Jane said:

My mother and I were each married on our birthday and each at the same age. I was born four years and a day after my mother’s marriage, and my twins were born four years and a day after my marriage to John, who is three years older than I am, and who shares a birthday with my mother.

If you write a date, say Christmas Day this year, in a continuous line of figures it looks like this – 25121961. Very well, if you write down our five dates of birth in that way and add the resultant numbers, the total is 8829685.

When was her wedding day?

[teaser31]

]]>The six rose bushes in my garden lie on a circle. When they were very small, I measured the six internal angles of the hexagon that the bushes form. These were three-digit whole numbers of degrees. In a list of them, of the ten digits from 0 to 9, only one digit is used more than once and only one digit is not used at all. Further examination of the list reveals that it contains a perfect power and two prime numbers.

In degrees, what were the smallest and largest angles?

[teaser3056]

]]>I have written down five positive whole numbers whose sum is less than 100. If you wrote the numbers in words, then you would find that each of them begins with a different letter of the alphabet. (Surprisingly, the same is true of the five numbers obtained by increasing each of my five numbers by one). If you write my five numbers in words and put them in alphabetical order, then they will be in decreasing order.

What (in decreasing order) are my five numbers?

[teaser2786]

]]>In a football tournament each country played each other country twice, the scores in all twelve matches being different.

The records for the top and bottom teams were:

England beat Wales twice by the same margin as she beat Ireland once.

The sum of the aggregate number of goals scored against Scotland, who finished second, and Ireland was 20.

What were the respective scores in the Ireland vs. Scotland matches?

[teaser30]

]]>Four friends and I live in the same town, one of us at the town centre, and the others at places due north, south, east and west of the town centre. Our names are North, South, East, West and Middle, but we do not necessarily live at the places which accord with our names.

In visiting one another we use the only connecting roads which run north-south and east-west through the town centre.

Before last year, when North and I exchanged houses (to accommodate his increasing family, mine by then having left home), I lived farther north than West, who lives farther east than Middle, who lives farther west than East. North lived farther east than South. (When visiting East, North had to turn right at the town centre, but I could go straight ahead when visiting North).

What is my name, and who lives in the north, east, south, west and middle positions respectively?

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.

[teaser736]

]]>The dartboard at the

Trial and Arrowpub is rather different from the standard one: there are only 3 sectors, each with a positive whole number label, and no central bullseye scoring region. There are still double and treble rings: for instance, if the sector label is 3, a dart in that sector can score 3, 6 or 9.As usual, scores are counted down from a starting value, the idea being to finish (“check out”) by reaching exactly zero. Players take turns throwing three darts, or fewer if they check out before that. Unusually, the checkout doesn’t have to finish on a double.

The lowest impossible checkout is the smallest value that can’t be achieved in one turn; on this board that value is 36.

What are the sector labels?

[teaser3055]

]]>Last month I told you about Uncle Bill’s gifts to his nephews. Uncle Ben has also sent a whole number of pounds (less than fifty) to be shared among his three nephews Tom, Dick and Harry. Each has received a different whole number of pounds, with Tom receiving the most and Harry the least, but with Tom getting less than twice as much as Harry. Each boy’s fraction of the total gift, when expressed as a decimal, consists of three different digits recurring (as in 0.

abcabc…), and each boy’s decimal uses the same three digits.How much did Tom, Dick and Harry get from Uncle Ben?

[teaser2789]

]]>Uncle Bill has sent a whole number of pounds (less than fifty) to be shared among his three nephews Tom, Dick and Harry. Each has received a whole number of pounds, with Tom receiving the most and Harry the least, but with Tom getting less than twice as much as Harry. Each boy’s fraction of the total gift, when expressed as a decimal, consists of three digits recurring (as in 0.

abcabc…), and the nine digits that appear in the three decimals are all different. (Uncle Ben also sent some money, but I’ll tell you about that next month).How much did Tom, Dick and Harry get from Uncle Bill?

[teaser2785]

]]>My birthday was on Sunday last. As I drove my car out of the garage in the morning I noticed that the mileage read the same as the date, 11061 or 1.10.61. The car is driven to the station and back twice every day of the year and is used for no other purpose, the total distance each week being exactly 178 miles. By a coincidence the next occasion when the mileage at some time during the day will read the same as the date will be on my husband’s birthday.

