I drove down a road with a number of petrol stations whose locations I saw on my map. I decided to check the price at the first station then fill up when I found one offering a lower price (or, failing that, the last one).

When I got home I noticed that I could arrange the prices (in pence per per litre) into an ascending sequence of consecutive whole numbers of pence, plus 0.9p (i.e. 130.9p, 131.9p, 132.2p, etc). I also worked out the average price that I would expect to pay using this strategy, if I were to encounter this set of prices in an unknown order, and I was surprised to find that this value turned out to be a whole number of pence per litre.

How many petrol stations were there?

When the puzzle was originally published in *The Sunday Times* it had been edited into a form that meant there were no solutions. Here I have rephrased the middle paragraph so that it works as the setter originally intended.

[teaser2566]

]]>A putting game has a board with eight rectangular holes, like the example (not to scale), but with the holes in a different order.

If you hit your ball (diameter 4cm) through a hole without touching the sides, you score the number of points above that hole. The sum of score and width (in cm) for each hole is 15; there are 2cm gaps between holes.

I know that if I aim at a point on the board, then the ball’s centre will arrive at the board within 12cm of my point of aim, and is equally likely to arrive at any point in that range. Therefore, I aim at the one point that maximises my long-term average score. This point is the centre of a hole and my average score is a whole number.

(a) Which hole do I aim at?

(b) Which two holes are either side of it?

[teaser3010]

]]>Do you have a little time to try this Teaser? I have taken a four-figure number and a six-figure number and I have consistently replaced digits by letters to give the words

LITTLEandTIME.If you take the digits of

TIMEand write down all the four-figure numbers which use exactly those digits in some order and then add up all those numbers, then your total will beLITTLE.What number is

TIME?

[teaser2863]

]]>“What’s inside it?” asked the Mole wriggling with curiosity.

“There’s cold chicken inside it”, replied the Rat briefly: “cold-tongue-cold-ham-cold-beef-pickled-gherkins-salad-french-rolls-cress-sandwiches-potted-meat-ginger-beer-lemonade-soda-water…”

“Oh, stop”, cried the Mole in ecstasies. “This is too much for one picnic. We can have another tomorrow on what’s left”.

“Do you really think so?” inquired the Rat seriously. “Let’s see. There’s only salad-pickled-gherkins-french-rolls-and-soda-water enough for two days: so if we have ham today we’ll have beef tomorrow; if we have potted meat today we’ll have cress sandwiches tomorrow; and if we have tongue today we’ll have lemonade tomorrow”.

“If we save the cress sandwiches for tomorrow we’ll have the beef today; if we keep the potted meat for tomorrow we’ll have the ginger beer today; and if we keep the lemonade for tomorrow we’ll have the ham today”. The Mole was entering into the spirit of the thing.

“In any event we’ll have the lemonade and ginger beer on different days, and likewise the beef and the chicken”, Rat shrieked excitedly.

“And if we have the chicken and cress sandwiches together, we’ll have the potted meat the day after we have the tongue”. The Mole rolled on his back at the prospect. “And we’ll eat every scrap”.

Which of the eight items did they save for the second day?

[teaser520]

]]>My grandson and I play a simple coin game. In the first round we toss seven coins and I predict how many “heads” there will be whilst my grandson predicts the number of “tails”. After the tossing I score a point for each head plus a bonus of ten if my prediction was correct — my grandson scores likewise for the tails. We then repeat this with six coins, then five, and so on down to a single coin. After each round we keep cumulative totals of our scores. In one game, for over half of the pairs of cumulative scores, my grandson’s total was like mine but with the digits in reverse order. In fact he was ahead throughout and at one stage his cumulative total was double mine — and he had an even bigger numerical lead than that on just one earlier occasion and one later occasion.

List the number of heads in each of the seven rounds.

