I have here two positive single figure numbers, each less than 9. Neither is a factor of the other. I add the larger number to the smaller.

Then, to that total I again add the original larger number, and to the new total I again add the original larger number and may, if I like, continue this process indefinitely, but never shall I obtain a total which is a “power” of any whole number whatsoever.

What are my two numbers?

This puzzle was included in the book *Brain Teasers* (1982, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser683]

]]>I have written down three even numbers and then consistently replaced digits by letters with different letters used for different digits. In this way I get:

ALL

THE

SAINTSIn fact multiplying together the first two of these numbers gives the third.

What number is my

SAINT?

[teaser2771]

]]>I wrote an odd digit in each of the sixteen cells of a four-by-four grid, with no repeated digit in any row or column, and with each odd digit appearing three or more times overall. Then I could read four four-figure numbers across the grid and four four-figure numbers down. I calculated the average of the four across numbers and the larger average of the four down numbers. Each was a whole number consisting entirely of odd digits, and each used an odd number of different odd digits.

What were those two averages?

[teaser3043]

]]>The clubs Barnet, Exeter, Gillingham, Plymouth, Southend and Walsall need to attract more fans. So each has persuaded one of the players Aguero, Ibrahimovic, Lampard, Neymar, Schweinsteiger and Suarez to join them. Also, each club has persuaded one of the managers Conte, Mourinho, Pellegrini, Terim, Van Gaal and Wenger to take control. For each club, if you look at the club, player and manager, then for any two of the three there are just two different letters of the alphabet that occur in both (with the letters possibly occurring more than once).

In alphabetical order of the teams, list their new players.

[teaser2770]

]]>My typewriter had the standard keyboard:

row 1:QWERTYUIOP

row 2:ASDFGHJKL

row 3:ZXCVBNMuntil I was persuaded by a time-and-motion expert to have it “improved”. When it came back I found that none of the letters was in its original row, though the number of letters per row remaining unchanged. The expert assured me that, once I got used to the new system, it would save hours.

I tested it on various words connected with my occupation — I am a licensed victualler — with the following results. The figures in parentheses indicate how many rows I had to use to produce the word:

BEER(1)

STOUT(1)

SHERRY(2)

WHISKY(3)

HOCK(2)

LAGER(2)

VODKA(2)

CAMPARI(2)

CIDER(3)

FLAGON(2)

SQUASH(2, but would have been 1 but for a single letter)Despite feeling a trifle

MUZZY(a word which I was able to type using two rows) I persevered, next essayingCHAMBERTIN.Which rows, in order, did I use?

This puzzle was included in the book *Brain Teasers* (1982, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser660]

]]>Mr Simpson, who lives at No. 1453 Long Street, is a keen mathematician, and so he was most interested when, [while delivering a letter], his postman mentioned a strange coincidence. If the numbers of [any] two houses to which he made consecutive deliveries were added together, the result came to the number of the next house to which he delivered a letter.

Mr Simpson asked him which houses he had visited, but the postman could only remember that some of them had single digits.

To which house did the postman deliver a letter immediately before delivering Mr Simpson’s letter?

I have changed the wording of this puzzle slightly for clarity.

[teaser18]

]]>The modernist music of Skaredahora eschewed traditional scales; instead he built scales up from strict mathematical rules.

The familiar major scale uses 7 notes chosen from the 12 pitches forming an octave. The notes are in (1) or out of (0) the scale in the pattern 101011010101, which then repeats. Six of these notes have another note exactly 7 steps above (maybe in the next repeat).

He wanted a different scale using 6 notes from the 12 pitches, with exactly two notes having another note 1 above, and one having another 5 above. Some notes could be involved in these pairings more than once.

His favourite scale was the one satisfying these rules that came first numerically when written out with 0s & 1s, starting with a 1.

What was Skaredahora’s favourite scale?

[teaser3042]

]]>A set of snooker balls consists of fifteen reds and seven others. From my set I put some [of the balls] into a bag. I calculated that if I picked three balls out of the bag simultaneously at random, then there was a one in a whole-number chance that they would all be red. It was more likely that none of the three would be red – in fact there was also a one in a whole-number chance of this happening.

How many balls did I put in the bag, and how many of those were red?

[teaser2768]

]]>“Moriarty speaking”, said the voice on the telephone to the Prime Minister. “As you have rejected my demands, a hidden bomb with destroy London. I’m particularly pleased with the detonating device”, he went on, chuckling fiendishly, “it’s designed to give me time to get away before the explosion. There are 60 switches (all turned OFF at the moment) arranged in a ring so that No. 60 is next to No. 1. Whenever any switch changes from ON to OFF it causes the following switch to change over virtually instantaneously (from OFF to ON or vice-versa). As soon as I put down this phone I’ll activate the device. This will automatically put switch No. 1 to ON, then one minute later to OFF, then one minute later still to ON, carrying on in this way after each minute changing switch No. 1 over. As soon as every switch has remained in the OFF position for 10 seconds simultaneously the bomb explodes. So goodbye now — for ever!”

The Prime Minister turned anxiously to Professor D. Fuse who had been listening in. “When will the activating device set off the bomb?” he asked.

What was the Professor’s reply?

This puzzle was included in the book *Brain Teasers* (1982, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser658]

]]>In “Amusements in Mathematics” (Nelson, 1917), the late Henry Ernest Dudeney published a magic knight’s tour of the chessboard. That is to say, a knight placed on the square numbered 1 could, by ordinary knight’s moves, visit every square of the board in the ordered numbered, and the numbers themselves in each row and column added up to 260. Yet it was not a fully magic square, for the diagonals did not add to the same constant. After much trying Dudeney came to the conclusion that it is not possible to devise such a square complete with magic diagonals, but, as he said, a pious opinion is not a proof.

You are invited to try your skill in devising a magic knight’s tour of a square 7×7, with or without magic diagonals.

Dudeney’s *Amusements in Mathematics* is available on Project Gutenberg [link].

[teaser17]

]]>For Christmas 1966 I got 200 Montini building blocks; a World Cup Subbuteo set; and a Johnny Seven multi-gun. I built a battleship on the “Wembley pitch” using every block, then launched seven missiles at it from the gun. The best game ever!

Each missile blasted a different prime number of blocks off the “pitch” (fewer than remained). After each shot, in order, the number of blocks left on the “pitch” was:

(1) a prime;

(2) a square;

(3) a cube;

(4) (a square greater than 1) times a prime;

(5) (a cube greater than 1) times a prime;

(6) none of the aforementioned; and

(7) a prime.The above would still be valid if the numbers blasted off by the sixth and seventh shots were swapped [with each other].

