixed proportion will remain red the next year, a fixed proportion will have white flowers, and a fixed proportion will be blue. Plants with flowers that are white or blue one year behave in a similar way, with fixed proportions of the flowers either remaining the same colour the following year or changing to one of the other two colours.

Initially, there were 75 plants with red flowers, 50 with white and 50 with blue. The next year the numbers were 75 red, 25 white and 75 blue, and last year 80 red, 30 white and 65 blue.

Assuming that they all survive, how many plants with flowers of each of the three colours would be expected this year?

[enigma929]

]]>I have in mind a three-figure square. Even if I now told you its first digit you could not work out the square. Even knowing that you could not do that, if I told you the sum of its digits you still could not work out the square. Even knowing that you could not do that, if I told you its middle digit you still could not work out the square.

Even knowing that you could not do that, if I told you its first digit you still could not work out the square. Even knowing that you could not do that, if I told you the sum of its digits you still could not work out the square. Even knowing that you could not do that, if I told you its middle digit you still could not work out the square.

Even knowing that you could not do that, if I told you its first digit you still could not work out the square. Even knowing that you could not do that, if I told you the sum of its digits you still could not work out the square. Even knowing that you could not do that, if I told you its middle digit you still could not work out the square.

Even knowing that you could not do that, if I told you its first digit you still could not work out the square. Even knowing that you couldn’t do that, if I told you the sum of its digits you still could not work out the square. Even knowing that you could not do that, if I told you its middle digit you still could not work out the square.

Knowing that you could not do that, if I told you its first digit you

couldnow work out the square.What is it?

I don’t have a source image for the puzzle as originally published, but I believe the above text is correct (although an actual source image would be appreciated).

[enigma928]

]]>“So, Maureen, how electable are they?” asked Tariq, the local party chairperson, counting up the number of people on the list of possible candidates for the by-election.

“It’s good news and bad news. The good news is that I have looked into their backgrounds and discovered that eight of them have proven leadership skills, seven of them have been completely honest about their expenses, and six of them are always loyal to the party when they tweet.”

“Good. And because I know how many candidates there are, I can be certain that at least one of them must have all three of these virtuous traits! What’s the bad news?”

“Um, well you are right, but unfortunately there is only one candidate who has all three virtues – and it is your nemesis Judy Prim.”

How many candidates are there in total?

[puzzle#154]

]]>Harry, Tom and I were trying to produce chains of perfect squares; we each started with a 1-digit perfect square and then attached another digit to form a 2-digit perfect square and then another to form a 3-digit perfect square, and so on. At each stage the newly attached digit could be placed at the front or end of the previous square (except that a leading zero was not allowed) or inserted at any point inside the previous square, but the order of the digits of the previous square had to remain unchanged. The newly attached digit could be one that already appeared in the chain.

We each got as far as a 4-digit perfect square; indeed our 4-digit squares were all different, but Tom and I had the same 3-digit square. Harry and Tom added 5-digit perfect squares to their chains and one of them added a 6-digit perfect square to his chain.

(1) Who had the 6-digit square, and what was it?

(2) What was the 5-digit square in the other chain?

(3) What was my 4-digit square?

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

[enigma923]

]]>The diagram shows a square dissected into 10 acute angled triangles but what is the smallest number of acute triangles into which a square can be dissected? An acute triangle is one whose angles are all less than (not equal to) 90°.

There are now 2600 puzzles available between **Enigmatic Code** and **S2T2**.

[enigma919]

]]>Harry, Tom and I were trying to find sets of three 2-digit positive integers such that if we added any two integers of the set together we formed a perfect square. Within each set the three integers had to be different, but an integer used in one set could be used again in another.

We [each] found two sets, such that the six perfect squares we each formed from our two sets were all different. One of my sets was the same as one of Harry’s and my other set was the same as one of Tom’s; their other sets were different.

(1) What were the integers in each of my two sets?

(2) Which integer appears in two sets that none of us found?

The following correction was published with **Enigma 920**:

In

Enigma 918“Six squares” the first line of the second paragraph should read: “We each found two sets …”. Apologies to all puzzlers.

I have made the correction in the puzzle text above.

