“See these eleven blocks?”, says a so-called friend. “Four of them of 8 inch thickness, two of them of 4 inch thickness, three of 3 inch and two of 1 inch thickness”.

“Pile them in a column 51 inches high with a 3 inch block at the bottom so that, remaining in position, individual blocks or combinations of adjacent blocks can be used to measure every thickness in exact inches from 1 inch to 48 inches”.

In what order do they stand?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser252]

]]>All the phone extensions at Mick’s work place are four-digit numbers with no zeros or twos, and no digit appears more than once in a number. He can enter all the extension numbers on the keypad by starting at key 2 (but not pressing it) then moving in any direction, including diagonally, to an adjacent key and pressing it; then similarly moving to and pressing adjacent keys until the number is entered. A couple of examples of his extension numbers are 3685 and 5148.

He phoned two different extension numbers this morning and later realised that one was an exact multiple of the other.

What was the larger extension number he dialled?

[teaser3141]

]]>I am organising a tombola for the fete. From a large sheet of card (identical on both sides) I have cut out a lot of triangles of equal area. All of their angles are whole numbers of degrees and no angle exceeds ninety degrees. I have included all possible triangles with those properties and no two of them are identical. At the tombola entrants will pick a triangle at random and they will win if their triangle has a right-angle. The chances of winning turn out to be one in a certain whole number.

What is that whole number?

[teaser2689]

]]>Five horses took part in a race, but they all failed to finish, one falling at each of the first five fences. Dave (riding Egg Nog) lasted longer than Bill whose horse fell at the second fence; Big Gun fell at the third, and the jockey wearing mauve lasted the longest. Long Gone lasted longer than the horse ridden by the jockey in yellow, Chris’s horse fell one fence later than the horse ridden by the jockey in green, but Fred and his friend (the jockey in blue) did not fall at adjacent fences. Nig Nag was ridden by Wally and Dragon’s jockey wore red.

Who was the jockey in yellow, and which horse did he ride?

[teaser2685]

]]>British Triangles, naturally, stand on horizontal bases with their points upwards. All their sides, never more than a sensible 60 inches, and their heights measure an exact number of inches. No B.T. is isosceles or right angled.

You can often put more than one B.T. on a common base. On a base of 28 inches 8 B.T.s are erected.

What are their heights?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser222]

]]>On rugby international morning, I found myself, along with eight friends, in a pub 5.8 miles from the match ground. We were enjoying ourselves, and so wished to delay our departure for the ground until the last possible minute. The publican, wishing to keep our custom for as long as possible, offered to help us get there by carrying us, one at a time, as pillion passengers on his motorbike.

We could walk at 2.5mph and the bike would travel at 30mph. We all left the pub together, and arrived at the ground in time for kick-off.

Ignoring the time taken getting on and off the bike, what was our minimum travelling time in minutes?

[teaser3140]

]]>There were six starters for the special handicap event for the youngest class at the school sports. All started at the gun, and none fell by the wayside or was disqualified.

Dave started quicker than Ed and finished before Fred. Colin finished two seconds after he would have finished if he had finished two seconds before Ed. Bob and Dave started quicker than Colin, and Bob was faster than Ed.

Alf, who won third prize, finished two seconds after he would have finished if he had finished two seconds after Colin.

Who won second prize?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser665]

]]>Today we are having a family get-together to celebrate Mother’s Day. My maternal grandmother, my mother and I have each written down our date of birth in the form “ddmmyy”. This gives us three six-figure numbers and, surprisingly, both of the ladies’ numbers are multiples of mine. Furthermore, all of the digits from 0 to 9 occur somewhere in the three six-figure numbers.

What is my mother’s six-figure date of birth?

[teaser2688]

]]>Uncle George occasionally wasted time on Brain-teasers, and it was expected that his will would take an awkward form. But his five nephews were surprised to learn, after his death, that he had dictated his will to a solicitor who had no use for punctuation. The will ran as follows:

maurice is not to sing at my funeral service if stephen receives the envelope containing three thousand pounds alec is to have five thousand pounds if thomas receives less than maurice alec is to have exactly twice what nigel has if thomas does not receive the envelope containing a thousand pounds stephen is to have exactly four thousand pounds

The nephews were confident that Uncle George always uttered complete sentences, none of which contained more than one conditional clause. They also felt sure that what he had said to the solicitor allowed one way, and one way only, of distributing the five envelopes (containing £5000, £4000, £3000, £2000, and £1000), which Uncle George had left for them.

Who receives each envelope? And may Maurice sing at the funeral service?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser204]

]]>Our chess club is divided into sections, with each section having the same number of players. The two oldest members will soon be retiring from playing and we will change the number of sections. The number of sections will change by one and so will the number of players in each section, but all the sections will still have the same number of players. This will result in there being a grand total of 63 fewer matches per year if each member plays all the other members of their section once per year.

How many players are in each section at the moment?

[teaser3139]

]]>In a new design of mobile phone each of the number buttons 1 to 9 is associated with 2 or 3 letters of the alphabet, but not in alphabetical order (and there are no letters on any other buttons). For example:

M,TandUare on the same button. Predictive software chooses letters for you as you type. The numbers to type forSUNDAY,TIMESandTEASERare all multiples of 495.What number should I type to make

SATURDAY?

[teaser2680]

]]>When my neighbour Giles found he could no longer look after his prize herds of cattle and sheep, he planned to divide the lot among his four sons in equal shares.

But when he started by counting up the cattle he soon found that their total was not exactly divisible by four, so he decided he would juggle with the numbers of beasts and achieve the four shares of equal value by making to the respective recipients quite arbitrary allocations of both cattle and sheep, taking into account that the value per head of the former was four times that of the latter.

The outcome was that:

(i) one son received 80% more beasts than another;

(ii) two sons each received a total of beasts which equalled the aggregate total of two of the other three sons;

(iii) one son received twice as many of one type of beast as of the other;

(iv) only one son received over 100 beasts in all.How many cattle were included in this last total of over 100 beasts?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser670]

]]>Just nine Prime Ministers held office in the Kingdom of Primea during the 20th century. No two Prime Ministers held office at the same time, none had more than one period in office, and the gap between successive Prime Ministers’ terms was never more than a month. Each held office for a period of time in which the number of whole years was a different prime number (e.g. holding office from 1910 to 1915 could cover four or five whole years) and no Prime Minister served for more than 30 years. Appropriately, they all took office in prime years, but there was no change of Prime Minister in 1973.

In which years during the 20th century did new Prime Ministers in Primea take up office?

[teaser3138]

]]>George and Martha are teaching their great-grandchildren some simple arithmetic. “If you add two thirties to four tens you get a hundred”, George was saying, and he wrote it like this:

“Strangely”, added Martha, there are nine different letters there, and if you allow each letter to stand for a different digit, there is a unique sum which works”.

Which digit would be missing?

