After the four teams in our local football league had each played each other once (3 points for a win, 1 for a draw), the league table was drawn up. It is presented below with the teams in decreasing order, but I’ve omitted some of the figures and replaced other digits consistently with letters, different letters being used for different digits.

Find the result of each match (for example, U beat H, C drew against D, and so on).

[enigma556]

]]>My daughter has made me a puzzle with eight square tiles that fit into a 2 × 4 grid. She tells me there is a connection between the symbols along the rows, and a simple rule that gets you from the symbols on the top row to the symbols below them.

She has put four of the tiles in place for me. Where do the other four go?

[puzzle#66]

]]>The houses in Acacia Avenue are numbered from 1 to 30. The residents of 12 of these have joined the local Neighbourhood Watch scheme, and are meeting in the local pub to discuss tactics. They decide to appoint a Street Committee of three from among themselves.

Someone suggested, for no convincing reason, that the committee should comprise three residents whose house numbers are in arithmetic progression — that is to say, the difference between the first two numbers is the same as the difference between the higher two.

This was agreed — until they discovered that no such group of three exists among the 12 numbers. This information uniquely determines the 12 numbers — what are they?

There are now 1392 *Enigma* puzzles available on the site, which leaves 400 puzzles remaining to post. Which means about 77.7% of all *Enigma* puzzles are now available.

[enigma953]

]]>With finals just round the corner, five friends in the Parapsychology Department were comparing notes.

“Mark and I will both get seconds”, predicted Matthew.

“Mark and I will both get firsts”, boasted Luke.

“Luke and I will both get seconds”, declared John.

“John will get a first and I will get a second”, asserted Peter.

Despite the fact that none of them was completely accurate in his contributions to this conversation, the examiners placed two in the first class and three in the second.

Which lucky lads got firsts?

[tantalizer396]

]]>Last year the famous treasure hunter, Archie Ologist, discovered an ancient tablet while excavating the pyramid of Pharaoh Unfair II. He managed to translate the tablet’s hieroglyphic inscription as follows:

“This year our great leader, Pharaoh Unfair II, has decided to introduce taxes in order to raise money to build pyramids. The tax system is very simple: after calculating your total income to the nearest talent, the Pharaoh’s tax inspectors will first take a glypth of the total and then a glipth of the remainder (all deductions are rounded up, of course).”

Unable to translate “glypth” and “glipth” from Egyptian, Ologist did some further research and discovered the following facts:

(a) Both words represent proper fractions in their lowest terms;

(b) Both fractions are formed out of four consecutive numbers (not counting zero) and have two-digit numerators and denominators;

(c) The sum of the numerator and denominator of either fraction is a perfect square;

(d) A glypth is the fraction which has the lowest numerator and complies with the above rules;

(e) A glipth is the fraction with the highest denominator and complies with the above rules.How much tax does Aleph the embalmer have to pay on his salary of 140,524 talents?

[enigma555]

]]>Jasmine has discovered that she has a special relationship with her dad. A few times, she has noticed that his age has been the reverse of hers: this year, she was 47 and her dad was 74. But this also happened when she was 36 and he was 63. It first happened when she was 14 and he was 41.

Jasmine thinks this might be unusual, so she checks out if it has happened to any of her 90 Facebook friends. How many of them do you think will discover this age reversal has happened with their own dad at some point?

[puzzle#65]

]]>In our local football league each team plays each of the others once in the course of a season. In each game the winning team gets three points or, in the event of a draw, each team gets one point. At the end of last season I looked at their league table and noted down details from four of the teams in decreasing order of points down that table.

Here is part of that information with digits consistently replaced by letters, different letters being used for different digits:

How many teams are there in this league?

[enigma954]

]]>The challenger for the world chess title succeeds only if he can score at least 12½ points from the title match. The match consists of 24 games but stops as soon as the reigning champion gets 12 or the challenger 12½ points. Scoring is 1 for a win and ½ for a draw and the reigning champion has the white pieces in the first game and every alternate game thereafter.

Speaking as a soothsayer of guaranteed accuracy, I can tell you that the next match will end with a score of 12-12. There will be nine wins scored with the black pieces and there will be exactly as many draws in total as the challenger wins games with the white pieces.

In that case what will happen in the final game?