[teaser3009]

]]>I started with a rectangle of paper. With one straight cut I divided it into a square and a rectangle. I put the square to one side and started again with the remaining rectangle. With one straight cut I divided it into a square and a rectangle. I put the square (which was smaller than the previous one) to one side and started again with the remaining rectangle. I kept repeating this process (discarding a smaller square each time) until eventually the remaining rectangle was itself a square and it had sides of length one centimetre. So overall I had divided the original piece of paper into squares. The average area of the squares was a two-figure number of square centimetres.

What were the dimensions of the original rectangle?

[teaser2864]

]]>The owner of the old curiosity shop repaired an antique mechanical fruit machine having three wheels of identical size and format. Afterwards each wheel was independently fair, just as when new. Each wheel’s rim had several equal-sized zones, each making a two-figure whole number of degrees angle around the rim. Each wheel had just one zone showing a cherry, with other fruits displayed having each a different single-figure number (other than one) of zone repetitions.

Inside the machine were printed all the fair chances (as fractions) of getting three of the same fruit symbol in one go. Each of these fractions had a top number equal to 1 and, of their bottom numbers, more than one was odd.

What was the bottom number of the chance for three cherries?

[teaser3008]

]]>I have a modern painting by the surrealist artist Doolali. It is called “Seventh Heaven” and it consists of a triangle with green sides and a red spot on each of its sides. The red spots are one seventh of the way along each side as you pass clockwise around the triangle. Then each of the red spots is joined by a straight blue line to the opposite corner of the triangle. These three blue lines create a new triangle within the original one and the new triangle has area 100 sq cm.

What is the area of the green triangle?

[teaser2865]

]]>James has decided to lay square block paving stones on his rectangular patio. He has calculated that starting from the outside and working towards the middle that he can lay a recurring concentric pattern of four bands of red stones, then three bands of grey stones, followed by a single yellow stone band. By repeating this pattern and working towards the centre he is able to finish in the middle with a single line of yellow stones to complete the patio.

He requires 402 stones to complete the first outermost band. He also calculates that he will require exactly 5 times the number of red stones as he does yellow stones.

How many red stones does he require?

[teaser3007]

]]>In 2009 George and Martha had a four-figure number of pounds in a special savings account (interest being paid into a separate current account). At the end of the year they decided to give some of it away, the gift being shared equally among their seven grandchildren, with each grandchild getting a whole number of pounds. At the end of the following year they did a similar thing with a different-sized gift, but again each grandchild received an equal whole number of pounds. They have repeated this procedure at the end of every year since.

The surprising thing is that, at all times, the number of pounds in the savings account has been a perfect cube.

What is the largest single gift received by any grandchild?

[teaser2869]

]]>My bank PIN consists of four different digits in decreasing order. I used this PIN to help me choose my six lottery numbers. I wrote down all the two-figure numbers that used two different digits from the PIN. Just six of those numbers were in the range from 10 to 49 and so they were my lottery choices. In fact the sum of the six is a perfect square. If you knew that square it would now be possible to work out my PIN.

What is my PIN?

[teaser2870]

]]>At the village of Badberg, hidden away in the Alps, there is a town clock of a certain antiquity. The maker was a wealthy but inexpert amateur. After the ornate instrument had been installed it was found that the great hands stood still during the equal intervals between each stroke of the hour on the massive bell. Naturally, the clock was always slow.

The placid villagers became accustomed to the erratic behaviour of their timepiece; only after the death of its donor did his nephew dare to tackle the problem. Finding it impossible to alter the striking mechanism, he ingeniously modified the movement to run at a higher constant speed so that the hands showed the correct time at least when the clock struck certain hours.

Unfortunately, the hands still stop for the same period between successive strokes of the bell, but the villages can now see and hear the correct time every six hours.

At what hours does the clock make its first stroke correctly?