How many blocks remained on the “pitch” after the seventh shot?

[teaser3041]

]]>At the fruit stall in our local market the trader built a stack of oranges using the contents of some complete boxes, each containing the same number of oranges.

He first laid out one box of oranges in a rectangle to form the base of a stack. He then added more oranges layer by layer from the contents of the other boxes. Each layer was a rectangle one orange shorter and narrower than the layer beneath it.

The top layer should have consisted of a single row of oranges but the trader was one orange short of being able to complete the stack.

How many oranges were there in each box?

This puzzle was included in the book *Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

This completes the 72 puzzles from the 2002 *Brainteasers* book. In the New Year I will start posting puzzles from the 1982 book *“The Sunday Times book of Brain Teasers (50 hard (very hard) master problems)”*, compiled by Victor Bryant and Ronald Postill. It is a selection of Teaser puzzles originally published in *The Sunday Times* between January 1974 and December 1979.

Happy New Year from *S2T2!*

[teaser1995]

]]>An Austin was pootling along a country lane at 30mph; behind were a Bentley doing 40mph and a Cortina doing 50mph. The Bentley and the Cortina braked simultaneously, decelerating at constant rates, while the Austin carried on at the same speed. The Bentley just avoided hitting the rear of the Austin, [while, at the same time,] the Cortina just avoided a collision with the Bentley. The Bentley and the Cortina continued to decelerate at the same rates, and stopped with a 45yd gap between them.

What was the gap between the Bentley and the Cortina at the moment they started to brake?

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

[teaser2508]

]]>I have three circular medallions that I keep in a rectangular box, as shown. The smallest (of radius 4cm) touches one side of the box, the middle-sized one (of radius 9cm) touches two sides of the blox, the largest touches three sides of the box, and each medallion touches both the others.

What is the radius of the largest medallion?

This puzzle was included in the book *Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1991]

]]>Jonny has opened a new bank account and has set up a telephone PIN. His sort code is comprised of the set of three two-figure numbers with the smallest sum which give his PIN as their product. He was surprised to find that the PIN was also the result of dividing his eight-figure account number by one of the three two-figure numbers in the sort code.

The PIN has an unusual feature which Jonny describes as a moving digit. If the number is divided by its first digit then the number which results has the same digits in the same order except that first digit is now at the end.

The account number does not contain the digit which moved.

What is the account number?

[teaser3040]

]]>My army number has eight digits, the third being the same as the sixth, all the others occurring only once. The sum of the digits is 33, and the difference between the sum of the first four and the sum of last four is 3. The first four digits have irregular ascending values. When out of their correct order, three only of my last four digits have consecutive numerical value; in correct order there is a difference of at least 2 between consecutive digits. The highest digit is 7 and army numbers never start with zero.

What is my number?

[teaser16]

]]>I asked my nine-year-old grandson Sam to set a Teaser for today’s special edition and the result was:

SAM

SET

NICE

CHRISTMAS

TEASERThose words are the result of taking five odd multiples of nine and consistently replacing digits by letters.

Given that

THREEis divisible by 3; What is the 9-digit numberCHRISTMAS?

This was not a prize puzzle.

[teaser2570]

]]>James is laying foot-square stones in a rectangular block whose short side is less than 25 feet. He divides this area into three rectangles by drawing two parallel lines between the longest sides and into each of these three areas he lays a similar pattern.

The pattern consists of a band or bands of red stones laid concentrically around the outside of the rectangles with the centre filled with white stones. The number of red stone bands is different in each of the rectangles but in each of them the number of white stones used equals the number of outer red stones used.

The total number of stones required for each colour is a triangular number (i.e., one of the form 1+2+3+…).

What is the total area in square feet of the block?

[teaser3039]

]]>To make an unusual paperweight a craftsman started with a cuboidal block of marble whose sides were whole numbers of centimetres, the smallest sides being 5cm and 10cm long. From this block he cut off a corner to create a triangular face; in fact each side of this triangle was the diagonal of a face of the original block. The area of the triangle was a whole number of square centimetres.

What was the length of the longest side of the original block?

[teaser2767]

]]>I have placed a full set of 28 dominoes on an eight-by-seven grid, with some of the dominoes horizontal and some vertical. The array is shown above with numbers from 0 to 6 replacing the spots at each end of the dominoes.

Fill in the outlines of the dominoes.

This puzzle was included in the book *Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1967]

]]>I have received three boxes of whatsits. They all look alike but those in one of the boxes weigh 6 grams each, those in another box all weigh 7 grams each, and all those in the remaining box weigh 8 grams each.

I do not know which box is which but I have some scales which enable me to weigh accurately anything up to 30 grams. I with to use the scales to determine which whatsit is which.

How can I do this with just one weighing?

The text of this puzzle is taken from the book *Brainteasers* (2002, edited by Victor Bryant), the wording differs only slightly from the puzzle originally published in the newspaper.

The following note was added to the puzzle in the book:

When this Teaser appeared in

The Sunday Times, instead of saying “some scales” it said “a balance”. This implied to some readers that you could place whatsits on either side of the balance — which opens up all sorts of alternative approaches which you might like to think about.

There are now 400 *Teaser* puzzles available on the site.

[teaser1984]

]]>George and Martha were participating in the local village raffle. 1000 tickets were sold, numbered normally from 1 to 1000, and they bought five each. George noticed that the lowest-numbered of his tickets was a single digit, then each subsequent number was the same multiple of the previous number, e.g. (7, 21, 63, 189, 567). Martha’s lowest number was also a single digit, but her numbers proceeded with a constant difference, e.g. (6, 23, 40, 57, 74). Each added together all their numbers and found the same sum. Furthermore, the total of all the digits in their ten numbers was a perfect square.

What was the highest numbered of the ten tickets?

[teaser3038]

]]>If you place a digit in each of the eight unshaded boxes, with no zeros in the corners, then you can read off various three-figure numbers along the sides of the square, four in a clockwise direction and four anticlockwise.

Place eight different digits in those boxes with the largest of the eight in the top right-hand corner so that, of the eight resulting three-figure numbers, seven are prime and the other (an anticlockwise one) is a square.

Fill in the grid.

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1966]

]]>This unusual instrument is operated by selecting one of the four switch positions: A, B, C, D, and turning the power on. The effects are:

A:The pratching valve glows and the queech obulates;

B:The queech obulates and the urfer curls up, but the rumption does not get hot;

C:The sneeveling rod turns clockwise, the pratching valve glows and the queech fails to obulate;

D:The troglodyser gives off hydrogen but the urfer does not curl up.Whenever the pratching valve glows, the rumption gets hot. Unless the sneeveling rod turns clockwise, the queech cannot obulate, but if the sneeveling rod is turning clockwise the troglodyser will not emit hydrogen. If the urfer does not curl up, you may be sure that the rumption is not getting hot.