[tantalizer397]

]]>I found myself on the stage if the Southwich Pavilion, helping the Great Marvello with his act. Marvello pointed to an endless row of boxes stretching off into the wings. The boxes had numbers on their lids, 1, 2, 3, … Marvello explained that each box contained a number of coins, at least one coin in each box.

As I stood in the centre of the stage, Marvello asked me to choose a number and I said 15. He told me to multiply my number by 6, and I gave him the answer, 90. He then told me to look in the box with 15 on its lid, count the coins in it and write the number of coins on the blackboard on the stage.

When I had done that, Marvello told me to open the box which had the number on the blackboard on its lid. I then had to count the number of coins in the box and subtract the number from the 90 I had obtained earlier. When I announced my answer to the audience, the great Marvello pointed with a flourish to the blackboard, for my answer was exactly the same as the number on the board. The audience applauded wildly.

After the show, Marvello explained that whatever number I had chosen he would have given precisely the same instructions, including multiplying by 6, and the trick would have worked out in the same way.

How many coins were in the box with the number 1990 on its lid?

[enigma554]

]]>The newly built reception room at the Ruritanian embassy is circular. In the middle, there is a large circular pillar. All very stylish, but now it needs to be carpeted. Of course, the carpet will need to be doughnut shaped, with a hole in the middle where the pillar is. Fernando the carpet fitter needs to know the area he has to cover. This would be easy if he knew the radius or diameter of both circles, but nobody seems to know where the plans are, and when he tries to measure the diameter of the room, the pillar gets in the way. Fernando doesn’t like measuring around corners, so the only thing he can think to do is to measure across the room with the tape measure touching the pillar. That distance is 10 metres.

How much carpet will he need?

[puzzle#64]

]]>You have to construct a puzzle by putting the numbers 2, 3, 5, 7, 11, 13 and 17 in some order into the boxes in the following framework:

I have a number

Xwhich is one of:2730, 4290, 5610, 6006, 6630, 10010, 13090, 14586, 15015, 15470, 24310, 34034, 36465, 39270, 46410, 255255, 510510.

Also

Xis divisible by , , , , and .Is

Xdivisible by ?Your puzzle should give a unique answer and be such that all the divisibility clues in it are needed to obtain the answer.

Which number should you put in the last box of the framework?

[enigma955]

]]>Smashing bazaar! The leading lights of our constituency party were all there. All four of them bought a pot of homemade queen’s foot jelly. The chairman and vice-chairman bought a teddy bear each too. The chairman and our MP each then bought one of Mrs Bratworthy’s fruit cakes. And finally the chairman, bless his heart, went back for an extra pot of jelly.

I suppose that makes it all sound a bit stuffy and deferential. Not a bit of it! I’m sure they won’t mind if I put it like this — Marmaduke spent 95p more than Clarence, whereas Algernon spent less than Jasper. A teddy costs twice as much as a pot of jelly and a cake can cost twice as much as a teddy.

Can you name the boys who bought the cakes?

[tantalizer398]

]]>One disadvantage of doing recurrence relations in your head, Kugelbaum told himself, was that all to often you were still calculating away when the clock showed it was time to give a class. To buy himself a little more time he jotted down on the whiteboard:

“In this relation”, explained Kugelbaum to the class, “A, B and X are all non-zero digits and AB/BA is a proper fraction in its lowest terms. The dot above the X in the numerator and that above the X in the denominator means that the digit X is used but it must be repeatable an arbitrary number of times without affecting the validity of the equation. So the X can be there once, twice, … 37 times … but, no matter how many times you use it, the equation must remain true. It is, or course, to be taken for granted that the number of Xs interposed between the A and B must be the same for both the numerator and the denominator”.

So saying, the professor returned to his mental calculations while the class set about solving the problem.

How many distinct solutions are there of Kugelbaum’s equation?

NB. A proper fraction has its numerator smaller than its denominator; if it’s in its lowest terms, numerator and denominator have no common factor other than 1.

[enigma553]

]]>“It’s Monday today, isn’t it?” pondered Angie.

“I seem to remember yesterday was Tuesday”, said Beatrice.

“I believe we’re as far from Sunday as we are from Wednesday”, said Carol.

“When the day after tomorrow becomes ‘yesterday’, ‘tomorrow’ will be Tuesday”, interjected Dorothy, cryptically.

