I like my eggs to be boiled for exactly 9 minutes. The problem is that I have no way to measure time except for two egg timers that are able to measure precisely 4 and 7 minutes respectively.

There is more than one way to set up the timers to measure exactly 9 minutes, but I am keen to eat my egg as soon as possible. Can you help?

[puzzle#22]

]]>I have constructed a cyclical chain of 4-digit perfect cubes such that each cube has at least two digits (but not necessarily the same digits) in common with each of its neighbours in the chain. The chain consists of as many different 4-digit cubes as is possible consistent with the stipulation that no cube appears in it more than once.

If a cube has a repeated digit, that digit only counts more than once in calculating the number of digits it has in common with another cube if it also appears more than once in the other cube; so 1000 has three digits in common with 8000, but only one digit in common with 4096.

Which two cubes are the neighbours of 9261 in the chain if:

1. Somewhere in the chain two neighbouring cubes have more than two digits in common?

2. Nowhere in the chain do two neighbouring cubes have more than two digits in common?

[enigma992]

]]>“I say”, said young Tommy on Christmas morning, when we had each seen our own presents but no one else’s, “here is a poser based on the tea-strainer, bath hat, bath soap and gloves I gave Jane, Kate, Lucy and Maud. I shall now put three questions and each of the girls must give at least two true answers out of the three”.

He first asked, “Did I give you the tea-strainer?” and got the answers: Yes, Yes, No, No. Then he asked: “Did I give you something for the bath?”, getting the answers: Yes, No, Yes, No. The he asked: “Did I give you something to wear?”, getting the answers: No, Yes, Yes, Yes. (Answers are in alphabetical order of girls).

“Now”, he said to the rest of us, “I can tell you that exactly one girl is in a position to deduce what each other girl got. Can you tell me what each girl did get?”

Due to a typo in *New Scientist* this puzzle was published as **Tantalizer No 42**.

[tantalizer421]

]]>Out there, somewhere in the night, is Elk Elloy, gunning for me. My only hope is to stay in the dark.

Stretching ahead of me is the Boulevard, all 3686.3 yards of it. If I can make the other end of it then I’ll be safe. But the whole length of the Boulevard is covered with neon strip lights. One hundred and ninety-three of them, each 19.1 yards long, set out end-to-end. They flash on and off steadily through the night. There go the 1st, 3rd, 5th, 7th, …, 193rd. They’re on for just an instant. Now there is a 12-second pause and then on come the 2nd, 4th, 6th, …, 192nd, for just an instant. Then another 12-second pause and we begin all over again with the odd numbered strips.

Fortunately, each strip only lights the ground directly below it, so there is a chance I can walk along the Boulevard and avoid ever being under a strip when it comes on.

There are just two catches. First, I must walk at a constant speed which is a whole number of yards per minute, otherwise I will arouse the suspicion of Patrolman Nulty who covers the Boulevard. Secondly, I cannot walk at more than 170 yards per minute.

What speed should I walk at, in yards per minute?

[enigma517]

]]>Which number is bigger:

The product of all the whole numbers from 1 to 10 inclusively, sometimes written as 10 factorial or 10!

or:

The number of seconds in six weeks?

Can you work it out without resorting to a calculator?

[puzzle#21]

]]>Each year the four football teams A, B, C and D play each other once, getting 3 points for a win and 1 for a draw. At the end of the year the teams are ordered by total points and those with equal points by goal difference. Any still not ordered are then ordered by goals scored and then, if necessary, by the result of the match between the two to be ordered. Any still not ordered draw lots. The top two teams with a prize.

The order the games are played in can vary, except that A always plays its opponents in the order B, C, D, and A vs B is always the very first match of the year.

By an amazing coincidence the following has happened in 1996, 1997 and 1998. One hour before A v C kicks off, team A’s manager/mathematician announces to the team that if they lose to C then they cannot possibly get a prize. Team A has gone on to win a prize in spite of losing to D.

1. Is it possible in 1996 A vs C was the 3rd game of the tournament?

2. In 1997, A vs C was the 4th games of the tournament. Name the two teams that you can say for certain met in the 2nd or 3rd game of the tournament.

3. In 1998, a total of 4 goals was scored in the tournament. What was the score in B vs C?

[enigma993]

]]>With the increasing standard of living, the members of my factory think more and more of the delights of abroad. And once they have tasted these delights there is no place to which they would rather go than the Island of Imperfection, where lies can be told proudly and as of right, as they were in the golden age.