**Enigma 40** is also called “Six squares”.

[enigma918]

]]>Tom Tightwad keeps his money in a safe, the code for which is a 10-digit number that uses every digit between 0 and 9.

Fearful of forgetting the code, Tom wrote it on the front of the safe, but disguised it. He wrote the first five digits along the top of a 5×5 grid, and the last five digits down the side. Then he multiplied each digit along the top by each digit down the side, filling the grid with 25 numbers. Finally, he erased his code number from the top and side so that only the grid remained.

Unfortunately, the clumsy fellow also erased most of the numbers in the grid. And of what remains, some digits are so hard to read that they have been replaced with an “X” (for example, in the diagram, the code’s first digit multiplied by its eighth digit is now “20-something”).

Can you crack the safe?

[puzzle#153]

]]>

The layout:Take 202 plates and lay them on the ground in a large circle. Take 202 coins and put one on each plate.

A move:Each move consists of moving two coins as follows. Take any coin and move it, in a clockwise direction to the next plate in the circle. Take any other coin and move it in an anticlockwise direction, to the next plate in the circle.Start with the layout and make a move. Then, starting with the position reached after the first move, make a second move. Continue in this way, making move after move.

Question 1:Can you make a sequence of moves so that eventually all the coins are on the same plate?

Question 2:Suppose that you had started with 303 plates and coins. What then is the answer to Question 1?

Question 3:As Question 2 but with 303 replaced by 404.

Question 4:Suppose that you had started with 202 plates and coins but that a move was changed so that, for each move, you could either move both coins in the same direction or in opposite directions to the next plates in the circle. What then is the answer to Question 1?

Question 5:As Question 4 but with 202 replaced by 303.

Question 6:As Question 4 but with 202 replaced by 404.

[enigma932]

]]>Tom, Dick and Harry sometimes recall their days as aces in the Battle of Britain:

“Do you realise, chaps”, says Harry, “that we bagged

PQjerries between us?”“How many did you get yourself, dear?” asks Maud dutifully.

“All but

RSof them”, Harry replies with a wink. “Tom gotQR(which is all butQPof the total) and Dick gotSRand I got, well, let me see now …”To make things harder for you. I have put

P,Q,R, andSfor the four different digits Harry mentions (none of which is zero).PQmeans 10×P+Q, and so on.So what was Harry’s personal bag?

[tantalizer323]

]]>John Major, Paddy Ashdown, Victoria Wood, Julie Walters, Tim Rice and Andrew Lloyd Webber met at a party. Among the six were John Major’s greatest fan, Paddy Ashdown’s greatest fan, Victoria Wood’s greatest fan, Julie Walter’s greatest fan, Tim Rice’s greatest fan, and Andrew Lloyd Webber’s greatest fan: and these six fans were all different.

Furthermore, given any two of the six people, their first names have at least one letter in common if and only if the surnames of their greatest fans have at least two letters in common.

Only one person there was their [own] greatest fan. Who? And who is Tim Rice’s greatest fan?

[enigma922]

]]>At my bedside, I have a traditional-style electric clock wired to the mains. It has a curious fault. Whenever there is a lightning strike on the village’s electricity substation, the power surge causes my clock to reverse direction.

When I went to bed last night, I could tell that a storm was brewing, but I managed to sleep through it. When I woke this morning, the clock said 7 o’clock, and a quick check of my wristwatch confirmed that this was indeed the correct time.

But I just heard on the local news that there had been three lightning strikes on the substation during the night with exactly 30 minutes between each strike.

At what time was the first lightning strike?

[puzzle#152]

]]>Here’s another enigmatic layout of dominoes for you using eight from a standard set, with two “vertical” and the rest “horizontal”, and using the “doubles” one-one, two-two and four-four.

If you now read each row as a number you get the four-figure squares 1024, 1156, 1225 and 1444.

Your task this week is to find a layout of eight dominoes from a standard set in a four-by-four array so that, once again, each row is a four figure square, not all the dominoes are vertical, and such that among your eight dominoes you use all seven doubles.

What are the four squares seen in your array?

[There now follows an additional but unnecessary clue: masochists should not read on.

Note that in the list of requirements I have not asked that the four squares are all different].