[teaser2611]

]]>Gold sovereigns were minted in London for most years from the great recoinage in 1817 until Britain left the gold standard in 1917. I have a collection of eight sovereigns from different years during that period, the most recent being an Edward VII sovereign (he reigned from 1902 until 1910). I noticed that the year on one of the coins is a perfect square and this set me thinking about other powers. Surprisingly, it turns out that the product of the eight years on my coins is a perfect cube.

What (in increasing order) are those eight years?

[teaser2695]

]]>The adjoining countries of Europhalia and Sopiculia have different standard units of weight and length, but both use the normal units of time. Although both countries use Arabic numerals, neither uses the denary (tens) method of counting, but each has a different integer less than ten as its counting base.

In reply to my request for more information a Europhalian friend wrote: “Our unit of weight is the Elbo, and there are 42 Elbos to 24 of their Solbos. The length of our common frontier is 21 Emils”. My Sopiculian correspondent replied: “16 Solbos weigh the same as 26 Elbos; the common frontier is 21 Somils long”.

I later discovered that in both countries there is a speed limit equivalent to our 50 mph. In Sopiculia this is defined by law as 104 Somils per hour.

What is the Europhalian speed limit?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser195]

]]>Eight friends met at a party; their ages in whole numbers of years were all different. They were Alan, Cary, James, Lucy, Nick, Ricky, Steve and Victor, with Lucy being the youngest. For each of them the square of their age was a three-figure number consisting of three different digits. Furthermore, for any two of them, the squares of their ages had at least one digit in common precisely when their names had at least one letter in common.

In alphabetical order of their names, what are the eight ages?

[teaser3137]

]]>In Church last Sunday I studied the three numbers on the “service-board”. The first was of the appointed psalm, and the other two of the hymns. They included all the digits from 1 to 9 inclusive and were all prime numbers. (Our service-book contains 666 hymns and the normal 150 psalms).

What were last Sunday’s numbers?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill).

[teaser653]

]]>My two pals and I have been considering “palprimes” (i.e. palindromic numbers that are also prime). In particular each of us tried to find a five-figure palprime and I managed to come up with 39293. Then each of my two pals found a five-figure palprime. On comparing them we were surprised to find that overall our three palprimes used all the digits from 1 to 9.

What were the other two five-figure palprimes?

[teaser2679]

]]>Old Andrew, our pensioner neighbour, was telling me about his equally well preserved brother, David.

“We weren’t twins, but we were both born on a Tuesday and born on the same date in different months. In fact, David was born on the first Tuesday after my birth which occurred on the same monthly date. I am hardier, of course, born in the winter time”.

“So, with these conditions, the interval between you couldn’t have been long”, I said.

“Well”, replied Andrew, “it couldn’t have been any longer”.

When was David born (day, month, year)?

[teaser797]

]]>Moriarty’s papers were alight. Holmes memorised a 3×3 grid of different 4-digit values in ascending order as illustrated, from

A(lowest) toI(highest), noticing that one numeral wasn’t used. Three cells, isolated from one another (no corner or edge contact), held squares of squares. Three others, similarly isolated, held cubes of non-squares. The other three held squares of non-squares. Holmes told Watson these features, specifying only the lowest value. “Not square”, remarked Watson.“True, and many grids match these facts. However, if I told you the positions of the squares of squares in the grid, you could deduce a majority of the other eight values (apart from the lowest one)”, replied Holmes.

In ascending order, which values did Watson know certainly?

[teaser3136]

]]>In the two sums displayed, digits have consistently

been replaced by letters, with different letters for

different digits:

GIRL+BOY=LOVE

GIRL−BOY=???Unfortunately, I have missed out the second answer,

but I can tell you that it is a three-letter word.Find my

LOVER‘s number.

[teaser2295]

]]>John bought some packs of pond plants consisting of oxygenating plants in packs of eight, floating plants in packs of four and lilies in packs of two, with each pack having the same price. He ended up with the same number of plants of each type. Then he sold some of these packs for twenty-five per cent more than they cost him. He was left with just fifty plants (with fewer lilies than any other) and he had recouped his outlay exactly.

How many of these fifty plants were lilies?

[teaser2698]

]]>I am the Managing Director of a factory and I have under me five employees. Their names are: Alf, Bert, Charlie, Duggie and Ernie. And their jobs are, not necessarily respectively: Doorkeeper, Doorknob Polisher, Bottle Washer, Welfare Officer and Worker.

There has been some dissatisfaction recently about wages which, in the past, I am bound to admit, have sometimes been rather haphazard. It is clearly very difficult to arrange things in such a way that merit is appropriately rewarded, but it seemed to me important that everybody’s position should at least be clear. After much thought, therefore, I put up the following notice:

Wages:1. Alf is to get more than Duggie.

2. Ernie is to get 12 per cent more than the Bottle Washer will when he receives the 10 percent rise that he will be getting next month.

3. The Doorknob Polisher is to get 30 per cent more than he used to.

4. Charlie is to get £12 a year less than 20 per cent more than the Welfare Officer.

5. No one is to get less than £200 or more than £600 a year.

6. The Doorkeeper is to get 5 per cent more than he would if he got 10 per cent less than Bert.

Everyone always has received in my factory, receives now, and as long as I am in charge will always receive an exact number of £s per year.

What are the various jobs of my employees, and what yearly wage is each of them to get?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill). The puzzle text above is taken from the book.

[teaser177]

]]>Skaredahora used three rhythmic patterns of quaver beats in a short, purely percussive, composition. His “Dotykam” rhythm has accents on beats 1, 4, 6, 7, 8, 10 & 11; “Kluc” has accents on beats 1, 8 and one particular beat in between; and “Omacka” has accents on beats 1, 2, 5, 6 & 10. Several percussion instruments are involved, each playing one of the three rhythms, but starting at different times. Overall the patterns overlap, with every beat in the composition being filled by an accent from exactly one of the instruments, and all the patterns finishing by the end of the composition.

What is the other beat of Kluc, and what is the order of appearance of the rhythmic patterns (e.g. DOOKD)?

[teaser3135]

]]>George and Martha have a five-figure code for their burglar alarm. George commented that the three-figure number formed by the first three digits of the code equalled the sum of the cubes of those first three digits.

Martha added that the three-figure number formed by the last three digits of the code equalled the sum of the factorials of those three digits. (She had to remind George what a factorial was — he remembered once she had pointed out that the factorial of 4 was 4×3×2×1 = 24).

What is their code?

This puzzle was *not* included in the book **The Sunday Times Brain Teasers Book 1** (2019).

This completes the archive of **Teaser** puzzles originally published in 2015. There is a complete archive of puzzles from 2015 – present available on **S2T2**, as wall as many older puzzles going back to 1949.

[teaser2746]

]]>Fabulé’s latest jewellery creation consists of a set of identically sized regular octahedra made of solid silver. On each octahedron, Fabulé has gold-plated four of its eight equilateral triangle faces. No two octahedra are the same, but if Fabulé had to make another, then it would be necessary to repeat one of the existing designs.