“It’s not the weekend”, said Ethel.

Clearly there was some disagreement – and if I told you how many of them had got it right, then you would be able to work out what day it is today. However, I’ve decided to keep it a secret. So who was right, and what day is it?

[puzzle#63]

]]>Adolf, Bernhard and Colin played a round of golf consisting of 18 holes. Each finished on par for the round, but in the course of the round each scored at least one eagle (2 under par), birdie (1 under par), par, bogey (1 over par) and double bogey (2 over par), but nothing either better or worse.

Each of the three had a different number of eagles, birdies, pars, bogeys and double bogeys in his round.

All three had the same number of birdies in their rounds, but no two of them had the same number of eagles, or the same number of pars, or the same number of bogeys, or the same number of double bogeys. Adolf had the same number of eagles as Bernhard had of pars.

How many eagle, birdies, pars, bogeys and double bogeys (in that order) did Colin have?

[enigma956]

]]>Uncle Ebeneezer, feeling suddenly mortal, summoned his four nephews and produced a brassbound box. “500 sovereigns!” he gloated. “Each of you is to have more than 100. Matthew, Mark and Luke will all get a different number and John will get the same as Luke. There will be none left over.”

He opened the box and took out four labelled leather bags, which he gravely distributed. Each nephew counted his own share and kept the number to himself. “Have you got a perfect square?” Luke then asked Mark. Mark answered but not, I regret to report, truthfully. Matthew deduced this but did not say so. Luke, however, taking Mark’s reply as gospel, presently thought to himself, “So Mark has the largest share”. Matthew, by the way, did not have a perfect square.

How had Uncle Ebeneezer split the boodle?

[tantalizer399]

]]>How many proper rectangles (which are not squares) can be seen in this grid?

In fact there are 22. And how many actual different-sized rectangles are there? Just three: the 1 × 2 occurs 12 times (six horizontally and six vertically); the 1 × 3 occurs six times; and the 2 × 3 occurs four times. So each different-shaped piece occurs on average 7 ⅓ times and the 1 × 3 piece’s number of occurrences is closest to that average.

For a larger square grid which I have in mind there is one size of rectangle which occurs exactly the average number of times, that average being a two-figure number.

How big is the grid, and what is the size of this “average piece”?

[enigma552]

]]>Three friends agree to drive from A to B via the shortest road route possible (driving down or right at all times). They are hungry, so also want to drive through a Big Burger restaurant, marked in red. They are arguing about how many shortest routes will pass through exactly one Big Burger.

Xenia:“I reckon there are 10.”

Yolanda:“I’d say more like 20.”

Zara:“No you’re both wrong, I bet there’s more than 50.”Who is right, or closest to right?

[puzzle#62]

]]>Ten executives each rented two of the 20 safe-deposit boxes shown:

Each of the 10 had the same number of gold coins to share out between her two boxes. The first executive put 1 coin in one of her boxes and the rest in her other box. The second executive put 2 coins in one of her boxes and the rest in her other box. The third executive put 3 coins in one of her boxes and the rest in her other box. And so on, up to the tenth executive, who put 10 coins in one of her boxes and the rest in her other box. The overall effect of this was that each of the four rows of boxes contained in total an equal number of the gold coins. Also each of the five columns of boxes contained in total an equal number of coins.

One night a burglar broke into and emptied four of the boxes, three of them being in the same row. His total haul of coins was a three-figure number with no zeros.

How many coins did each executive have in the first place?

[enigma957]

]]>A Trappist fête has its drawbacks and its triumphs too. You will see what I mean from an incident at one I attended last week. There was a sea-food stall, selling small items on sticks, where you could buy (in ascending order of price) a whelk, a mussel, a fishcake, a slice of eel and an oyster. While I watched, a party of five monks approached and gave a sign meaning that each of them wanted a different item. (I hasten to add that this was the only untoward sign they made throughout).

Each then placed the same amount of money of the counter. The stall-holder reflected a moment, and then handed each monk the exact item he had wanted. A total of 50 pence then passed across the counter, followed by the return of a total of 16 pence in change. No “tanners” (2½p pieces) were involved.

Assuming that everyone used his loaf and not any extraneous knowledge, what was the price of a fishcake?

[tantalizer400]

]]>