When is this?

[teaser29]

]]>My kitchen floor is tiled with identically-sized equilateral triangle tiles while the floor of the bathroom is tiled with identically-sized regular hexagon tiles, the tiles being less than 1m across. In both cases the gaps between tiles are negligible. After much experimenting I found that a circular disc dropped at random onto either the kitchen or bathroom floor had exactly the same (non-zero) chance of landing on just one tile.

The length of each side of the triangular tiles and the length of each side of the hexagon tiles are both even triangular numbers of mm (i.e., of the form 1+2+3+…).

What are the lengths of the sides of the triangular and hexagonal tiles?

[teaser3054]

]]>Spiders Beth and Sam wake up in the bottom corner of a cuboidal barn (all of whose sides are whole numbers of metres). They want to reach the opposite bottom corner without actually walking across the floor. Beth decides to walk on one of five possible shortest routes, two of them being around the edge of the floor and the other three being over the walls and ceiling. Sam decides instead to spin a web directly to the point on the ceiling diagonally opposite the starting point and then to drop down into the corner. The total length of his journey is within five centimetres of a whole number of metres.

How high is the barn?

[teaser2782]

]]>Five girls entered our Village Beauty Contest. Each of the five judges had to put them into a definite order and then allot fifteen marks between them. The aggregate marks for each girl decided the issue. The judges each gave a different maximum mark and also chose a different girl to be top. No girl had two identical marks in her score.

Joe gave the maximum possible to Rose, and he placed Gwen above Linda. Tom gave Ann one more than Sam did. Pam had no zero in her score and although she finished ahead of Rose she didn’t win. Ann scored the only five that was allotted. Dan placed the girls as closely together as he could.

The judge who put Gwen first put Rose last. Brian succeeded in putting them all in their correct final order.

Who won, and what were her individual marks from each judge?

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.

[teaser730]

]]>Your problem this week is to find an unusual nine-digit number. It comprises the digits from 1 to 9, in some order, each used once and only once.

The number formed by the first digit (reading from the left) is exactly divisible by 1 (which doesn’t tell you a lot!), the number formed by the first two digits is exactly divisible by 2, that formed by the first three digits is exactly divisible by 3, and so on, which the number formed by the first eight digits being divisible by 8, and with the complete number of nine digits being divisible by 9.

It is perhaps surprising that such a number exists, and even more surprising that is should be unique.

What is the number?

[teaser1040]

]]>Kublis Ghen was very particular about his army formations. Originally, each company consisted of a certain number of men who could be drawn up in the form of a perfect square. Nor was this all, for when the companies were drawn up one behind the other (each company being spread out to form a single row) the entire army itself thus constituted a square. It was an army to be proud of, but when the great conqueror determined to attack Thalbazzar he was not content, and summoned his chief of staff: “My army is not big enough”, he declared. “Double it”.

Knowing the temper of his master, the chief of staff saw to it that the size of the army was doubled — exactly. But an unforeseen difficulty arose: the army could no longer form a perfect square — there was just one man too many.

“Kill him”, ordered the conqueror on hearing the news, and the offending supernumerary was duly dispatched, so that the army marched into battle in the form of a square though, of course, its company formations had been completely disorganised.

A million men did Kublis Ghen

Against Thalbazzar thow;says the poet, but that is an exaggeration.

How many men were in the army that Kublis Ghen threw against Thalbazzar?

[teaser28]

]]>George and Martha have a telephone number consisting of nine digits; there is no zero and the others appear once each. The total of the digits is obviously 45, so that the number is divisible by nine. Martha noticed that, if she removed the last (i.e., the least significant) digit, an eight-digit number would remain, divisible by eight. George added that you could continue this process, removing the least significant digit each time to be left with an

n-digit number divisible bynright down to the end.What is their telephone number?

[teaser3053]

]]>Waking in the night I glanced at my bed-side clock and thought it indicated just after 2:20. Putting on my spectacles and looking more carefully I saw that it was actually just after 4:10. I had, of course, interchanged the hands at my first glance.