How many octahedra are there in the set?

[teaser2538]

]]>An artist hammered thin nails from a pack of 40 into a board to form the perimeter of a rectangle with a 1 cm gap between adjacent nails. He created a work of art by stringing a long piece of wire from one nail to another, such that no section of wire was parallel to an edge of the rectangle. The wire started and ended at two different nails, no nail was used more than once and the length of the wire was a whole number of cm. No longer wire was possible that satisfied the above conditions.

What were the dimensions of the rectangle and the length of the wire chain (in cm)?

This is a revised version of the puzzle posted as **Teaser 3134**. The underlined text has been changed from the previously published version on *The Sunday Times* website.

This is the version of the puzzle that appeared in the printed newspaper.

[teaser3134]

]]>An artist hammered thin nails from a pack of 40 into a board to form the perimeter of a rectangle with a 1 cm gap between adjacent nails. He created a work of art by stringing a long piece of wire from one nail to another, such that consecutive nails were on different edges of the rectangle. The wire started and ended at two different nails, no nail was used more than once and the length of the wire was a whole number of cm. No longer wire was possible that satisfied the above conditions.

What were the dimensions of the rectangle and the length of the wire chain (in cm)?

The wording of this puzzle was later revised. The underlined section was changed to a more restrictive condition.

See: **Teaser 3134: Stringing along [revised]** for the revised puzzle.

[teaser3134]

]]>George and Martha’s bank PIN is an odd four-figure number. George did some calculations and wrote down three numbers: the first was the sum of the four digits in the PIN; the second was the average of the four digits; and the third was the product of the four digits.

Then Martha multiplied these three numbers together and was surprised that the answer equalled the original PIN.

What is their PIN?

This puzzle was *not* included in the book **The Sunday Times Brain Teasers Book 1** (2019).

[teaser2737]

]]>My diary has this design on the cover:

In this 2-by-3 grid there are 12 junctions (including the corners), some pairs of which are joined by a straight line in the grid. In fact there are 30 such pairs.

The diary publisher has been using such grids of various sizes for years and the 1998 diary was special because its grid had precisely 1998 pairs of junctions joined by lines. Within the next 10 years they will once again be able to produce a special diary where the number of joined pairs equals the year.

(a) What was the grid size on the 1998 diary?

(b) What is the year of this next special diary?

[teaser2676]

]]>Cardiff and London share a line of latitude; Cardiff and Edinburgh share a line of longitude.

The Archers, the Brewers, the Carters and the Drews are four married couples born and married in London, Cardiff, Edinburgh and Belfast. One of each sex was born in each city; one marriage took place in each city. No one was married in the city of his or her birth. Mrs Archer was the only woman married east of where she was born; Mr Archer was the only man married south of where he was born; Mr Brewer was the only man to marry a woman born north of him. Mr Carter and Mrs Drew were twins.

Where were the Carters married?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser190]

]]>I was recently studying a large map that showed all the towns and major roads in a country. Every town had at least one road leading to it and every road led from one town to another. The roads only met at towns and all joined together to make a network with lots of blank areas in between, which I happily coloured in, needing just four different colours.

I counted up the number of areas (excluding the area around the outside of the network) and the number of towns, and discovered that both numbers were prime. Also, when I took these two numbers with the number of roads, the three numbers together used every digit from 0 to 9 precisely once.

In increasing order, what were the three numbers?

[teaser3133]

]]>At Granny’s birthday this year she was telling us some surprising things about some past birthdays. She told us that each year she writes down the date of her birthday (in the eight-digit form

dd/mm/yyyy) and her age in years.On two occasions in her lifetime it has turned out that this has involved writing each of the digits 0 to 9 exactly once. The first of these occasions was in 1974.

What is Granny’s date of birth (in the eight-digit form)?

Note that in order to solve this puzzle it is important to be aware of the date it was originally set.

All 200 puzzles included in the book **The Sunday Times Brain Teasers Book 1** (2019) are now available on **S2T2**.

[teaser2750]

]]>I removed an even number of red cards from a standard pack and I then divided the remaining cards into two piles. I drew a card at random from the first pile and it was black (there was a whole-numbered percentage chance of this happening). I then placed that black card in the second pile, shuffled them, and chose a card at random from that pile. It was red (and the percentage chance of this happening was exactly the same as that earlier percentage).

How many red cards had I removed from the pack?

[teaser2751]

]]>I wrote to an American friend on the 3rd February 1964, and told him of the coincidence of our family birthdays. My wife and I, our three sons, and our four grandsons all have our birthdays on the same day of the week every year, though all our birthdays are different. When I wrote, I used the usual English form of the date — 3/2/64 — and I gave all our birthdays in that form also.

My third son received a cable from my friend on his birthday in 1964, but later I was surprised to get a cable from him myself on my eldest son’s birthday. Next my eldest grandson received a cable on his right birthday, and I realised that we were all going to receive cables, but that my friend was, quite reasonably, reading the dates in the American form, i.e. he assumed that the letter had been sent on 2nd March 1964.

However, I did not write to put him right, and my wife was the next person to get a cable; this was not on her birthday.

What was the day and date of her birthday in 1964?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser188]

]]>My son, at a loose end after A-levels, asked me for a mental challenge. As we’d been discussing palindromes, I suggested he try to find an arrangement of the digits 1 to 9 with the multiplication symbol “×” to give a palindrome as the answer, and he came up with:

29678 × 1453 = 43122134.

I said: “Now try to find the smallest such palindromic product starting in 4, where the last digit of the smallest number is still 3”. He found that smallest product, and, interestingly, if he added the two smaller numbers instead of multiplying them, then added 100, he also got a palindrome.

What is the smallest product?

[teaser3132]

]]>At our local fete one of the games consisted of guessing the number of marbles in a jar: some of the marbles were red and the rest were blue. People had to guess how many there were of each colour.

The organiser gave me a couple of clues. Firstly, he told me that there were nearly four hundred marbles altogether. Secondly, he told me that if, when blindfolded, I removed four marbles from the jar, then the chance that they would all be red was exactly one in a four-figure number.

How many red marbles were there, and how many blue?

[teaser2747]

]]>My school holds “Round the river” runs — a whole number of metres to a bridge on the river and then the same number of metres back. Some years ago I took part with my friends Roy, Al, David and Cy. We each did the outward half at our own steady speeds (each being a whole number of centimetres per minute). For the return half I continued at my steady speed, Roy increased his speed by 10%, Al increased his speed by 20%, David increased his by 30%, and Cy increased his by 40%. We all finished together in a whole number of minutes, a little less than half an hour.

What (in metres) is the total length of the race?

[teaser2749]

]]>“Now”, says Bell at the pub, “look intelligent and imaginative even if you don’t look beautiful by any means”. We swallow the insult and look solemn. Bell likes his jokes and we like his puzzles.