In order to get milk chocolate from the machine, you must ensure:

(a) that the sneeveling rod is turning clockwise AND;

(b) that if the troglodyser is not emitting hydrogen, the queech is not obulating.

1. Which switch position would you select to get milk chocolate?If, tiring of chocolate, you wish to receive the Third Programme, you must take care:

(a) that the rumption does not get hot AND;

(b) either that the urfer doesn’t curl and the queech doesn’t obulate or that the pratching valve glows and the troglodyser fails to emit hydrogen.

2. Which switch position gives you the Third Programme?

No setter was given for this puzzle.

This puzzle crops up in several places on the web. (Although maybe it’s just because it’s easy to search for: “the queech obulates” doesn’t show up in many unrelated pages).

And it is sometimes claimed it “appeared in a national newspaper in the 1930s” (although the BBC Third Programme was only broadcast from 1946 to 1967 (after which it became BBC Radio 3)), but the wording always seems to be the same as the wording in this puzzle, so it seems likely this is the original source (at least in this format).

“Omnibombulator” is also the title of a 1995 book by Dick King-Smith.

[teaser44]

]]>In the annual cross-country race between the Harriers and the Greyhounds each team consists of eight men, of whom the first six in each team score points. The first man home scores one point, the second two, the third three, and so on. When these are added together, the team with the lower total wins the match.

In this year’s match, the Harriers’ captain came in first and as his team followed he totted up the score. When five more Harriers and a number of Greyhounds had arrived, he found that it would be possible still for his team either to lose or to draw or to win, depending on the placings of the two Harriers yet to come.

The tension was relieved slightly when the seventh Harrier arrived, since now the worst that could happen was a draw. Then, in an exciting finish, the eighth Harrier just beat one of his rivals to gain a win for his site by a single point.

What were the scores? And what were the placings of the 16 runners assuming that no two runners tied for a place?

[teaser14]

]]>Last year I was given a mathematical Advent calendar with 24 doors arranged in four rows and six columns, and I opened one door each day, starting on December 1. Behind each door is an illustrated prime number, and the numbers increase each day. The numbers have been arranged so that once all the doors have been opened, the sum of the numbers in each row is the same, and likewise for the six columns. Given the above, the sum of all the prime numbers is as small as it can be.

On the 24th, I opened the last door to find the number 107.

In order, what numbers did I find on the 20th, 21st, 22nd and 23rd?

[teaser3037]

]]>We have a large rectangular field with a wall around its perimeter and we wanted one corner of the field fenced off. We placed a post in the field and asked the workment to make a straight fence that touched the post and formed a triangle with parts of two sides of the perimeter wall. They were to do this in such a way that the area of the triangle was as small as possible. They worked out the length of fence required (less than 60 metres) and went off to make it.

Meanwhile, some lads played football in the field and moved the post four metres further from one side of the field and two metres closer to another.

Luckily when the men returned with the fence it was still the right length to satisfy all the earlier requirements. When they had finished erecting it, the triangle formed had each of its sides equal to a whole number of metres.

How long was the fence?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1956]

]]>George, Martha and their daughter all drive at their own steady speeds (whole numbers of mph), the daughter’s speed being 10mph more than Martha’s. One day George left home to drive to his daughter’s house at the same time as she left her house to visit her parents: they passed each other at the Crossed Keys pub. Another time Martha left her daughter’s to return home at the same time as her daughter started the reverse journey: they too passed at the Crossed Keys. The distance from George’s to the pub is a two-figure number of miles, and the two digits in reverse order give the distance of the pub from their daughter’s.

How far is it from George’s home to the Crossed Keys?

[teaser2769]

]]>The schoolchildren run around in a walled regular pentagonal playground, with sides of 20 metres and with an orange spot painted at its centre. When the whistle blows each child has to run from wherever they are to touch each of the five walls, returning each time to their starting point, and finishing back at the same point.

Brian is clever but lazy and notices that he can minimize the distance he has to run provided that his starting point is within a certain region. Therefore he has chalked the boundary of this region and he stays within in throughout playtime.

(a) How many sides does Brian’s region have?

(b) What is the shortest distance from the orange spot to Brian’s chalk line?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1958]

]]>Clearing out an old drawer I found a wrinkled conker. It was my magnificent old 6709-er, a title earned by being the only survivor of a competition that I had had with friends. The competition had started with five conkers, veterans of many campaigns; each had begun at a different value between 1300 and 1400.

We used the rule that if an

m-er beat ann-er in an encounter (by destroying it, of course!) them-er would become anm+n+1-er; in effect, at any time the value of a conker was the number of destroyed conkers in all confrontations in its “ancestry”.I recall that at the beginning of, and throughout, the competition, the value of every surviving conker was a prime number.

What were the values of the five conkers at the start?

[teaser3036]

]]>Animals board the ark in pairs.

EWEandRAM

HENandCOCKIn fact these are numbers with letters consistently replacing digits; one pair of the numbers being odd, the other pair being even, and both pairs have the same sum. The three digits of the number

ARKare consecutive digits in a muddled order. All this information uniquely determines the numberNOAH.What is the number

NOAH?

[teaser2766]

]]>In my fantasy football league each team plays each other once, with three points for a win and one point for a draw. Last season Aberdeen won the league, Brechin finished second, Cowdenbeath third, and so on, in alphabetical order. Remarkably each team finished with a different prime number of points. Dunfermline lost to Forfar.

In order, what were Elgin’s results against Aberdeen, Brechin, Cowdenbeath, and so on (in the form WWLDL…)?

[teaser2765]

]]>

Your task is to place a non-zero digit in each box so that:

- the number formed by reading across each row is a perfect square, with the one in the top row being odd;
- if a digit is used in a row, then it is also used in the next row up;
- only on one occasion does the same digit occur in two boxes with an edge in common.
Fill in the grid.

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1948]

]]>My friend, “Skeleton” Rose, rambled on with me and my uncle (“The Devil” and “Candyman”) about Mr Charlie, who gave, between us, three identical boxes of rainbow drops.

Each identical box’s card template had a white, regular convex polygonal base section with under ten sides, from each of which a similar black triangular star point extended. All these dark star points folded up to an apex, making an enclosed box.

The number of sweets per box equalled the single-figure sum of its own digits times the sum of the star points and the box’s faces and edges. If I told you how many of the “star point”, “face” and “edge” numbers were exactly divisible by the digit sum, you would know this number of sweets.

How many sweets were there in total?