I began to wonder what time around then that my observation would have occurred, to the exact fraction of a minute, if the hands could have been exactly interchanged.

What do you think?

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.

[teaser717]

]]>I have written down some numbers and then consistently replaced digits by capital letters, with different letters used for different digits. In this way my numbers have become:

TRIPLE(which is a multiple of three)

EIGHT(which is a cube)

NINE(which is divisible by nine)

PRIME(which is a prime)What is the number

TIME?

[teaser2781]

]]>Mr and Mrs Herbert Fogg, in celebration of Herbert’s forty-third birthday, invited three friends, Ann, Bill, and Cuthbert, to dinner.

“Of course, Herbert is quite a bit older than any one of you”, Mrs Fogg declared to the three guests. Herbert frowned.

“It’s odd”, Ann quickly interposed, “if you multiply Bill’s and Cuthbert’s ages together, then divide by Herbert’s age the remainder is just my age”.

“Whereas if you multiply together Ann’s and Cuthbert’s ages, again divide by Herbert’s age, then the remainder is just how old I was a year ago”, added Bill.

“Not so simple for me”, said Cuthbert, the youngest present. “If you multiply together Ann’s and Bill’s ages, divide by Herbert’s, then the remainder is just one and a half times my age”.

“Very interesting”, Herbert reflected. “In fact, from what you have told me it may be possible to find all three of your ages, if I knew how to start!”

Show that Herbert was almost right and find the exact ages of Ann, Bill, and Cuthbert.

[teaser27]

]]>

“Argent bend sinister abased sable in dexter chief a hog enraged proper” blazons our shield (shaped as a square atop a semi-circle, with a 45° diagonal black band meeting the top corner). We’ve three shields. For the first, in centimetres, the top width (

L) is an odd perfect cube and the vertical edge height of the band (V) is an odd two-figure product of two different primes. The others have, in inches, whole-numberL(under two feet) andVvalues (all different). For each shield, the two white zones have almost identical areas. All threeV/Lvalues, in percent, round to the same prime number.Give the shortest top width, in inches.

[teaser3052]

]]>Last time it was vertical. But no one could accuse Uncle Bungle of being consistent and this time it was horizontal. The way, I mean, in which he tore the piece of paper on which were written the details of the matches between 4 local football teams,

A,B,C, andD, who are to play each other once.All that was left was:

It is not known whether all the matches have been played. And not more than 7 goals were scored in any game.

With the information that it is possible to discover the score in each match, you should be able to discover them.

What was the score in each match?

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

[teaser702]

]]>A letter to

The Timesconcerning the inflated costs of projects read:When I was a financial controller, I found that multiplying original cost estimates by π used to give an excellent indication of the final outcome.

Interestingly, I used the same process (using 22/7 as a good approximation for π). On one occasion, the original estimate was a whole number of pounds (less than £100,000), and this method for the probable final outcome gave a number of pounds consisting of exactly the same digits, but in the reverse order.

What was the original estimate?

When originally published the amount was given as “less than £10,000”, which was raised to “less than £100,000” in the following issue. But even with this change the puzzle is still open to interpretation.

[teaser2529]

]]>Farmer Green was in the red. On studying his statement (which was entirely in complete £s) he found that the aggregate of the figures in his deposits total was only one-fourth of the aggregate of the figures in his withdrawals total. He also noted that, coincidentally, the latter aggregate equals the balance, in red, that would be shown when he has paid in the £749 cheque which he, fortunately, had received that morning.

What was the overdraft shown on the statement?

[teaser26]

]]>“Here is the shipping forecast for the regions surrounding our island.

First, the coastal regions:

Hegroom:E 6, rough, drizzle, moderate.

Forkpoynt:E 7, rough, rain, good.

Angler:NE 7, rough, drizzle, moderate.

Dace:NE 7, smooth, drizzle, moderate.Now, the offshore regions:

Back:E gale 8, rough, rain, moderate.

Greigh:E 7, smooth, drizzle, poor.

Intarsia:SE 7, rough, drizzle, moderate.

Catter:E gale 8, high, drizzle, moderate.

Eighties:E 7, rough, fair, good.”In this forecast, no element jumps from one extreme to another between adjacent regions.