“Imagine you have some scales, but only three weights. However, you have a heap of Grade A sand, and a couple of bags; so you make two extra weights, one as heavy as all the first three put together, and the other twice as heavy as all the first three. Whereupon all the remaining sand is removed to a great distance”.

“With these five weights you must be able to weigh 1 ounce, 2 ounces, 3, 4, and so on, as far as possible. No gaps in that lot. It’s how far you can to the first gap I’m after. Usual prize — one pint for the best score before closing time”.

What should Bell’s five weights be to give the highest possible score?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser418]

]]>Little Spencer saves 5p coins in a jar, and when they reach £10, deposits them in his savings account. He likes playing with the coins. In one game, after first turning them all heads up, he places them in a row on the table. Starting from the left, he then turns over every 2nd coin until he has reached the end of the row. He then again starts from the left, and this time turns over every 3rd coin. He repeats this for every 4th, 5th coin etc, until finally he turned over just one coin, the last in the row.

At the end of the game I could see that if Spencer had exactly 75 per cent more coins he would have an increase of 40 per cent in the number showing heads. However, if he had exactly 50 per cent fewer coins, he would have a decrease of 40 per cent in the number showing heads.

What is the value of his 5p savings?

There are now 750 **Teaser** puzzles available on the **S2T2** site.

[teaser3131]

]]>Peter became bored during the Sunday service, so his mind turned to the three three-figure hymn numbers displayed on the board, all chosen from the five hundred hymns in the hymnal. He noticed that the sum of the digits for each hymn was the same, that one hymn number was the average of the other two, and that no digit appeared more than once on the board.

What (in increasing order) were the three hymn numbers?

[teaser2742]

]]>Each of the three houses of Merryhouse School entered four students in the cross-country race. Points were awarded with 12 for the winner, 11 for second, and so on down to 1 for the tail-ender (from Berry House). When the points were added up, all houses had equal points. Three of the runners from Cherry House were in consecutive positions, as were just the two middle-performers from Derry House.

Which house did the winner come from, and what were the individual scores of its runners?

[teaser2744]

]]>“Puzzle here”, says Bell at the pub. “Chap has a ribbon shop, sells the stuff by the inch, no commercial sense. He’s barmy anyway; look how he measures it. His counter is exactly 100 inches long and he’s marked it off into 16 bits; 6 of 11 inches, 2 of 6 inches, 3 of 5 inches, 1 of 3 inches and 4 of 1 inch, and he can measure any number of inches up to a hundred, that is, by picking the right pair of marks.

“You have to sort the spaces out; but I’ll tell you, all the 11 inches are together round about the middle — well, a bit to the right, but not as much as 4 inches off centre. You get the idea? For most measurements he’s using a kind of feet and inches with eleven inches to the foot”.

“Young Green is nearly right: he can’t measure 99 inches unless there’s a 1-inch space at one end, but he doesn’t need a 1-inch at the other end for 98 inches; he does it with two 1-inch spaces at the same end; but there might be a 1-inch at the other end, all the same, and there might not”.

“In answer to two foolish questions, the ribbon must be measured single thickness, no folding; and it’s a straight counter, it’s not circular”.

“Usual prize, one pint”.

How were the spaces arranged from left to right?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser560]

]]>Liam has nine identical dice. Each die has the usual numbers of spots from 1 to 6 on the faces, with the numbers of spots on opposite faces adding to 7. He sits at a table and places the dice in a 3×3 square block arrangement.

As I walk round the table I see that (converting numbers of spots to digits) each vertical face forms a different three-figure square number without a repeating digit.

As Liam looks down he sees six three-digit numbers (reading left to right and top to bottom) formed by the top face of the block, three of which are squares. The total of the six numbers is less than 2000.

What is that total?

[teaser3130]

]]>I told Sue a two-figure number and I told Terry another two-figure number, one of which was a multiple of the other. I explained this to them but knew that neither of them would be able to work out the other number. (In fact, if they had to guess the other number Sue had three times as many choices as Terry — but I did not tell them that).

When Sue confirmed that it was impossible for her to work out Terry’s number, he was then able to work out her number.

What were their numbers?

[teaser2735]

]]>I have a square garden with sides a whole number of metres in length. It is surrounded by a fence with posts at the corners and then at one metre intervals. I wish to make the garden into four triangular beds surrounding a lawn that has four sides of different lengths. To mark out the lawn I choose one post on each of the sides of the garden and I stretch a length of string around those four posts. I can create my lawn in various ways but the length of string needed is always one of two possible values. I have chosen one arrangement using the smaller of the two lengths.

What is the area of my lawn?

[teaser2728]

]]>In the time of the Great Caliph a large annual tax was one day levied on shopkeepers for each weight used in their shops. The ingenious Al Gebra contrived to use very few weights but he often had weights in both his scale pans. The exchequer hit back by making it compulsory to make every weighing by using two weights only, one in each pan. Al now contrived with 20 weights to weigh up to 118 lb. in 1 lb. steps. Using 1 lb. as the least weight he found various ways of doing this. “But”, he said, “I’m getting old and I’m going to have the lightest possible set”.

Which set was this?

[teaser119]

]]>At the local arcade, Claire and David played an air hockey game, using a square table with small pockets at each corner, on which a very small puck can travel 1m left-right and 1m up-down between the perimeter walls. Projecting the puck from a corner, players earn a token for each bounce off a wall, until the puck drops into a pocket.

In their game, one puck travelled 1m further overall than its left-right distance (and the other, travelled 2m further). Claire’s three-digit number of tokens was a cube, larger than David’s number which was triangular (1 + 2 + 3 + …). Picking up an extra token, they found they could split their collection into two piles, one consisting of a cube number of tokens and the other a square.

How many tokens did they end up with?

I’ve modified the wording slightly to remove a typo and improve clarity.

[teaser3129]

]]>In this long multiplication sum, I am multiplying a three-figure number by itself. Throughout the workings one particular digit has been replaced by

Xwherever it occurs: all other digits have been replaced by a question mark.What is the three-figure number being squared?

[teaser2740]

]]>Peter likes to note “pandigital” times, such as 15.46, 29/03/78, that use all ten digits. Here the five individual numbers (15, 46, 29, 3 and 78) have a product that is divisible by the perfect square 36, and they also have a sum that is two more than a perfect square.

He has been watching for pandigital times ever since and remembers one where the product of the five numbers was not divisible by any perfect square (apart from 1): this has never happened since! He is also looking out for a pandigital time where the sum of the five numbers is a perfect square:

(a) When was that last “non-square” pandigital time?

(b) When will that first “square-sum” pandigital time be?

[teaser2752]

]]>“On our last expedition to the interior”, said the famous explorer, “we came across a tribe who had an interesting kind of Harvest Festival, in which every male member of the tribe had to contribute a levy of grain into the communal tribal store. Their unit of weight was roughly the same as our pound avoirdupois, and each tribesman had to contribute one pound of grain for every year of his age”.