[teaser3035]

]]>I have chosen five different numbers, each less than 20, and I have listed these numbers in three ways. In the first list the numbers are in increasing numerical order. In the second list the numbers are written in words and are in alphabetical order. In the third list they are again in words and as you work down the list each word uses more letters than its predecessor. Each number is in a different position in each of the lists.

What are my five numbers?

[teaser2764]

]]>The numbers 1 to 9, in any order and using each once only, are to be placed one at a time in the nine squares A to J. As each number replaces a letter in a square, any numbers standing at that moment in adjacent squares (left, right, up or down, but

notdiagonally) are to be multiplied by three.Thus, if we decided to begin with 4 in A, then 9 in E, 7 in B and 2 in D, etc., we should have:

and so on. On completion, the nine final numbers are added together to find the score.

There are obviously 81 ways of making the first move, and there are 131,681,894,400 ways of completing the array; yet the number of possible scores in quite small.

What is the smallest possible score?

[teaser13]

]]>Do you remember all that fuss over the “Millennium bug”?

On that New Year’s Day I typed a Teaser on my word processor. When I typed in 2000 it actually displayed and printed 1900. This is because whenever I type a whole number in figures the machine actually displays and prints only a percentage of it, choosing a random different whole number percentage each time.

The first example was bad enough but the worrying this is that is has chosen even lower percentages since then, upsetting everything that I prepare with numbers in it. Luckily the percentage reductions have not cut any number by half or more yet.

What percentage did the machine print on New Year’s Day?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1946]

]]>A straight track from an observation post, O, touches a circular reservoir at a boat yard, Y, and a straight road from O meets the reservoir at the nearest point, A, with OA then extended by a bridge across the reservoir’s diameter to a disembarking point, B. Distances OY, OA and AB are whole numbers of metres, with the latter two distances being square numbers.

Following development, a larger circular reservoir is constructed on the other side of the track, again touching OY at Y, with the corresponding new road and bridge having all the same properties as before. For both reservoirs, the roads are shorter than 500m, and shorter than their associated bridges. The larger bridge is 3969m long.

What is the length of the smaller bridge?

[teaser3034]

]]>Mark and John played 18 holes of golf: the holes consisting of six each of par 3, par 4 and par 5. Each player finished the round in 72, consisting of six 3s, six 4s and six 5s. In fact each of them had six birdies (one under par), six on par, and six bogies (one over par). At no hole did the two players take the same number of strokes, and Mark beat John on ten of the holes.

How many of Mark’s winning holes were:

(a) on par 3 holes?

(b) on par 4 holes?

(c) on par 5 holes?

[teaser2763]

]]>Ruritania is reluctant to adopt the euro as it has a sensible currency of its own. The mint issues the coins in four denominations, the value of each being proportional to its radius. The total value of the four, in euros, is 28.

The four coins are available in a clever presentation pack. It consists of a triangular box of sides 13 cm, 14 cm and 15 cm. The largest coin just fits into the box, touching each of its sides, roughly as shown:

Then there are three straight pieces of thin card inside the box. Each touches the large coin and is parallel to a side of the box. This creates three smaller triangles in the corners of the box. The three remaining coins just fit into the box, with one in each of these small triangles. Each coin touches all three sides of the triangle.

Unfortunately I have lost the smallest coin from my presentation pack.

What, in euros, is its value?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1942]

]]>“Yet more storms” is a gigantic painting in the State Gallery. It is currently on the wall of the 27-foot-wide “Modern masters” corridor, but the curator feels that it would look better on the 64-foot-wide “Britain’s impressionists” corridor, which meets the “Modern masters” one at right angles.

So he instructs his staff to slide the painting around the corner without tilting it. His staff manage to turn the painting as requested, but had it been any wider it would not have fitted around the corner.

How wide is the painting?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1935]

]]>In the bears’ villa there are three floors, each with 14 rooms. The one switch in each room bizarrely toggles (on off) not only the single light in the room but also precisely two other lights on the same floor; moreover, whenever A toggles B, then B toggles A.

As Goldilocks moved from room to room testing various combinations of switches, she discovered that on each floor there were at least two separate circuits and no two circuits on a floor had the same number of lights. Furthermore, she found a combination of 30 switches that turned all 42 lights from “off” to “on”, and on one floor she was able turn each light on by itself.

(a) How many circuits are there?

(b) How many lights are in the longest circuit?

[teaser3033]

]]>A friend showed me a beautiful gem with shiny flat faces and lots of planes of symmetry. After a quick examination I was able to declare that it was “perfectly square”. This puzzled my friend because none of the faces had four edges. So I explained by pointing out that the gem’s number of faces was a perfect square, its number of edges was a perfect square, and its number of vertices was a perfect square.

How many faces did it have, and how many of those were triangular?

[teaser2762]

]]>My car has an odometer, which measures the total miles travelled. It has a five-figure display (plus two decimal places). There is also a “trip” counter with a three-figure display.

One Sunday morning, when the car was nearly new, the odometer showed a whole number which was a perfect square and I set the trip counter to zero. At the end of that day the odometer again showed a whole number that was a perfect square, and the trip counter showed an odd square.

Some days later, the display on the odometer was four times the square which had been displayed on that Sunday evening, and once again both displays were squares.

What were the displays on that last occasion?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1929]

]]>My wife and I, my son and daughter, my two grandsons, and my granddaughter (the youngest of the family, who was fifteen last birthday) were all born on the same day of the week, and we all have our birthdays on the same date, but all in different months. [I won’t be able to say this if there are any further additions to the family.]

My grandsons were born nine months apart, my daughter eighteen months after my son, and I am forty-one months older than my wife.

What are all our birth dates?

The puzzle was originally published in the 14th May 1961 edition of *The Sunday Times*, however the condition in square brackets was omitted, and the corrected version (and an apology) was published in the 21st May 1961 edition.

[teaser12]

]]>I noticed a dartboard in a sports shop window recently. Three sets of darts were positioned on the board. Each set was grouped as if the darts had been thrown into adjacent numbers (e.g., 5, 20, 1) with one dart from each set in a treble. There were no darts in any of the doubles or bulls.

The darts were in nine different numbers but the score for the three sets was the same. If I told you whether the score was odd or even you should be able to work out the score. The clockwise order of numbers on a dartboard is:

20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5

What was the score that all three sets of darts made?

[teaser3032]

]]>Today my daughter was looking through her old scrapbook and came across a rhyme I had written about her when she was a little girl:

Here’s a mathematical rhyme:

Your age in years is a prime;

Mine is too,

And if you add the two

The answer’s a square — how sublime!She was surprised to find that this is also all true today. Furthermore is will all be true again some years hence.