The extremes are:

wind direction:SE & NE;

wind strength:6 & gale 8;

sea state:smooth & high;

weather:fair & rain;

visibility:good & poor.Each region adjoins four others (meeting at just one point doesn’t count).

Which of the regions does Angler touch?

[teaser3051]

]]>I have tried to make a calendar using some dice. To display the month I want to use three dice, with a capital letter on each of the faces to display:

[

JAN] or [FEB] or [MAR] etc.I chose the capital letters on the dice to enable me to go as far as possible through the year. Furthermore, it turned out that one particular die contained four vowels.

(a) What was the last month that I was able to display?

(b) What were the two other letters on that particular die?

[teaser2780]

]]>The Lotaseetas have a rather casual attitude to commerce. Every Monday morning, Ming, the rice-planter, gathers from the grove outside his bungalow a number of mangoes of uniform weight. For the rest of the week he uses these as weights for measuring out the rice on his scales, charging each customer according to the number of mangoes required to balance the weight purchased.

Unfortunately, the mangoes themselves lose a fixed percentage of their weight each day by evaporation. However, Ming roughly compensates for this by using two mangoes as the unit on Wednesdays and Thursdays and three on Fridays. Clearly, Wednesday is a good day for buying rice, and Tuesday is a bad day.

His first customer at precisely 10 o’clock each morning (they are late risers) is Fung, the rice merchant, who always buys the same quantity of rice for his shop. On Monday Fung’s rice cost him 243 cowries. On Friday it cost him 256 cowries.

How much did it cost him on Tuesday?

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

[teaser699]

]]>“These examination results show that either the knowledge of mathematics, physics, and chemistry throughout the school is deplorable weak, or the papers were very stiff”, said the headmaster to the staff concerned.

“Unfortunately, I have mislaid the detailed list, but some of the figures are easy to remember. The number of pupils taking the three subjects was 440. In chemistry 200 passed; in physics 210 failed; and in mathematics 220 passed. Of those who passed in chemistry 66 failed in mathematics, whereas of those who passed in mathematics 22 failed in physics. I cannot recall the number of pupils who passed in all three subjects, or the number who failed in all three, but both these numbers were perfect squares”. At this stage the senior mathematics master got out his pencil and paper and started to puzzle it out.

(1) How many pupils failed in all three subjects?

(2) Of those who passed in chemistry, how many also passed in physics?

[teaser25]

]]>Given any number, one can calculate how close it is to a perfect square or how close it is to a power of 2. For example, the number 7 is twice as far from its nearest perfect square as it is from its nearest power of 2. On the other hand, the number 40 is twice as far from its nearest power of 2 as it is from its nearest square.

I have quite easily found a larger number (odd and less than a million!) for which one of these distances is twice the other.

What is my number?

[teaser3050]

]]>We are having a “cold turkey” party on Boxing Day. Fewer than 100 people have indicated that they are coming, and the percentage of them choosing the vegetarian option is (to the nearest whole number) a single-digit number. My vegetarian aunt might also come. If she does, then (to the nearest whole number) the percentage having the vegetarian option will remain the same.

If she does come, how many people will be there and how many of them will have the vegetarian option?

[teaser2778]

]]>You see, Inspector, the combination of my safe is a six-figure number. In case anyone needed to get into it while I was away, I gave each of my clerks (Atkins, Browning and Clark) one of the two-figure numbers which make up the combination. I also told each the position in the combination of the number of another clerk, but not the number itself.

Browning must have overheard me telling a friend that it is a coincidence that two of these numbers are squares and if you put them together you get a four-figure number that equals the other clerk’s number squared. I remember I also said something about whether or not the combination is divisible by this clerk’s number.

When he was caught, Browning said, “I can’t understand why the alarm went off; I know Clark’s is the first number”. I later realised that what I’d told my friend about whether or not that other number was a factor was wrong, which was lucky for me as Browning had got his own number in the right place.

What was the combination?

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

[teaser696]

]]>Someone here in Land’s End Lane of the name of Roger Gray (or is it “Grey”? I never can remember which), seems gratified to find that by allotting distinctive digits against appropriate letters of the alphabet he can, by substituted figures for letters, make an addition sum of his name, the answer to which, converted back into letters in similar manner, spells his house number.