“The contributions were weighed on the tribe’s ceremonial scales, using a set of seven ceremonial weights. Each of these weighed an integral number of pounds, and it was an essential part of the ritual that not more than three of them should be used for each weighing, though they need not all be in the same pan”.

“We were told that if ever a tribesman lived to such an age that his contribution could no longer be weighed by using three weights only, the levy of grain would terminate for ever. And in the previous year, one old man had died only a few months short of attaining this critical age, greatly to the relief of the headman of the tribe”.

“The scientist with our expedition confirmed that the weights had been selected so that the critical age was the maximum possible”.

What was the age of the old man when he died?

And what were the weights of the seven ceremonial weights?

[teaser118]

]]>I have a large number (but fewer than 2000) of identical spherical bonbons, arranged exactly as a tetrahedral tower, having the same number of bonbons along each of the six edges of the tower, with each bonbon above the triangular base resting on just three bonbons in the tier immediately below.

I apportion all my bonbons between all my grandchildren, who have different ages in years, not less than 5, so that each grandchild can exactly arrange his or her share as a smaller tetrahedral tower, having the same number of tiers as his or her age in years.

The number of my grandchildren is the largest possible in these circumstances.

How many tiers were in my original tower? And how old in years are the eldest and youngest of my grandchildren?

[teaser3128]

]]>Mark has recently converted from vegetarianism. John sent him a coded message consisting of a list of prime numbers. Mark found that by systematically replacing each digit by a letter the list became the message:

EAT BEEF AT TIMES

IT IS A PRIME MEATWhat number became

PRIME?

[teaser2732]

]]>I have arranged the numbers from 1 to 27 in a 3-by-3-by-3 cubical array. I have noticed that the nine numbers making up one of the faces of the cube are all primes.

Also, I have searched through the array and written down the sum of any three numbers that are in a straight line. I have then calculated the grand total of all those line-sums. It turns out that the grand total is itself a perfect cube!

What is that grand total?

[teaser2743]

]]>Seated happily one Sunday morning in my local pub, solving the Brain-teaser, I was accosted by the slightly inebriated resident bore. He said: “I have a puzzle for you which concerns seven people who were either engineers or salesmen and, of course, salesmen always tell the truth and engineers always lie. I will make four statements from which you must deduce how many salesmen there are:

1. B and E are salesmen;

2. A and C have different occupations;

3. D says that G is an engineer;

4. A says that B declares that C insists that D asserts that E affirms that F states that G is a salesman”.I said: “That’s a very old one”, and told him the answer. Grinning, he then replied, “Yes, that would be the correct answer if all the four statements were true, but it is not the correct answer because I intentionally made one of the four statements false. Further, if were to tell you how many salesmen there were under these new circumstances, you would be able to tell me which statement was false”.

“Ah”, I said, “that’s a much more interesting problem”. I thought for a moment and indeed I was able to tell him not only which statement was false but also how many salesmen there really were.

Which statement was false and how many salesmen were there really?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser906]

]]>Next month’s four week rota for Monday to Friday dinner duties starting on Monday 1st is covered by five teachers each having the following duty allocations. Ann, Ben and Ed each have four, Celia six and Dave has two. Strangely, all the prime number dates (1 is not a prime) are covered by Ben, Celia and Dave, while the others are covered by Ann, Celia and Ed. After working a duty, nobody works on either of the following two shifts, so anyone working on a Friday will not work on the following Monday or Tuesday. Celia has no duties on Mondays while Ben and Ed have none on Wednesdays.

In date order, who is on duty from Monday to Friday of the first week?

[teaser3127]

]]>Jorkens, the wily old cricketer, is faced with a new type of square cut. His house has three square bedrooms, all of different sizes. He has just bought a new carpet for the largest bedroom and has cut up its old carpet into four rectangular pieces, the smallest of which has an area of four square metres. He is able to use the four pieces to carpet the other two bedrooms exactly.

What is the area of the largest bedroom?

[teaser2745]

]]>St Patrick’s Day is March 17 and it is a prime day in many ways:

What number month? 3;

What number day? 17;

How many letters in “March”? 5;

How many days in March? 31.I asked Pat the same questions about his birthday this year — but I simply asked whether the four answers were prime or not. When he had told me he said: “Now, if I told you its day of the week this year, you should be able to work out my birthday”.

Then, without me actually being told the day, I was indeed able to work out his birthday.

What is his birthday?

[teaser2738]

]]>George possesses a nine-ton cruising and racing yacht, and its class rules do not allow him to exceed 500 sq. ft. of sail area. When racing George uses a tall Bermudan mainsail and a single overlapping Genoa jib.

George has designed his own sails, and after considerable experiment he has found that the best shape for the mainsail hangs the boom at right-angles to the mast, and the best shape for the jib also takes the form of a right-angled triangle.

On his plans for a new suit of sails George found that all the dimensions worked out as an exact number of feet (no inches or fractions). He recently approached two different sailmakers for estimates for the new sails. The first sailmaker quoted him at 4s. per square foot of sail, while the second said he always charged on the length of the perimeter edge of a sail, and he quoted at 12s. per foot run of edge.

To George’s astonishment both quotations were identical not only in total for the pair of sails but also for each separate sail. George was keen on racing and had designed his sail area as close to the limit as possible.

By how many square feet were George’s new sails below the limit of 500 sq. ft. allowed by the class rules?

[teaser47]

]]>My five nieces Abby, Betty, Cathy, Dolly and Emily each had some sweets. I asked them how many they had but they refused to answer directly. Instead, in turn, each possible pair from the five stepped forward and told me the total number of sweets the two of them had. All I remember is that all ten totals were different, that Abby and Betty’s total of 8 was the lowest, and that Cathy and Dolly’s total of 18 was the second highest. I also remember one of the other totals between those two but I don’t remember whose total it was. With that limited information I have worked out the total number of sweets.

In fact it turns out that the other total I remember was Betty and Cathy’s.

In alphabetical order of their names, how many sweets did each girl have?

[teaser3126]

]]>The “factorials” of numbers (denoted by !) are defined by:

1! = 1

2! = 2 × 1

3! = 3 × 2 × 1

4! = 4 × 3 × 2 × 1

etc.It is possible to take eleven of the twelve factorials 1!, 2!, 3!, 4!, 5!, 6!, 7!, 8!, 9!, 10!, 11!, 12! and to split them into groups of three, four and four, so that in each group the product of the factorials in that group is a perfect square.

What are the factorials in the group whose product is the smallest?

[teaser2729]

]]>The staff of a certain international organisation, in their day-to-day work, use four languages — English, French, German and Italian — and every employee is required to be fluent in two of them, viz. his or her own, and one of the other three.

The four heads of branches of the Circumlocution Division are of different nationalities; their second languages are also all different. Each has a woman secretary whose second language is the native language of her chief, but in no case is either language of the chief the native language of his secretary.

All eight are colleagues of mine. John and Mary are English, Jules and Adèle French, Otto and Heidi German, and Nino and Gina Italian.