How old are my daughter and I?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1928]

]]>George and Martha have nine cards with a different non-zero digit on each. To teach their nephew to count they lined up the cards in increasing order. He then rearranged the order of the line and Martha was impressed when she noticed that no digit was in its original position. George was even more impressed when he found that the six-figure number formed by the last six cards was the square of the three-figure number formed by the first three.

What was that three-figure number?

[teaser2761]

]]>I met a nice girl at a party and asked for her phone number. To prove that she was no pushover she made me work for it.

“My number has seven digits, all different”, she told me. “If you form the largest number you can with those seven digits and subtract from it the reverse of that largest number, then you get another seven-digit number”, she added.

“Then if you repeat the process with that new seven-digit number, you get another seven-digit number”, she added. “And if you repeat the process enough times you’ll get back to my phone number”.

This information did enable me to get back to her!

What is her telephone number?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1923]

]]>Jenny is using her calculator, which accepts the input of numbers of up to ten digits in length, to prepare her lesson plan on large numbers. She can’t understand why the results being shown are smaller than she expected until she realizes that she has entered a number incorrectly.

She has entered the number with its first digit being incorrectly entered as its last digit. The number has been entered with its second digit first, its third digit second etc. and what should have been the first digit entered last. The number she actually entered into her calculator was 25/43rds of what it should have been.

What is the correct number?

[teaser3031]

]]>I have a rectangular garden of area just over one hectare. It is divided exactly into three parts — a lawn, a flower bed, and a vegetable patch. Each of these three areas is a right-angled triangle with sides a whole numbers of metres in length. A fence runs along two adjacent sides of the rectangular garden. The length of the fence is a prime number of metres.

What are the dimensions of the rectangular garden?

**Note:** 1 hectare = 10,000 square metres.

[teaser2760]

]]>Joe’s billiard table is of high quality but slightly oversized. It is 14 ½ feet long by 7 feet wide, with the usual six pockets, one in each corner and one in the middle of each long side.

Joe’s ego is also slightly oversized and he likes to show off with his trick shots. One of the most spectacular is to place a ball at a point equidistant from each of the longer sides and 19 inches from the end nearest to him. He then strikes the ball so that it bounces once off each of the four sides and into the middle pocket on his left.

He has found that he has a choice of directions in which to hit the ball in order the achieve this effect.

(a) How many different directions will work?

(b) How far does the ball travel in each case?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1917]

]]>Lazy Jack was engaged to deliver a circular to every house in the district. He found the supply of circulars would divide into fourteen equal batches of rather more than 300 each, so he decided to deliver one batch each day and thus spread the job over a fortnight.

On the first day he faithfully distributed circulars one to a house, but that proved very tiring, so the next day he delivered two at each house he visited. With a fine sense of fairness, he never delivered to the same house twice, and one each succeeding day he chose the next smaller number of houses to visit that would enable him exactly to exhaust the day’s batch by delivering an equal number of circulars at each house. The fourteenth day’s batch of circulars all went through one letter box.

To how many houses was delivery made?

[teaser11]

]]>In snooker, pot success (PS) is the percentage of how many pot attempts have been successful in that match (e.g. 19 pots from 40 attempts gives a PS of 47.5%). In a recent match, my PS was a whole number at one point. I then potted several balls in a row to finish a frame, after which my improved PS was still a whole number. At the beginning of the next frame, I potted the same number of balls in a row, and my PS was still a whole number. I missed the next pot, my last shot in the match, and, remarkably, my PS was still a whole number.

If I told you how many balls I potted during the match, you would be able to work out those various whole-number PS values.

How many balls did I pot?

[teaser3030]

]]>Dancers numbered from 1 to 9 were about to perform a square dance: five were dressed in red and the rest in blue. They stood in a 3-by-3 array with all three dancers in the first row wearing red and all three in another row wearing blue. Their numbers formed a magic square (i.e. the sum of the three numbers in any row, column or diagonal was the same). One of the dancers in red looked around and noted that the sum of the numbers of the four other dancers in red was the same as the sum of the numbers of the four dancers in blue.

One of the dancers in blue was number 8, what were the numbers of the other three dancers in blue?

[teaser2758]

]]>In each of the four houses in a small terrace lives a family with a boy, a girl and a pet rabbit. One of the children has just mastered alphabetical order and has listed them thus:

I happen to know that this listing gave exactly one girl, one boy and one rabbit at her, his or its correct address. I also have the following correct information: neither Harry nor Brian lives at number 3, and neither Donna nor Jumper lives at number 1. Gail’s house number is one less than Mopsy’s house number, and Brian’s house number is one less than Cottontail’s.

Who lives where?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1915]

]]>David’s train left at 11 am. The previous night David had set his bedroom clock and watch in such a way that they would read the correct time when he intended to leave the following morning. His clock lost regularly, while his watch gained regularly. The next morning he actually left when the clock said 8:15 and his watch said 8:10. When the train left the station (on time) David looked at his watch and noticed that it said 11:10. David had planned to leave at an exact number of minutes past 8 and he worked out afterwards that he had left at an exact number of minutes past 8.

At what time did David intend to leave, and at what time did he actually leave?

[teaser10]

]]>I chose a whole number and asked my grandson to cut out all possible rectangles with sides a whole number of centimetres whose area, in square centimetres, did not exceed my number. (So, for example, had my number been 6 he would have cut out rectangles of sizes 1×1, 1×2, 1×3, 1×4, 1×5, 1×6, 2×2 and 2×3). The total area of all the pieces was a three-figure number of square centimetres.

He then used all the pieces to make, in jigsaw fashion, a set of squares. There were more than two squares and at least two pieces in each square.

What number did I originally choose?

[teaser3029]

]]>At our DIY store you can buy plastic letters of the alphabet in order to spell out your house name. Although all the A’s cost the same as each other, and all the B’s cost the same as each other, etc., different letters sometimes cost different amounts with a surprisingly wide range of prices.

I wanted to spell out my house number:

FOURand the letters cost me a total of £4. Surprised by this coincidence I worked out the cost of spelling out each of the numbers from

ONEtoTWELVE. In ten out of the twelve cases the cost in pounds equalled the number being spelt out.For which house numbers was the cost different from the number?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1908]

]]>A sports quiz featured one footballer, one cricketer and one rugby player each week. Over the six-week series the footballers featured were (in order)

Gerrard,Lambert,Lampard,Rooney,SmallingandWelbeck. The cricketers were (in some order)Carberry,Compton,Robson,Shahzad,StokesandTredwell. The rugby players (in some order) wereCipriani,Launchbury,Parling,Robshaw,TrinderandTwelvetrees. Each week, for any two of the three names, there were just two different letters of the alphabet that occurred in both names (with the letters possibly occurring more than once).List the cricketers in the order in which they appeared.

