A circus elephant standing on one end of a see-saw pivoted at the centre was balanced by a troupe of fewer than 20 acrobats, each of equal weight, standing on the other end. The elephant moved forwards and several acrobats jumped off to maintain balance. The elephant moved backwards and some of them climbed back on to the end to rebalance.

The elephant always moved a prime number of feet and there was always a prime number of acrobats on the see-saw. If I told you how far backwards the elephant moved you could work out the numbers of acrobats.

(In equilibrium, the product of weight and distance from pivot point must be the same on both sides)

How far did the elephant move backwards, and how many acrobats are there in the troupe?

]]>I have written down an above-freezing temperature, a whole number of degrees Celsius, in which the digits are all different and are in decreasing order. I have then calculated the Fahrenheit equivalent. It is also a whole number whose digits are all different, but here the digits are in increasing order.

If I told you the first digit of the Celsius temperature, then you would not be able to calculate the temperature. However, bearing that in mind, if I now told you the final digit of the Celsius temperature, then it would be possible to calculate it.

You should now be able to work out the Celsius and Fahrenheit temperatures.

What are they?

]]>Jack told Gill “I have found three equally-spaced prime numbers 29, 41, and 53. The difference between the first and second is the same as the difference between the second and third, and there are no repeated digits in the six digits of my primes”. Gill told Jack she had also found three equally-spaced primes, each having three digits and with no repeated digits in the nine digits of her primes. She said, “If I told you that the three-digit sum of each of my primes is an odd number then you should be able to find them”.

In ascending order what are Gill’s three primes?

]]>Jeb’s 25x25km square ranch had his house’s flagpole at the ranch’s pole of inaccessibility (the point whence the shortest distance to a boundary fence was maximised).

At 50, Jeb gave each of his four sons a triangular tract of his land, with corners whole numbers of km along boundary fences from each corner post (as illustrated, not to scale). Each tract’s area (in square km) equalled that son’s age last birthday (all over 19, but under 30). All the tracts’ perimeters differed, and each son set his hacienda’s flagpole at his own tract’s pole of inaccessibility.

Curiously, for Jeb’s new octagonal ranch the pole of inaccessibility and the shortest distance from this to a boundary fence were unchanged.

Give the eldest son’s shortest distance from his flagpole to a boundary fence.

]]>I really like the postcards that a friend of mine has sent me each of the last three years from her annual holiday in Farflung. Each card had six stamps on it, involving the same four denominations, and at least one of each denomination was present on each card. These four denominations were all different and coloured differently, and of course represented whole numbers of pecahans. I can’t read what the denominations are, but my friend had remarked the postage totals for the three years were 23, 32 and 36 pecahans.

What were the four denominations, in decreasing order?

]]>A garden centre bought 500 plants of four different varieties from its supplier. The price per plant of each variety was a whole number of pence and their total average price worked out at exactly one pound.

The number of plants of variety 2 purchased was “d” greater than that of variety 1, and its price per plant was “d” pence less than that of variety 1. Similarly, the number of variety 3 plants equalled the number of variety 2 plus “d” and its price equalled the variety 2 price less “d” pence. Finally, the number of variety 4 plants equalled the number of variety 3 plus “d” and its price equalled the variety 3 price less “d” pence.

What, in pence, is the most that a plant could have cost?

]]>Edward, the sports shop owner, had an annual display of tennis balls. He arranged the balls in four identical pyramids, with a square base holding each in place (one ball on the top of each pyramid, four on the layer below, nine below that and so on).

However, this year he wanted to display the same number of balls but reduce the total footprint of the bases by at least 55 per cent, to allow for other stock. His son Fred suggested arranging all the balls in one large pyramid with an equilateral triangular base (one ball on the top, three on the layer below, six below that and so on). Edward realised that this would work, but if there were any fewer balls, it wouldn’t work.

How many balls did Edward display?

]]>I have a set of ten cards, each of which has a different digit written on it. All the cards have been used to make a set of prime numbers. After discarding the smallest prime, and without changing the order of any cards, I have placed the remaining primes in order of decreasing size to give a large number. It is possible, without changing the order of any cards, to break this number into a set composed entirely of cubes. Neither set contains a number with more than four digits.

List, in order of decreasing size, my set of prime numbers.

]]>Each of four contending couples in a quiz game has equal probability of elimination at the end of each of the first three rounds, one couple going after each round. In the fourth round, the remaining couple has a constant probability p, less than ½, of winning the jackpot, which consists of £1000 in the first game; if the jackpot is not won, it is added to the £1000 donated in the next game. Each couple may enter three successive games of the quiz, except that any couple having played for the jackpot in the fourth round of any game then withdraws altogether, being replaced by a new couple in the next game.

If the probability that a couple, competing from the first game, wins £2000 is 7/96, what is the value of p as a fraction?

]]>Apparently in Costa Lotta a single-digit percentage of banknotes are forgeries and so I have designed a marker pen which tests whether notes are genuine. I thought it would be quite useful to the banks because, on average, for every N uses it only gives an incorrect result once (where N is some whole number).

Unfortunately my design has been abandoned by the banks because it turns out that on average for every N occasions on which the pen indicates a forgery, only one of the notes will in fact be forged!

What is N?

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