The man of the same nationality as John’s secretary has German as his second language.

The secretary of the man whose native language is Jules’s second language is of the same nationality as Heidi’s chief.

Who is Jules’s secretary? And which man has Italian as his second language?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill).

[teaser732]

]]>Don and Derek live on intermediate floors in neighbouring blocks of flats each provided with a lift. Neither lift is subject to direction change by users and they both travel continuously and steadily from the ground floor to the top floor and back, stopping at each floor, and waiting at the top and bottom twice as long as at other stops. During the year the number of floors in each block is increased, though not necessarily by the same number. Don, who lives on a higher floor than Derek, notices that if his lift is not on his floor, it is twice as likely to be going down when it first comes to him as it used to be. Derek makes a similar observation on his lift.

The following year each moves four floors up and they each notice that if the lift is not on his floor, it has the same likelihood of going down when it first comes to him as it had originally.

On which floors do Don and Derek live now?

[teaser46]

]]>At Teaser Tor trig. point I found a geocaching box. The three-figure compass bearings (bearing 000=north, 090=east, etc.) from there to the church spires at Ayton, Beeton and Seaton were needed to decode the clue to the next location.

Each spire lay in a different compass quadrant (eg 000 to 090 is the North-East quadrant). Curiously, each of the numerals 1 to 9 occurred in these bearings and none of the bearings were prime values.

Given the above, if you chose one village at random to be told only its church spire’s bearing, it might be that you could not calculate the other two bearings with certainty, but it would be more likely you could.

Give the three bearings, in ascending order.

[teaser3125]

]]>I have written down five positive whole numbers whose sum is a palindromic number. Furthermore, the largest of the five numbers is the sum of the other four. I have reproduced the five numbers below, but have consistently replaced digits by letters, with different letters used for different digits.

We are about to enter an odd-numbered new year and so it is appropriate that my numbers have become:

A

VERY

HAPPY

NEW

YEARWhat is the odd-numbered

YEAR?

This puzzle was *not* included in the book **The Sunday Times Brain Teasers Book 1** (2019).

[teaser2727]

]]>Last year I went to calligraphy lessons. They were held weekly, on the same day each week, for nine consecutive months. Actually I only went to 15 of the lessons, and after the course was over I listed the dates of those lessons that I had attended. In order to practise my new skills I wrote the dates in words (in the format “First of January” etc.) and I found to my surprise that each date used a different number of letters.

What were the dates of the first and last lessons that I attended?

[teaser2733]

]]>A gardener was laying out the border of a new lawn; he had placed a set of straight lawn edging strips, of lengths 16, 8, 7, 7, 7, 5, 4, 4, 4 & 4 feet, which joined at right angles to form a simple circuit. His neighbour called over the fence, “Nice day for a bit of garden work, eh? Is that really the shape you’ve decided on? If you took that one joined to its two neighbours, and turned them together through 180°, you could have a different shape. Same with that one over there, or this one over here — oh, look, or that other one”. The gardener wished that one of his neighbours would turn through 180°.

What is the area of the planned lawn, in square feet?

[teaser3124]

]]>Andrew, Alexander, Austin, Anthony, Benjamin, Charles, Christopher, Elijah, Jacob, Jayden, Jackson, James, Jason, Mason, Michael, Nathan, Newman, Robert, Samuel and William entered a darts competition, arranged into five teams of four players. It turned out that, for any pair of members of any team, there were just two letters of the alphabet that occurred (once or more) in both their names. The names in each team were arranged alphabetically, the first name being the captain and the last name the reserve. Then the teams were numbered 1 to 5 in alphabetical order of the captains.

In the order 1 to 5, who were the reserves?

[teaser2725]

]]>We have recently celebrated the 50th anniversary of the Teaser column. At the party for the setters they gave each letter of the alphabet a different number from 1 to 26 (e.g. they made

A= 7). Appropriately, this was done in such a way that, for each setter present, the values of the letters of their surname added up to 50. Angela Newing was there (soN+E+W+I+N+G= 50), as were Nick MacKinnon and Hugh Bradley. Only two of Graham Smithers, Danny Roth, Andrew Skidmore, John Owen and Victor Bryant could make it.Which two?

[teaser2533]

]]>Harry, Kenneth, Lionel, Martin, Nicholas and Oliver were, the competitors in the 100-yard race on Sports Day. They wore cards numbered 1, 2, 3, 4, 5, 6 but not in that order. In no case was the position of any of the competitors the same as his card number but two of the competitors had positions equal to each other’s card number.

Nicholas was 5th and his card number was the same as Kenneth’s position. Harry’s card number was the same as Oliver’s position which was 4th. Martin’s card number was 1.

It was found that the sum of the products of each competitor’s position and card number was 61.

Place the competitors in the order in which they finished the race and give their card numbers.

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser161]

]]>

A factory makes six types of cylindrical pipe, A to F in decreasing size, whose diameters in centimetres are whole numbers, with type A 50 per cent wider than type B. The pipes are stacked in the yard as a touching row of As with an alternating row of touching Bs and Cs in the next layer, with each B touching two As. Type Ds fill the gap between the As and the ground; Es fill the gap between As and the Bs; and Fs fill the gap between As, Ds and the ground. Finally another row of As is put on top of the stack, giving a height of less than 5 metres.

What is the final height of the stack in centimetres?

[teaser3123]

]]>I asked Peter to place the numbers 1 to 10 at the ten intersection points of the Christmas star so that in each of the five lines the four numbers added to the same total. He found that this was impossible so instead he did it with the numbers 1 to 9 together with just one of those digits repeated. In his answer there was just one line in which that digit did not appear.

[In ascending order] what were the four numbers in that line?

[teaser2726]

]]>When South, who was bored with being dummy, had left the card room of the Logic Club for good, West extracted four kings, three queens and two jacks from the pack, showed them round, shuffled them, and dealt three cards to each of the three.

“Forget the suits”, he said. “Let each of us look at his own three cards and see if he can make a sure statement about any kings, queens or jacks in the next man’s hand; we will use as evidence what we see in our own hands and what we hear each other say.

They played with the full rigours of logic. West began by announcing that he could say nothing about North’s hand; North then said that he could say nothing about East’s hand; thereupon East said that he could deduce the value of one card — and one only — in West’s hand.

What can you say about the cards that were dealt to East?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill).

[teaser727]

]]>Three leaves fall from a book. The total of the page numbers remaining in the second half of the book is now three times the total of the page numbers remaining in the first half.

The total of the page numbers on the fallen leaves lies between 1050 and 1070, and is the highest which could have produced this effect.

How many numbered pages did the book contain originally?

I found many solutions, which improve on the published answer. So I have marked the puzzle as *flawed*.

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser159]

]]>Five witnesses were interviewed following a robbery at the bank in the High Street. Each was asked to give a description of the robber and his actions.

The details given were: height; hair colour; eye colour; weapon carried; escape method.

Witness 1:short; fair; brown; cricket bat; motorbike.