There seems nothing particularly surprising in this, for if it really is “Gray” and if he happened to live at No. 7, as I do, all he would have to do would be to write:

and substitute say:

However, he doesn’t live at No. 7, and moreover, I happen to know that there’s an 8 in his sum.

So where does Roger live, and what is his sum?

[teaser24]

]]>Jed farms a large, flat, square area of land. He has planted trees at the corners of the plot and all the way round the perimeter; they are an equal whole number of yards apart.

The number of trees is in fact equal to the total number of acres (1 acre is 4840 square yards). If I told you an even digit in the number of trees you should be able to work out how far apart the trees are.

How many yards are there between adjacent trees?

[teaser3049]

]]>It was a frightened, breathless, but very charming young lady who knocked on my office door late one night.

“They have gone”, she said: “The Rajah’s rubies are no longer in their ancestral home”.

I visited the scene of the crime and discovered a tattered piece of paper on which was written:

Where were the jewels hidden?

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

[teaser846]

]]>I asked Harry and Tom to write down three numbers that between them used nine different digits and which added to 2015. They each succeeded and one of Harry’s three numbers was the same as one of Tom’s. I noticed that Harry’s three numbers included a perfect square and Tom’s included a higher perfect square.

What were those two squares?

[teaser2777]

]]>A man divided 8s. 4d. among his three sons so that the difference between the shares of the eldest and youngest, expressed in halfpence, was a perfect square. The total of Alan’s and Bob’s shared was an exact multiple of 5½d.; the total of Bob’s and Clive’s shares an exact multiple of 6½d.; and the total of Alan’s and Clive’s shares an exact multiple of 9½d.

What was the name of the second son, and what was his share?

[teaser23]

]]>Our holiday rep, Nero, explained that in Carregnos an eight-digit total of car registrations results from combinations of three Greek capital letters after four numerals (e.g. 1234 ΩΘΦ), because some letters of the 24-letter alphabet and some numerals (including zero) are not permitted.

For his own “cherished” registration the number tetrad is the rank order of the letter triad within a list of all permitted letter triads ordered alphabetically. Furthermore, all permitted numeral tetrads can form such “cherished” registrations, but fewer than half of the permitted letter triads can.

Nero asked me to guess the numbers of permitted letters and numerals. He told me that I was right and wrong respectively, but then I deduced the permitted numerals.

List the permitted numerals in ascending order.

[teaser3048]

]]>I have won three Premium Bond prizes and noted the number of non-winning months between my first and second wins, and also the number between my second and third wins. Looking at the letters in the spelling of the months, I have also noted the difference between the numbers of letters in the months of my first and second wins, and also the difference between those of the months of my second and third wins. All four numbers noted were the same, and if you knew that number then it would be possible to work out the months of my wins.

What (in order of the wins) were those three months?

[teaser2776]

]]>On Trafalgar Day each of the five Sea Lords will take over a post now occupied by one of the others. Each predicts what will happen; those whose predictions are right will get more senior posts, and those whose predictions are wrong will get more junior posts.

The most junior speaks first:

Fifth Sea Lord:“Two Sea Lords will make a direct exchange.”

Fourth Sea Lord:“The Third Sea Lord will become the Second Sea Lord.”

Third Seal Lord:“A man who makes a true prediction will take over my job.”Of the First and Second Sea Lords each predicts the same future new post for himself.

Which post is that?

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

[teaser692]

]]>

Across:

1a.Square of a multiple of 11.

5a.A fourth power.

8a.A multiple of2d.

9a.A cube reversed.

11a.If added to4dit gives a square.

12a.3 times a prime number.

14a.A multiple of 11.

16a.A prime number.

17a.A cube, which is also the sum of three cubes.

18a.An even number.

20a.A multiple of 12.

21a.A prime number.

22a.The square of a palindromic number.

Down:

1d.The product of two consecutive numbers (neither a prime).

2d.See8a.

3d.A cube.

4d.See11a.

6d.The sum of the digits of3d.

7d.The square of (6dreversed).

10d.The product of two consecutive odd numbers.

13d.The product of two consecutive even numbers.