[teaser2757]

]]>Mrs. A, Mrs. B and Mrs. C and their three daughters bought materials for dresses. Each lady spent as many shillings per yard as she bought yards. Each mother spent £5 5s more than her daughter. Mrs. A bought 11 yards more than Florence, and Mrs. C spent £6 15s less than Patricia. The other daughter’s name was Mary.

Whose daughter was each?

**Note:** £1 = 20s.

[teaser9]

]]>Dai had seven standard dice, one in each colour of the rainbow (ROYGBIV). Throwing them simultaneously, flukily, each possible score (1 to 6) showed uppermost. Lining up the dice three ways, Dai made three different seven-digit numbers: the smallest possible, the largest possible, and the “rainbow” (ROYGBIV) value. He noticed that, comparing any two numbers, only the central digit was the same, and also that each number had just one single-digit prime factor (a different prime for each of the three numbers).

Hiding the dice from his sister Di’s view, he told her what he’d done and what he’d noticed, and asked her to guess the “rainbow” number digits in ROYGBIV order. Luckily guessing the red and orange dice scores correctly, she then calculated the others unambiguously.

What score was on the indigo die?

I’ve changed the wording of the puzzle slightly to make it clearer.

[teaser3028]

]]>I live in a long road with houses numbered 1 to 150 on one side. My house is in a group of consecutively numbered houses where the numbers are all nonprime, but at each end of the group the next house number beyond is prime. There are a nonprime number of houses in this group. If I told you the lowest prime number which is a factor of at least one of my two next-door neighbours’ house numbers, then you should be able to work out my house number.

What is it?

[teaser2544]

]]>I have started with a four-figure number with its digits in decreasing order. I have reversed the order of the four digits to give a smaller number. I have subtracted the second from the first to give a four-figure answer, and I have seen that the answer uses the same four digits — very neat!

Substituting letters for digits, with different letters being consistently used for different digits, my answer

wasNEAT!What, in letters, was the four-figure number I started with?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1904]

]]>I have received an astonishing letter from a fellow puzzle-setter. He writes:

“I am sending you a puzzle in the form of a parcel consisting of an opaque box containing some identical marbles, each weighing a whole number of grams, greater than one gram. The box itself is virtually weightless. If I told you the total weight of the marbles, you could work out how many there are.”

He goes on:

“To enable you to work out the total weight of the marbles I am also sending you a balance and a set of equal weights, each weighing a whole number of grams, whose total weight is two kilograms. Bearing in mind what I told you above, these weights will enable you to calculate the total weight of the marbles and hence how many marbles there are.”

I thought that he was sending me these items under seperate cover, but I had forgotten how mean he is. He went on:

“I realise that I can save on the postage. If I told you the weight of each weight you would still be able to work out the number of marbles. Therefore I shall not be sending you anything.”

How many marbles were there?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1892]

]]>Callum and Liam play a simple dice game together using standard dice (numbered 1 to 6). A first round merely determines how many dice (up to a maximum of three) each player can use in the second round. The winner is the player with the highest total on their dice in the second round.

In a recent game Callum was able to throw more dice than Liam in the second round but his total still gave Liam a chance to win. If Liam had been able to throw a different number of dice (no more than three), his chance of winning would be a whole number of times greater.

What was Callum’s score in the final round?

[teaser3027]

]]>I have allocated a numerical value (possibly negative) to each letter of the alphabet, where some different letters may have the same value. I can now work out the value of any word by adding up the values of its individual letters. In this way

NONEhas value 0,ONEhas value 1,TWOhas value 2, and so on up toELEVENhaving value 11. Unfortunately, looking at the words for the numbersTWELVEtoNINETEEN, I find that only two have values equal to the number itself.Which two?

[teaser2756]

]]>In the village of Alphaville, the number of females divided by the number of males was a certain whole number. Then one man and his wife moved into the village and the result was that the number of females divided by the number of males was one less than before. Now today two more such married couples have moved into the village, but the number of females divided by the number of males is still a whole number.

What is the population of the village now?

[teaser2755]

]]>In the block of flats where I live there are 4 dogs and 4 cats. The 4 flat numbers, where the dogs are kept, multiplied together = 3,570, but added = my flat number. The 4 numbers, where the cats are kept, multiplied together also = 3,570, but added = 10 less than mine. Some flats keep both a dog and a cat, but there is only one instance of a dog and a cat being kept in adjacent flats.

At what number do I live, and where are the dogs and cats kept?

[teaser8]

]]>A four-digit number with different positive digits and with the number represented by its last two digits a multiple of the number represented by its first two digits, is called a

PAR.A pair of

PARs is aPARTYif no digit is repeated and eachPARis a multiple of the missing positive digit.I wrote down a

PARand challenged Sam to use it to make aPARTY. He was successful.I then challenged Beth to use my

PARand the digits in Sam’sPARto make a differentPARTY. She too was successful.What was my

PAR?

[teaser3026]

]]>Last night I dreamt that I made a train journey on the HS2 line. The journey was a whole number of miles in length and it took less than an hour. From the starting station the train accelerated steadily to its maximum speed of 220 mph, then it continued at that speed for a while, and finally it decelerated steadily to the finishing station. If you took the number of minutes that the train was travelling at a steady speed and reversed the order of its two digits, then you got the number of minutes for the whole journey.

How many miles long was the journey?

[teaser2736]

]]>In a woodwork lesson the class was given a list of four different non-zero digits. Each student’s task was to construct a rectangular sheet whose sides were two-figure numbers of centimetres with the two lengths, between them, using the four given digits. Pat constructed the smallest possible such rectangle and his friend constructed the largest possible. The areas of these two rectangles differed by half a square metre.

What were the four digits?

[teaser2719]

]]>I had to drive the six miles from my home into the city centre. The first mile was completed at a steady whole number of miles per hour (and not exceeding the 20mph speed limit). Then each succeeding mile was completed at a lower steady speed than the previous mile, again at a whole number of miles per hour.

After two miles of the journey my average speed had been a whole number of miles per hour, and indeed the same was true after three miles, after four miles, after five miles, and at the end of my journey.

How long did the journey take?

[teaser2731]

]]>Ann, Beth and Chad start running clockwise around a 400m running track. They run at a constant speed, starting at the same time and from the same point; ignore any extra distance run during overtaking.

Ann is the slowest, running at a whole number speed below 10 m/s, with Beth running exactly 42% faster than Ann, and Chad running the fastest at an exact percentage faster than Ann (but less than twice her speed).