Witness 2:tall; fair; grey; gun; car.

Witness 3:tall; dark; brown; crowbar; motorbike.

Witness 4:short; ginger; blue; knife; car.

Witness 5:tall; dark; blue; stick; pushbike.When the police caught up with the perpetrator, they found that each of the five witnesses had been correct in exactly two of these characteristics.

What was the robber carrying, and how did he get away?

[teaser3122]

]]>In this letters-for-digits substitution puzzle, each letter consistently represents a different digit. In the display, each letter in the top row is the sum of the two letters directly below it:

What number is

LOPSIDED?

[teaser2533]

]]>A football tournament has four groups each of four teams, with the teams in the same group playing each other once. So far the teams have each played two games and in each group the distribution of points is different. Also, in each group just one pair of teams are level on points and their positions have been determined by their “goals scored”. Milliner has scored four (including a hat-trick), Orlando two, and nine other players have scored one goal each. Despite his success, Orlando’s team is not the top of their group.

What are the results of the two games that Milliner’s team have played? And the two games that Orlando’s team have played?

[teaser2721]

]]>A Harshad number (or H-number) is any number that is divisible by the sum of its digits. Using each non-zero digit just the once, I have written down a 9-figure H-number. Reading from left to right, it also consists of three 2-figure H-numbers followed by a 3-figure H-number. Again working from left to right through the 9-figure number, the last five digits form a 5-figure H-number. Reversing the order of the first five digits of the 9-figure number also gives a 5-figure H-number.

What is the 9-figure number?

[teaser2531]

]]>A teacher is preparing her end of term class test. After the test she will arrive at each pupil’s score by giving a fixed number of marks for each correct answer, no marks for a question that is not attempted, and deducting a mark for each incorrect answer. The computer program she uses to prepare parents’ reports can only accept tests with the number of possible test scores (including negative scores) equal to 100.

She has worked out all possible combinations of the number of questions asked and marks awarded for a correct answer that satisfy this requirement, and has chosen the one that allows the highest possible score for a pupil.

What is that highest possible score?

[teaser3121]

]]>George and Martha are jogging around a circular track. Martha starts at the most westerly point, George starts at the most southerly point, and they both jog clockwise at their own steady speeds. After a short while Martha is due north of George for the first time. Five minutes later she is due south of him for the first time. Then George catches up with her during their second laps at the most northeasterly point of the track.

What is Martha’s speed (in degrees turned per minute)?

[teaser2532]

]]>My grandson and I play a coin game. First we toss seven coins and I have to predict in advance the number of heads whilst he has to predict the number of tails. I then get a number of points equal to the number of heads, he gets a number of points equal to the number of tails, and anyone whose prediction was correct gets a fixed bonus number of points (less than 40). We repeat this with six coins in the second round, then five, and so on down to two. In a recent game we noticed that, after each round, the total of all the points so far awarded was equal to a prime number.

What is the “fixed bonus” number of points? And what was the total of all the points at the end of the game?

[teaser2724]

]]>I have eight paving stones whose dimensions are an exact number of inches. Their areas are 504, 420, 324, 306, 270, 130, 117 and 112 square inches. Four of these are red and four green. There should be a ninth stone coloured yellow and square so that all nine stones can be fitted together to form a square in such a way that the red stones completely enclose the other five and the green stones completely enclose the yellow one.

What are the dimensions of the red stones?

[teaser150]

]]>While waiting for buses, I often look out for interesting number plates on passing cars. From 2001 the UK registration plate format has been 2 letters + a 2-digit number + 3 more letters, the digits being last two of the year of registration with 50 added after six months (for example in 2011, the possible numbers were 11 and 61). I spotted one recently with its five letters in alphabetical order, all different and with no vowels. Looking more closely, I saw that if their numerical positions in the alphabet (A = 1, B = 2 etc.) were substituted for the 5 letters, their sum plus 1 was the 2-digit number and the sum of their reciprocals was equal to 1.

Find the 7-character registration.

[teaser3120]

]]>Bearing in mind today’s date, I have written down two numbers in code, with different letters being used consistently to replace different digits. The addition of the two numbers is shown below, appropriately leading to today’s date as a six- figure number.

What is the value of

SEND?

[teaser2723]

]]>(1) In this list there is a true statement and a false statement whose numbers add up to give the number of a false statement.

(2) Either statement 4 is false or there are three consecutive true statements.

(3) The number of the last false statement subtracted from the product of the numbers of the first and last true statements gives the number of a statement which is false.

(4) The number of even-numbered true statements equals the number of false statements.

(5) One of the first and last statements is true and the other is false.

(6) When I first sent this problem to the editor, thanks to a typing error no sixth statement was included. However the answer to the following question was the same then as it is now:

Which statements are false?

**The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser724]

]]>At a certain stage in our all-against-all Rugby football competition the table of results read as follows. There had been no matches in which neither side had scored at all:

What was the result of the match between

AandB?

Note:This problem was set when a try was worth 3 points, a penalty goal 3 points and a converted try 5 points. Match points are on the usual basis of 2 for a win and 1 for a draw.

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser120]

]]>My grandson is studying “History since the Battle of Hastings”. I made him a game, which consisted of a row of nine cards, each with a different non-zero digit on it. Throw a standard die, note the number of spots displayed, count that number of places along the row and pause there. Throw the die again, move the corresponding number of places further along and pause again. Repeat this until you come off the end of the row, noting the digit or digits you paused on and put these together in the same order, to produce a number.

Keeping the cards in the same order I asked my grandson to try to produce a square or cube or higher power. He eventually discovered that the lowest possible such number was equal to the number of one of the years that he had been studying.

What is the order of the nine digits along the row?

[teaser3119]

]]>I have four tetrahedral dice. Each has four identical triangular faces and on each face of each die is one of the numbers 1, 2, 3 or 4 (repeats being allowed on a die). No die uses the same four numbers and, for each die, the sum of its four numbers is ten.

I play a game with my friend. My friend chooses a die first and then I choose one of the remaining three dice. We each throw our chosen die and score the face-down number: sometimes it’s a draw, otherwise the higher number wins. I can choose my die so that I am always more likely to win.

So my friend changes the rules. We now throw our chosen die twice, add the two numbers, and the higher total wins.

Now he knows that there is one die he can choose that makes him more likely to win the game.

What are the four numbers on the winning dice?

The wording for this puzzle has been modified from the originally published version.

[teaser2720]

]]>In a primitive eastern country a shepherd was counting his

sheep into four separate pens. In the first were 75 sheep, so he wrote in his records:In the second were 255 and he wrote:

In the third were 183 and he wrote:

After counting the sheep in the fourth pen he wrote:

How many sheep were in the fourth pen?