[enigma915]

]]>The trendy Wotherspoons have just bought a second-hand public convenience, described by the agent as: “Victorian — ripe for conversion”. They worked out a scheme for making it into a family home, which included plenty of conversation booths, ceramic surrounds, large cuspidors, and indoor water gardens, and should have been very cheap. But the builder’s estimate was a nasty shock.

“Well, I suppose it was bound to run to four figures”, said Wotherspoon.

“Yes, dear”, his wife replied, but surely they have been written in reverse order?”

The builder, however stuck to his guns, insisting that he would barely make a penny at the price quoted.

This is far from true. He makes estimates by writing down what he thinks it will actually cost and quadrupling it. The result in this case is that Mrs W. is right — the actual cost has been reversed.

So what will the Wotherspoons have to pay for their modishness?

[tantalizer324]

]]>In this problem each letter stands for a different digit and the same letter represents the same digit wherever it appears.

What is the value of

OVERFINE?

[enigma920]

]]>When the aliens landed in my back garden, my first thought was that they use the same number system as we do, because there was a two-digit number written in our Earthly digits on the side of their spaceship.

But when the aliens emerged from the ship, I saw that they each had 16 fingers. I intuited that they therefore used a hexadecimal (base 16) method of counting. Their first ten digits, from 0 to 9, work just like our terrestrial decimal system, but 10-15 are expressed as A to F, respectively. After F (our 15) comes 10 (our 16), 11 (our 17) and so on. What can I say, I am very intuitive.

In hexadecimal, 1E is the same as our decimal 30 (one 16 and fourteen 1s) and their 25 is our 37 (two 16s and five 1s).

The number painted on the side of their ship was therefore written in hexadecimal. Yet by a remarkable coincidence, it could be translated into our number system by simply reversing the order of the digits.

What was the alien number in Earthly notation?

[puzzle#151]

]]>The number 102564 can be multiplied by 4 simply by moving the last digit to the front:

102564 × 4 = 410256

Can you find the smallest number (with no leading zeros) which can be multiplied by 2 using this method?

[enigma924]

]]>The little Thingummies, conscious of inflation, have just written their annual letters to Father Christmas. Alice has asked for two mice, three puzzles and a whistle. Brenda has put in for three mice and a revolver. Caspar has requested a mouse, two revolvers and three puzzles. Desmond has listed a mouse and three whistles. Finally, Edward, always a bit of a ditherer, has written: “I’d like just one revolver please, and then, to fill my stocking up, some of the same as the others”.

Father Christmas is always very good to the Thingummies and supplies what they ask for. They, for their part, observe a strict protocol whereby Alice does neither better nor worse than Brenda; Desmond does neither better nor worse than Edward; Desmond and Edward together do neither better nor worse than Caspar; and the three boys collectively do neither better nor worse than the two girls collectively. It all depends, of course, on working out suitable weightings for each kind of present.

Before the old gentleman tugs his beard out in despair, can you tell him what else to give Edward?

[tantalizer325]

]]>The Pet Food Department of Wee Chums plc. has just ended its New Year’s bargain sale of tins of BowWowser, MoggieMunch, BudgieBits and GoldieGrubs.

The department’s sales recorded show that nobody bought more than one tin of any particular brand, and every possible combination of tins was bought. A fair number bought the 4 tins. Twice as many bought each combination of three tins, and three times as many bought each combination of two tins. Different numbers bought only a single tin, though I forget how many.

However, I do remember that half of all the BowWowser tins sold were singles: and the corresponding proportions for MoggieMunch, BudgieBits, and GoldieGrubs were one quarter, one sixth and one eighth respectively. Altogether, getting on for 10,000 tins were sold.

How many tins of BowWowser were sold altogether?

[enigma917]

]]>Logan pulled an old jigsaw box off the shelf and gazed at the picture of a carousel on the lid. Under the picture in big, bold type it stated “468 pieces”.

He tipped out the contents. It didn’t look like 468 pieces. He started to count, but realised it would take a long time. What if he just counted the edge and corner pieces? That might be quicker, and if any of those pieces were missing, that would confirm that it wasn’t a complete jigsaw.

If only Logan knew how many edge pieces there were, including corners. Since there was nothing unusual about the shape of the jigsaw, how many should he expect to find?

[puzzle#150]

]]>