After 4625 seconds, one runner is 85m clockwise around the track from another runner, who is in turn 85m clockwise around the track from the third runner.

They decide to continue running until gaps of 90m separate them, irrespective of which order they are then in.

For how long in total do they run (in seconds)?

[teaser3025]

]]>I regularly entered the Lottery, choosing six numbers from 1 to 49. Often some of the numbers drawn were mine but with their digits in reverse order. So I now make two entries: the first entry consists of six two-figure numbers with no zeros involved, and the second entry consists of six entirely different numbers formed by reversing the numbers in the first entry. Interestingly, the sum of the six numbers in each entry is the same, and each entry contains just two consecutive numbers.

What (in increasing order) are the six numbers in the entry that contains the highest number?

[teaser2730]

]]>The field near to our home is rectangular. Its longer sides run east to west. In the south-east corner there is a gate. In the the southern fence there is a stile a certain number of yards from the south-west corner. In the northern fence there is a stile, the same certain number of yards from the north-east corner. A straight path leads across the field from one stile to the other. Another straight path from the gate is at right angles to the first path and meets it at the old oak tree in the field.

When our elder boy was ten, and his brother five years younger, they used to race from opposite ends of a long side of the field towards the stile in that as the target. But later, when they were a little less unevenly matched, they raced from opposite stiles towards the oak as the target. Of course the younger boy raced over the shorter distances. This change of track gave the younger boy 9 yards more and the elder boy 39 yards less than before to reach the target.

The sides of the field and the distances of the oak tree from each of the stiles are all exact numbers of yards.

How far apart are the two stiles, and how far is the oak tree from the gate?

[teaser1108]

]]>Many years ago there lived in Addison Street four sisters, June, Alice, Mary and Dawn. In that street there lived also four young men. Harry, Graham, Richard and Tom. On a bright, spring day in 1911 each of the four sisters married the young man of her choice. Each of the four couples had children: June and Mary together had as many children as Alice.

When all the children grew up and, in the course of time, married to become parents themselves each of them had the same number of children as there had been in his or her own family (for instance, had June borne five children, each of these five would have had five, too, and so on).

Last week, the four original couples celebrated their Golden Wedding at a party attended by all their 170 grandchildren. Richard, who despite the fact that his was the smallest family had always been fascinated by numbers, pointed out that June and Alice together had as many grandchildren, as Mary and Dawn. He also noticed that Harry had four times as many grandchildren as Graham had children.

Which sisters married which young men on that bright spring day in 1911?

[teaser7]

]]>From a set of playing cards, Tessa took 24 cards consisting of three each of the aces, twos, threes, and so on up to the eights. She placed the cards face up in single row and decided to arrange them such that the three twos were each separated by two cards, the threes were separated by three cards and so forth up to and including the eights, duly separated by eight cards. The three aces were numbered with a one and were each separated by nine cards. Counting from the left, the seventh card in the row was a seven.

In left to right order, what were the numbers on the first six cards?

[teaser3024]

]]>In my latest effort to produce a neater Easter Teaser I have once again consistently replaced each of the digits by a different letter. In this way:

NEATER LATEST EASTER TEASERrepresent four six-figure numbers in increasing order. Furthermore, the following addition sum is correct:

What is the value of

BONNET?

[teaser2741]

]]>The “Sacred Sign of Solomon” consists of a pentagon whose vertices lie on a circle, roughly as shown:

Starting at one of the angles and going round the circle the number of degrees in the five angles of the large pentagon form an arithmetic progression; i.e., the increase from the first to the second equals the increase from the second to the third, etc.

As you can see, the diagonals form a small inner pentagon and in the case of the sacred sign one of its angles is a right angle.

What are the five angles of the larger pentagon?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1885]

]]>We play a variation of a famous puzzle game using coins instead of rings. We start with a pile of coins consisting of at least one of each of the 1p, 2p, 5p, 10p, 20p, 50p and £1 coins, with no coin on top of another that is smaller in diameter. So the 5p coins are on the top, then the 1p, 20p, £1, 10p, 2p and 50p coins respectively, with the 50p coins being on the bottom. One typical pile is illustrated:

The object of the game is to move the pile to a new position one coin at a time. At any stage there can be up to three piles (in the original position, the final position, and one other). But in no pile can any coin ever be above another of smaller diameter.

We did this with a pile of coins recently and we found that the minimum number of moves needed equalled the value of the pile in pence. We then doubled the number of coins by adding some 1p, 5p and 50p coins totalling less than £3, and we repeated the game. Again the minimum number of moves equalled the value of the new pile in pence.

How many coins were in the pile for the first game?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1852]

]]>George and Martha possess two digital “clocks”, each having six digits. One displays the time on a 24-hour basis in the format hh mm ss, typically

15 18 45, and the other displays the date in the format dd mm yy, typically18 07 14.On one occasion, George walked into the room to find that the two “clocks” displayed identical readings. Martha commented that the long-term (400-year) average chance of that happening was 1 in just over a six-digit number. That six-digit number gives the birth date of one their daughters.

On what date was that daughter born?

[teaser3023]

]]>An “arithmetic” sequence is one in which each number is a fixed amount more than the previous one. So, for example, 10, 29, 48, … is an arithmetic sequence. In this case its ninth number is 162, which happens to be divisible by 9. I have in mind another arithmetic sequence whose ninth number is divisible by 9. This time it starts with two three-figure numbers, but in this case I have consistently replaced digits by letters, with different letters used for different digits.

The arithmetic sequence then begins

ONE,TWO, …, and its ninth number isNINE.To win, find the number to

WIN.

[teaser2681]

]]>George and Martha have been looking into tests for divisibility, including one for the elusive number seven. George wrote down a thousand-figure number by simply writing one particular non-zero digit a thousand times. Then he replaced the first and last digits by another non-zero digit to give him a thousand-figure number using just two different digits. Martha commented that the resulting number was divisible by seven. George added that it was actually divisible by exactly seven of 2, 3, 4, 5, 6, 7, 8, 9 and 11.

What were the first two digits of this number?

**Note:** This puzzle is marked as *flawed*, as under the intended interpretation there is no solution.

[teaser2677]

]]>Living next door to each other are two families each having three children. The product of the three ages in one family is equal to the product of those in the other family. Next month three of the six children, including the eldest, have birthdays, and at the end of the month the two products will again be equal.

None of the children has the same age either now or at the end of next month, and no child is older than twelve.

What are the ages in each family now?

[teaser6]

]]>There are 100 members of my sports club where we can play tennis, badminton, squash and table tennis (with table tennis being the least popular). Last week I reported to the secretary the numbers who participate in each of the four sports. The digits used overall in the four numbers were different and not zero.