**The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser724]

]]>Midsummer Day, 1962 (June 24) was my youngest grandson John’s first birthday, and I was then able to claim that my nine grandchildren were aged 0, 1, 2, 3, 4, 5, 6, 7, and 8 years old (neglecting, of course, the odd days). They were all born in June, and if they are arranged in birthday order through the month the following facts are true:

John is the middle grandchild;

The sum of the dates of the last four is an exact multiple of the sum of the dates of the first four;

The sum of the ages of the last four is two-thirds of the sum of the ages of the first four;

The sum of the years of birth of the first three is equal to the sum of the years of birth of the last three;

The intervals between birthdays are 0, 1, 2, 3, 4, 5, 6, and 7 days, but not in that order;

Also:

My eldest son’s two daughters are exactly two years apart;

The twins were born four hours apart;

Two children are as old as their dates of birth.

What was the date of birth of the grandchild born in 1954?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

There are now **700** *Teaser* puzzles available on the site!

[teaser101]

]]>If a date during the current millennium is day

Din monthMduring year (2000+N), it is said to be a product date if the product ofDandMequalsN(for example 11 February 2022).My daughter and I have been investigating the numbers of days from one product date to the next product date. I was able to establish the longest such interval

L, while my daughter worked out the shortest such intervalS. We were surprised to find thatLis a whole number multiple ofS.What is that multiple?

[teaser3118]

]]>“Sign Works” make rectangular signs of all sizes. Pat ordered a sign for the pub with the following instructions:

“The length must be twice the width. Furthermore, the dimensions should be such that if you take its length (in centimetres), square the sum of its digits and take away the length itself, then you get the width. On the other hand, if you take its width (in centimetres), square the sum of its digits and take away the width itself, then you get the length”.

This was enough information for the sign-maker to calculate the dimensions of the sign.

What were they?

[teaser2753]

]]>On the Island of Imperfection there are three tribes; 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 is about three inhabitants; Ugly, Stupid and Toothless, whose names give some indication of the nature of their imperfections.

The island currency is a Hope. The weekly wage of a Pukka is between 10 and 19 Hopes, of a Shilli-Shalla between 20 and 29, and of a Wotta-Woppa between 30 and 39 (in each case a whole number of Hopes). They make three statements each, anonymously:

Asays: (1)Bis a Wotta-Woppa. (2) My wages are 25% less than 20% more than one of the others. (3) I get 10 hopes more thanB.

Bsays: (1) The wages of one of us is different from the sum of those of the other two. (2) We all belong to the same tribe. (3) More ofC‘s statements are true than mine.

Csays: (1) Ugly earns more than Toothless. (2) The wages of one of us are 15% less than the wages of another. (3) Stupid is a Shilli-Shalla.Find

C‘s name, tribe and wages.

**The Sunday Times Book of Brain-Teasers: Book 2** (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser718]

]]>Ali and Baba are playing the Chay game, Ali has a bag containing 12 numbered tickets, 3 each of numbers 1, 2, 3, 4 (all numbers represented by strokes). Baba has 6 double-sided counters containing the same 12 numbers, and a board of 6 squares.

Ali draws 6 tickets from his bag one at a time, calling out the number as he does so. At each call Baba selects a counter with that number on one of its sides and places it face up in a square. If in 6 calls he fills 6 squares he wins. Once a counter is laid it stays. The counter-couplings are so arranged that 5 squares could always be filled if the numbers were known beforehand.

But, unnoticed by Baba, Ali has slyly added 1 stroke each to 2 of his opponent’s counters. As a result, Baba frequently places no more than 3 or 4 counters, and at last comes a deal when, after Ali has called “Two”, “One”, he calls a third number and Baba cannot fill it.

It is the last straw.

Baba, having lost many pice and much temper, angrily examines the four remaining counters. Three of them are identical couplings!

“Ah! wicked one”, he cries, “you have forged my counters!”. And, throwing them down, he clears for action.

What couplings are on these 4 counters?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser77]

]]>A square, a triangle and a circle went into a bar.

The barman said: “Are you numbers over 18?”

They replied: “Yes, but we’re under a million”

The square boasted: “I’m interesting, because I’m the square of a certain integer”

The triangle said: “I’m more interesting — I’m a triangular number, the sum of all the integers up to that same integer”

The circle said: “I’m most interesting — I’m the sum of you other two”

“Well, are you actually a circular number?”

“Certainly, in base 1501, because there my square ends in my number exactly. Now, shall we get the drinks in?”

The square considered a while, and said: “All right, then. You(‘)r(e) round!”

In base 10, what is the circular number?

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]]>Joe was talking to the supermarket manager about shelf stocking, and she mentioned a recent exercise with milk. Each morning, they added sufficient new stock to ensure there were 1,200 litres on the shelves. They found that 56% of the new stock was sold on the first day, half that amount the next day and half that new amount the day after. Any remaining stock was regarded as out of date and removed next morning before restocking. Hence, they calculated the number of litres to be added each morning.

What is that number?

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]]>After the tea-party Alice persuaded the Mad Hatter, the March Hare and the Dormouse to run with her round a nearby circular track, promising that they should all four win the race by reaching the winning-post at the same moment — so long as they did not vary their speeds!

Round the track were twelve posts equally-spaced a whole number of feet apart, No. 12 being at the start, which was also the finishing-post. At each post one of the Flamingoes was stationed as umpire. We will call them F1, F2, …, F12. F12 acted as starter. The umpires reported as follows:

(1) All four runners maintained their own constant speeds.

(2) F2 noted that Hatter passed Dormouse at his post exactly 30 seconds after the start.

(3) F3 reported that Hare passed Hatter at his post exactly 45 seconds after the start.

(4) F8 said that Hare passed Alice at his post, at which time Alice was passing his post for the third time and Hare for the sixth time.

(5) The umpires reported no more overtakings, although obviously there were others.

The speeds of the four runners, in feet per second, were whole numbers between 5 and 20.

How many laps had they all completed when they all won. And how many seconds did the race last?

*The Sunday Times Book of Brain-Teasers: Book 2* (1981, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

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]]>Mr Robinson is an elderly gentleman who seeks to combat insomnia by juggling with the number of days he has lived. On April 25 last he noted that this was the product of three numbers each consisting of ten times a single figure less one. The following night he found to his surprise that the total of his days was the perfect cube of the product of the three figures of the previous day.

How many days had he lived on April 26?

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]]>In an election for golf-club president, voters ranked all four candidates, with no voters agreeing on the rankings. Three election methods were considered.

Under first-past-the-post, since the first-preferences order was

A,B,C,D, the president would have beenA.Under Alternative Vote, since

Ahad no majority of first preferences,Dwas eliminated, with his 2nd and 3rd preferences becoming 1st or 2nd preferences for others. There was still no majority of 1st preferences, andBwas eliminated, with his 2nd preferences becoming 1st preferences for others.Cnow had a majority of 1st preferences, and would have been president.Under a Borda points system, candidates were given 4, 3, 2, or 1 points for each 1st, 2nd, 3rd or 4th preference respectively.

DandCwere equal on points, followed byBthenA.How many Borda points did each candidate receive?

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