There are three tribes on this Island — the Pukkas who always tell the truth, the Wotta Woppas who never tell the truth, and the Shilli Shallas who make statements which are alternately true and false (or false and true). This story deals with four members of our Factory: Alf, Bert, Charlie and Duggie, whose jobs — not necessarily respectively — were, when they were last working in this country, those of: Door-Opener, Door-Shutter, Welfare Officer and Bottle Washer. Each of the three tribes has at least one representative, but I am afraid I can give no more information about the tribe to which the fourth man belongs.

The make statements as follows:

Alf:

(1) The Welfare Officer is a Wotta Woppa;

(2) Only one of us is a Pukka.

Bert:

(1) Alf is a Pukka;

(2) Charlie is the Welfare Officer.

Charlie:

(1) I am a Wotta Woppa;

(2) Duggie is the Door-Shutter.

Duggie:

(1) The Door-Opener is a Shilli Shalla;

(2) Charlie is not a Wotta Woppa.Find the jobs which Alf, Bert, Charlie and Duggie had when they were last in this country, and the tribes to which they belong on the Island of Imperfection.

[puzzle14]

]]>In the following division sum each letter stands for a different digit. Rewrite the sum with the letters replaced by digits.

[enigma516]

]]>How might Caesar get you from 3 to 47?

A bit of general knowledge might help you here, or some numerology, because there are two neat solutions to this puzzle.

[puzzle#20]

]]>George is contemplating buying a farm which is a very strange shape, comprising a large triangular lake with a square field on each side. The area of the lake is exactly seven acres, and the area of each field is an exact whole number of acres.

Given that information, what is the smallest possible total area of the three fields?

[enigma994]

]]>When the five ministers at St. Saviour’s all got divorced, it was a relief. When all announced a remarriage, it was a surprise. When the brides were revealed to be the five ex-wives, it was a sensation. Still, the priggish Dinah was not the first to remarry and there were no direct swaps, so I daresay the decencies were preserved.

The weddings were held on successive Saturdays. Peter’s took place earlier than Anne’s and later than Quentin’s. Barbara’s was later than Tristram’s and earlier than Celia’s.

Peter married Simon’s ex-wife. Barbara got hitched to the man whose former wife married Emily’s ex-husband. Quentin paired up with the lady whose former husband married Dinah. Ronald was spliced with the lady whose ex-husband married Celia.

Who, pray, is now married to whom?

[tantalizer422]

]]>The Anglo-Slovak club had its meeting last week. Those present were Tom, Vyctur, Ted, Tago, Ray, Min, Wex, Olav, Russ and Cy.

Some of the members stood up and took part in an old Slovakian dance, rather like a Morris dance. The dancers stood around the floor with no three in a straight line and between each pair a taut piece of ribbon was stretched across the floor. Some ribbons were pink and the rest were blue. I noticed that there was a pink ribbon between two of them precisely when their Christian names had an odd number of letters in common. (So, for example, had a Jane, David and Victor been dancing, there would have been a pink ribbon from Jane to David, a blue from David to Victor, and a blue from Jane to Victor).

As soon as I saw how many dancers there were I realised that two of the ribbons would have to cross. But they had arranged themselves in such a way that there was no pink triangle and no blue triangle of ribbons.

Who was dancing?

[enigma515]

]]>A bishop visited his friend the vicar on her birthday. Knowing the bishop liked number puzzles, the vicar told him about a family that had just joined her church.

“If you multiply their three ages together, you get 2450, and if you add their ages together, you get your own age, your grace.”

The bishop, after some thought, said: “I can’t be certain how old everyone in the family is.”

The vicar responded: “I am older than everyone in that family.”

The bishop could then tell how old everyone was.

How old was the vicar on that day?

[puzzle#19]

]]>This is the layout of the digit buttons on my telephone. My boss’s telephone number uses each of the ten digits once and it starts with 0. Furthermore each pair of adjacent digits in her telephone number is also adjacent (horizontally, vertically or diagonally) on the telephone keypad. (By coincidence my own number, 0895632147, has the same properties).

I have just looked through my boss’s telephone number and written down a list of all the two-figure numbers that can be seen in it by reading a pair of adjacent digits (which would be 89, 95, 56… in my number). In that list, some of the numbers are special in that they consist of two digits which are also consecutive (such as 89, 56, 32 and 21 in my number). I have worked out the product of those special numbers and it is the year in which my boss will be 50. And if I look at the number of the year in which she was born, no two of its adjacent digits are adjacent (either way round) in her telephone number.