15d.A cube.

16d.A square.

19d.An even multiple of (18a+21a).

[teaser21]

]]>I gave Robbie three different, non-zero digits and asked him to add up all the different three-digit permutations he could make from them. As a check for him, I said that there should be three 3’s in his total. I then added two more [non-zero] digits to the [original three digits] to make [a set of] five digits, all being different, and asked Robbie’s mother to add up all the possible five-digit permutations of these digits. Again, as a check, I told her that the total should include five 6’s.

Given the above, the product of the five digits was as small as possible.

What, in ascending order, are the five digits?

I have changed the text of this puzzle slightly to make it clearer.

[teaser3047]

]]>It was typical of Uncle Bungle that he should have torn up the sheet of paper which gave particulars of the numbers of matches played, won, lost, drawn, etc. of four local football teams who were eventually going to play each other once. The only piece left was as shown (as usual there are 2 points for a win and 1 for a draw):

It will not surprise those who know my Uncle to hear that one of the figures was wrong, but fortunately it was only one out (i.e. one more or less than the correct figure).

Each side had played at least one game, and not more than seven goals were scored in any match.

Calling the teams

A,B,CandD(in that order), find the score in each match.

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

[teaser691]

]]>Four celebrities entered a dance competition. Five judges each shared out their eight marks among the four dancers, with each getting a non-zero whole number. Each judge split the eight marks in a different way and then allocated them as follows. Amanda’s marks to Lana and Natasha added to the same total as Barry’s marks to Madge and Paula. Barry gave more marks to Madge than to any other dancer, Charles gave more to Paula than to any other, and Doris gave more to Natasha than to any other. Lana scored more from Edna than from Amanda. All dancers had the same total so the head judge’s scores were used, giving a winner and runner-up.

Who was the head judge, who was the winner and who was the runner-up?

[teaser2775]

]]>Some problems that look simple enough at first can prove to be remarkably tricky. Consider, for instance, the kite pictured above. The shaded areas are squares of equal size, the sides of each square being 15 inches. The width of the kite from A to B is exactly 42 inches.

What is the length of the kite from C to D?

[teaser20]

]]>George and Martha run a business in which there are 22 departments. Each department has a perfect-square three-digit extension number, i.e., everything from 100 (10²) to 961 (31²), and all five of their daughters are employees in separate departments.

Andrea, Bertha, Caroline, Dorothy and Elizabeth have extensions in numerical order, with Andrea having the lowest number and Elizabeth the highest. George commented that, if you look at those of Andrea, Bertha and Dorothy, all nine positive digits appear once. Martha added that the total of the five extensions was also a perfect square.

What is Caroline’s extension?

[teaser3046]

]]>Ashley, Bill, Charles, David, and Edward are (not necessarily in that order), a dustman, a grocer, a miner, a blacksmith, and an artist, and all live on the right hand side of Strife Lane, in even numbered houses. All five are of different ages and no man has reached the age of retirement (65). All of course are upright and honest citizens, and never tell lies. However, I had forgotten what job each man did, where he lived, and how old he was, and so, to help me, each man volunteered the following statements:

Ashley:

(1) The artist lives at No. 10, next to Charles;

(2) Nobody lives next to the grocer, although Bill is only two doors away.

Bill:

(3) I am the only man whose age is indivisible by 9;

(4) I am 4 years older than Ashley;

Charles:

(5) The blacksmith’s age is 5 times his house number;

David:

(6) The miner lives 4 houses higher up the road from me;

(7) The miner’s age is 3 times the dustman’s house number, but he is two-thirds the dustman’s age;

Edward:

(8) The dustman is twice as old as David;

(9) I am the oldest man in the street.At what number does Ashley live?

How old is the grocer?

Who is the artist?

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

[teaser688]

]]>I have two traditional-looking dice, but only one of them is fair. In the other there is an equal chance of getting a 1, 2, 3, 4 or 5, but the die is loaded so that a 6 is thrown more than half the time. I threw the two dice and noted the total. It turned out that with my dice the chance of getting that total was double what it would have been with two fair dice.

What (as a simple fraction) is the chance of getting a 6 with the loaded die?