The secretary wondered how many of the members were keen enough to play all four sports, but of course he was unable to work out that number from the four numbers I had given him. However, he used the four numbers to work out the minimum and the maximum possible numbers playing all four sports. His two answers were two-figure numbers, one being a multiple of the other.

How many played table tennis?

[teaser3022]

]]>In arithmetic, the zero has some delightful properties. For example:

ANY+NONE=SAME

X.ZERO=NONEIn that sum and product, digits have been replaced with letters, different letters being used for different digits. But nothing should be taken for granted: here the equal signs are only approximations as, in each case, the two sides may be equal or differ by one.

What is your

NAME?

[teaser2551]

]]>Four football teams — Albion, Borough, City and District — each play each other twice, once at home and once away. They get three points for a win and one for a draw. Last season, each team did worse in the away match than in the corresponding home match, scoring fewer goals and getting fewer points. The final position was as follows:

What were the two scores when Borough played District?

This is the final puzzle to use the title **Brainteaser**. The following puzzle is **Teaser 1848.**

[teaser1847]

]]>We wish to just completely cover a 6-by-6 square grid with T-shaped pieces made up of 4 squares as shown. No overlapping is allowed. We wish to do this in as many different ways as possible. Any pattern that can be obtained from another by rotation or reflection is regarded as the same pattern.

(a) In how many ways can the 6-by-6 grid be filled?

(b) In how many ways can a 10-by-10 grid be filled?

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1852]

]]>Farmer Giles had a rectangular field bordered by four fences that was 55 hectares in size. He divided the field into three by planting two hedges, from the mid-point of two fences to two corners of the field. He then planted two more hedges, from the mid-point of two fences to two corners of the field. All four hedges were straight, each starting at a different fence and finishing at a different corner.

What was the area of the largest field when the four hedges had been planted?

[teaser3021]

]]>George and Martha are in the pub. He has ordered a glass of lager for her and some ale for himself. He has written three numbers in increasing order, none involving a zero, then consistently replaced digits by letters to give:

DRANK

GLASS

LAGERGeorge explains this to Martha and tells her that the third number is in fact the sum of the previous two. From this information she is able to work out the value of each letter.

What is the number of George’s

ALE?

[teaser2705]

]]>Sixteen teams, A-P, entered a knockout football competition. In none of the matches did any team score more than five goals, no two matches had the same score, and extra time ensured that there were no draws.

The table shows the total goals, for and against, for each team:

My own team, K, was knocked out by the team who were the eventual champions.

Who played whom in the semi-finals, and what were the scores.

*Brainteasers* (2002, edited by Victor Bryant). The puzzle text above is taken from the book.

[teaser1839]

]]>This problem and its result illustrate why in courts of Law leading questions are frowned on and why two independent witnesses constitute strong evidence.

A pack of cards is shuffled and cut at random. The cut card is shown to two persons who tell the truth with probability of a half. They are asked to name the card, and independently, the both answer: “Five of Diamonds”.

What is the probability that the card is the Five of Diamonds?

[teaser5]

]]>“Bizarrers” dartboards have double and treble rings and twenty sectors ordered as on this normal dartboard [shown above]. However, a sector’s central angle (in degrees) is given by (100 divided by its basic score). The 20 sector incorporates the residual angle to complete 360º.

Each player starts on 501 and reduces this, eventually to 0 to win. After six three-dart throws, Baz’s seventh could win. His six totals were consecutive numbers. Each three-dart effort lodged a dart in each of three clockwise-adjacent sectors (hitting, in some order, a single zone, a double zone and a treble zone). [Each] three-sector angle sum (in degrees) exceeded that total.

The sectors scored in are calculable with certainty, but not how many times hit with certainty, except for one sector.

Which sector?

This puzzle uses a different spelling for “arraz” from the previous puzzle involving Baz (**Teaser 2934**).

[teaser3020]

]]>I have two rectangular maps depicting the same area, the larger map being one metre from west to east and 75cm from north to south. I’ve turned the smaller map face down, turned it 90 degrees and placed it in the bottom corner of the larger map with the north-east corner of the smaller map touching the south-west corner of the larger map. I have placed a pin through both maps, a whole number of centimetres from the western edge of the larger map. This pin goes through the same geographical point on both maps. On the larger map 1cm represents 1km. On the smaller map 1cm represents a certain whole number of kilometres …

… how many?

[teaser2678]

]]>This is Teaser 2700. The number 2700 is not particularly auspicious, but it does have one unusual property. Notice that if you write the number in words and you also express the number as a product of primes you get:

TWO THOUSAND SEVEN HUNDRED = 2 2 3 3 3 5 5

The number of letters on the left equals the sum of the factors on the right! This does happen occasionally. In particular it has happened for one odd-numbered Teaser in the past decade.

What is the number of that Teaser?

[teaser2700]

]]>The following is the test paper which Tom showed up:

8 + 7 = 62

59 + 4 = 50

5 + 3 = 5

11 × 1 = 55

12 + 8 = 23Naturally Miss Monk put five crosses on it. But Tom was indignant; for he was a self-taught boy and proud of his achievement. After some persuasion his teacher agreed to test him orally in addition and multiplication. This time Tom answered every question correctly.

Miss Monk was puzzled and then horrified when she discovered that although Tom had learnt to use arabic symbols he had ascribed to each an incorrect value. Fortunately he knew the meaning of + and × [and =].

If Tom were presented with the written problem: 751 × 7, what would his answer be?

[teaser4]

]]>William was searching for a number he could call his own. By consistently replacing digits with letters, he found a number represented by his name:

WILLIAM.He noticed that he could break

WILLIAMdown into three smaller numbers represented byWILL,IandAM, whereWILLandAMare prime numbers.He then calculated the product of the three numbers

WILL,IandAM.If I told you how many digits there are in the product, you would be able to determine the number represented by

WILLIAM.What number is represented by

WILLIAM?

[teaser3019]

]]>I have written down an addition sum and then consistently replaced digits by letters, with different letters used for different digits. The result is:

KILO+WATT=HOURAppropriately enough, I can also tell you that

WATTis a perfect power.Find the three-figure

LOWnumber.

[teaser2699]

]]>The five teams in our local league each play each other once in the course of the season. After seven of the games this season a league table was calculated, and part of it is shown below (with the teams in alphabetical order). Three points are awarded for a win and point [to each side] for a draw. In our table the digits 0 to 6 have been consistently replaced by letters.

Over half the matches so far have had a score of 1-1.

List all the results of the games (teams and scores).

[teaser1312]

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