What is her telephone number?

**Enigma 486** is also called “Number please”.

[enigma995]

]]>We have recently started on the Island of Imperfection to build up a modern society. We have come to realise some of the inestimable advantages that a pay claim, with all its colour and friendly disagreements, can bring to life.

There are three tribes on the Island. The Pukkas, who always tell the truth, the Wotta Woppas, who never tell the truth, and the Shilli-Shallas who make statements which are alternately true and false, or false and true. Our story deals with one representative from each tribe.

Psychologists of the future may learn much about thinking on the Island at this time from the simple fact that the main units of currency were called

Hopes, and that a Hope was made up of 100Fears.The three (

A,BandC) speak about their tribes and wages:

A:

(1) My wages are greater thanB‘s;

(2) The weekly bill for wages (ofA,BandC) is 80 Hopes;

(3)Cis a Shilli-Shalla.

B:

(1)Ais a Pukka;

(2)A‘s wages are six Hopes less than mine;

(3) The weekly bill for wages (ofA,BandC) is 92 Hopes.

C:

(1) My wages are not a multiple of five Hopes;

(2)A‘s wages are nine Hopes greater than mine.(Their wages are all different and are greater than 20 Hopes and less than 35 Hopes per week; and in each case they are exact multiples of 50 Fears).

Find to which tribe each man belongs, and his weekly wage.

[puzzle15]

]]>I wrote to a mathematical friend in Utopia and asked him to send me the results of the recent general election there. He decided to make me work for it, as you can see from his reply:

Dear Friend,

(1) The Dextrous Party lost control of the Scitting (our 600-seat parliament) and now has fewer seats than the Sinistrals. The Other and Indeterminate Parties remained third and fourth respectively.

(2) No new party was elected to the Scitting and none was removed.

(3) No party has an overall majority in the new Scitting.

(4) The Other Party lost almost half its seats, while the Indeterminate Party exactly doubled its seats.

(5) In the last Scitting all four parties held a perfect square of seats (the Other’s figure was also a perfect cube). In the new Scitting, two have perfect squares while the other two have perfect fifths (a whole number raised to the fifth power). No party had or has only one seat.

So now you can determine the composition of both old and new Scittings.

Can you?

[enigma514]

]]>It is the year 2100, and the Moon Colonisation Programme is well-advanced.

A power cable is being laid all the way around the moon’s equator. The original plan was to put the cable on the moon’s surface, but it has been suggested that instead it should be buried in a trench that is 1-metre deep. This will make it safer and will also save on the amount of cable needed.

How much shorter will the cable be if it is buried in this way?

[puzzle#18]

]]>You may think you have seen this puzzle before but the solution this time is different. Just as before, each letter stands for a different digit, the same letter represents the same digit wherever it appears and no number starts with zero. But this time THREE is an even number.

What is your WEIGHT?

[enigma996]

]]>Aunt Edith sweetened her departure by giving each of our five children a bag containing 10 fat stripey humbugs. She made them all promise to hold off till after breakfast but Barbara is as persuasive as she is unprincipled and she and the others all arrived at the table chewing.

Taking stock over the porridge, I found that half the surviving humbugs belonged to children with an even number [remaining] and that the twins still had 14 between them. Only Anne had more than one less than average, but even she still had more than one humbug. Charles still had less than William but more than Pat. The girls still had exactly 10 between them.

How many did each child have left?

**Note:** Both this puzzle and the following puzzle were labelled **Tantalizer No 423** when published in *New Scientist*. So I’ve labelled this one **423a** to distinguish them.

[tantalizer423a] [tantalizer423]

]]>The fruit stall proclaimed, “Our fruit is so cheap it is even less than a bargain”, and so it had a good number of customers.

Hannah bought an apple and two bananas and yet spent less than Sarah who bought an orange and a 10-pence lemon. Joan bought 10 apples, 11 bananas and two oranges and yet did not spend all the 107 pence in her purse. Alan bought three apples, two bananas and an orange and his bill was less than 30 pence. Only Mot was unlucky: he tried to buy eight apples, seven bananas and two oranges, but they came to more than the 79 pence in his pocket.

Each piece of fruit cost a whole number of pence.

What was the cost of each apple, banana and orange?

[enigma513]

]]>