[teaser519]

]]>At our local bridge club dinner we were each given a raffle ticket. The tickets were numbered from 1 to 80. There were six people on our table and all our numbers were either prime or could be expressed as the product of non-repeating primes (e.g. 18 = 2×3×3 is not allowed). In writing down the six numbers you would use each of the digits 0 to 9 once only. If I told you the sum of the six numbers (a perfect power) you should be able to identify the numbers.

List the numbers (in ascending order).

[teaser3006]

]]>In this subtraction sum I have consistently replaced digits with letters, different letters being used for different digits:

BONFIRE–TOFFEE=TREATSWhat is the number of

ENTRIES?

[teaser2876]

]]>I have written down a very large number (but with fewer than twenty-five digits). If you spell out each of its digits as a word, then for each digit its last letter is the same as the first letter of the next digit (the last letter of the last digit being the same as the first letter of the first digit). One example of such a number would be 83821.

Neither my number nor the sum of its digits is a palindromic number (but in fact the original number is one more than a palindrome).

What is my number?

[teaser2593]

]]>“I have affixed a card to each of your foreheads”, explained the professor of logic to his three most able students. All three were perfect logicians capable of instantly deducing the logical consequences of any statement.

He continued: “You can see the others’ cards, but none of you can see your own. On each card is written a positive whole number and one of the numbers is the sum of the other two”.

The professor turned to the first student, Albert: “From what you can see and what you have heard, can you deduce the number on your card?”. Albert could not.

The professor then asked Bertrand the same question. He could not deduce the number on his card.

Likewise, when asked next, Charles could not deduce the number on his card.

The professor then asked Albert the same question again and this time Albert was able to deduce that the number on his card was 50.

What number was on Bertrand’s card, and what number was on Charles’s card?

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

[teaser1814]

]]>Ten equal-length, rigid tubes, each a different prime-valued external radius from 11 mm to 43 mm, were baled, broadside, by placing the 43 mm and 11 mm tube together and the third tube, not the largest remaining, touching both of these. Each subsequent tube touched the previous tube placed and the 43 mm tube. A sub-millimetre gap between the final tube placed and the 11 mm tube, made a near perfect fit.

The radius sum of the first three tubes placed against the 43 mm tube was a multiple of one of the summed radii. Curiously, that statement remains true when each of “four”, “five”, “seven” and “eight” replaces “three”. For “two” and “six” tubes placed their radius sum was a multiple of an as yet unplaced tube’s radius.

What radius tube, in mm, was placed last?

[teaser3005]

]]>George and Martha are running a holiday camp and their four daughters are staying there. To keep the peace they have been given chalets in different areas. Amelia’s chalet number is between 1 and 99, Bertha’s is between 100 and 199, Caroline’s between 200 and 299, and Dorothy’s between 300 and 399.

George commented that the difference between any two of the four chalet numbers is either a square or a cube. Martha added that the same could be said for the sum of the chalet numbers of the three youngest daughters.

Who is the eldest daughter and what is her chalet number?

[teaser2591]

]]>An interesting fragment tells us what took place at the meeting of the Five Prophets:

“If Hosea hath prophesied truth, Micah and Obadiah will die in the same city; if Micah hath prophesied truth, Joel and Obadiah will die in the same city. It is the saying of Joel that three of the five prophets shall die in Babylon; and Nahum declareth to Hosea that the two of them shall die in different cities.”

It is well known that those who prophesy correctly die in Jerusalem, whereas those who make false prophecies die in Babylon.

So how many of these five will die in Jerusalem?

[teaser518]

]]>In our football league, the teams all play each other once, with three points for a win and one for a draw. At the end of the season, the two teams with most points are promoted, goal difference being used to separate teams with the same number of points.

Last season’s climax was exciting. With just two games left for each team, there were several teams tied at the top of the league with the same number of points. One of them, but only one, could be certain of promotion if they won their two games. If there had been any more teams on the same number of points, then none could have guaranteed promotion with two wins.

How many teams were tied at the top of the league, and how many of the remaining matches involved any of those teams?

[teaser3004]

]]>