[teaser2774]

]]>Awaiting guests on poker night, Tel placed (using only clubs, face-up, in order left-to-right) the Ace (=1) to 9 (representing numerals), then interspersed the Ten, Jack, Queen and King (representing –, +, × and ÷ respectively) in some order, but none together, among these.

This “arithmetic expression” included a value over 1000 and more even than odd numbers. Applying

BEDMASrules, as follows, gave a whole-number answer. “NoBrackets or powErs, so traverse the expression, left-to-right, doing eachDivision orMultiplication, as encountered, then, again left-to-right, doing eachAddition orSubtraction, as encountered.”Tel’s auntie switched the King and Queen. A higher whole-number answer arose. Tel displayed the higher answer as a five-card “flush” in spades (the Jack for + followed by four numeral cards).

Give each answer.

[teaser3045]

]]>We were visiting the island state of Kimbu and had come to the post-office to send off some parcels to friends at home. The island’s currency is the pim, and the postmaster told us that he had only stamps of five different face-values, as these had to be used up before a new issue of stamps was introduced.

These stamps were black, red, green, violet, and yellow, in descending order of values, the black being the highest denomination and the yellow the lowest.

One parcel required stamps to the value of 100 pims and we were handed 9 stamps; 5 black, one green, and 3 violet. The other two parcels required 50 pims’ worth each, and for these we were given two different sets of 9 stamps.

One consisted of 1 black and 2 of each of the other colours, and the other set contained 5 green and 1 of each of the others.

What would be been the smallest number of stamps needed for a 50-pim parcel, and of which colours?

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

[teaser685]

]]>Adam noticed that today’s Teaser number was a four-figure palindrome. He told me that he had written down [a] four-figure palindrome and he asked me to guess what it was. I asked for some clues regarding its prime factors and so he told me, in turn, whether his number was divisible by 2, 3, 5, 7, 11, 13, … Only after he had told me whether it was divisible by 13 was it possible to work out his number.

What was his number?

When this was originally published “another” was used where I have placed “[a]”, but this makes the puzzle unsolvable. However, as presented above, there is a unique solution.

[teaser2772]

]]>My friend, Mr. Little, is a farmer whose chief interest is in sheep, but keeps a number of Shorthorn and Redpoll (polled) cattle too.

Mr. Little is six years older than his wife and 33 years older than his son, George. George is twice as old as Mary, the daughter of the house, whose birthday happens to be today.

There is no tractor at Hillside Farm and Mr. Little does the ploughing behind his two horses leaving a furrow nine inches wide. The number of acres ploughed this year happens to be the same as the number of cats on the farm.

Readers are invited to determine the information required to complete the following “cross-number” puzzle:

Across:

1a:Number of Mr. Little’s Redpoll cows.

3a:Square of the number of Redpoll cows.

5a:Total number of cows.

6a:Mr. Little’s age.

7a:Number of ewes per ram in Mr. Little’s flock. This also happens to be the number of miles walked by Mr. Little in doing this year’s ploughing.

9a:Acreage of rough grass land. (1.5 acres per ewe).

11a:Age of Mrs. Little.

12a:Number of ewes in flock (in scores).

13a:Cube of the number of collies on the farm.

Down:

1d:Area of rectangular farmyard in square yards.

2d:Length of farmyard in yards.

3d:Number of ewes on the farm.

4d:Age at which Mr. Little intends to retire.

7d:Seven times Mr. Little’s age.

8d:Total number of sheep.

10d:Number of rams on the farm.

12d:Total number of horns possessed by all the cattle.

[teaser19]

]]>The kids had used all the blocks each of them owned (fewer than 100 each) to build triangular peaks — one block on the top row, two on the next row, and so on.

“My red one’s nicer!”

“My blue one’s taller!”

“Why don’t you both work together to make one bigger still?” I said.

I could see they could use all these blocks to make another triangle.

This kept them quiet until I heard, “I bet Dad could buy some yellow blocks to build a triangle bigger than either of ours was, or a red and yellow triangle, or a yellow and blue triangle, with no blocks ever left over.”

This kept me quiet, until I worked out that I could.

How many red, blue and yellow blocks would there be?

[teaser3044]

]]>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?

*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 remained 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?

*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?

*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]

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