*by Mark Valentine*

#### Published Sunday February 05 2023 (link)

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?

## Sunday Times Teaser 3149 – Cube Route

*by Andrew Skidmore*

#### Published Sunday January 29 2023 (link)

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.

*by Edmund Marshall*

#### Published Sunday January 22 2023 (link)

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?

## Sunday Times Teaser 3147 – Noteworthy

*by Victor Bryant*

#### Published Sunday January 15 2023 (link)

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?

*by John Owen*

#### Published Sunday January 08 2023

In our curling league (between 4 and 26 teams), each team plays each other once. Teams are ranked according to the number of wins (draws are impossible). If any teams are tied on wins, ranking is only possible if those teams have different numbers of wins in their mutual games. For example, in a three-way tie if A beats B, B beats C and A beats C, the ranking is ABC, but if C beats A (or A has not yet played C), then ranking is impossible, as A and B have one win each.

At one point (each team had played G games), ranking the teams as above was possible. However, if each team had played G-1 games, a ranking would have been impossible, irrespective of results. With one more team in the league, the minimum number of games needed to allow a ranking is G+2.

How many teams are in the league and what was the value of G?

*by Danny Roth*

#### Published Sunday January 01 2023 (link)

“I have forgotten the phone number!” complained Martha, about to phone a friend. “So have I!” replied George, “but I have some vague memories of it. It is a perfect square with all the digits different, and the last digit is equal to the number of digits to be dialled. The last-but-one digit is odd and one of the digits is zero. Also the second and third and last-but-one digits are all exact multiples of the first digit. Maybe you can work it out.”

Martha proceeded to dial the number correctly.

What number did she dial?

*by Stephen Hogg*

#### Published Sunday December 25 2022 (link)

On the NHS vaccine booking website, my queue position updated every minute. My initial position was an odd value in the 9000s. This value had four different digits, and its leftmost digit and just two others in it were factors of it. Curiously, these properties applied to each of the decreasing four-figure values after the next eight updates. I also noticed that no value had a digit in common with its predecessor, no zeroes occurred before the fourth update and no nines after this.

My son, told just the above, found all the possible values from which a complete set could be selected. From these he was certain of more than one of the nine queue positions.

Give these certain positions in displayed order.

## Sunday Times Teaser 3143 – Pipe Fittings

*by Peter Good*

#### Published Sunday December 18 2022 (link)

A plumber had three thin metal pipes with square, rectangular and elliptical cross-sections. In order to fit them into his van, he slid the rectangular pipe inside the elliptical pipe and the elliptical pipe inside the square pipe, before placing the pipe assembly in the van. There are four points where the pipes all touch, as shown in the diagram. The maximum and minimum widths of the elliptical and rectangular pipes and the diagonal width of the square pipe were all even numbers of mm less than 1,000, of which one was a perfect square.

What were the five widths (in increasing order)?

*by Colin Vout*

#### Published Sunday December 11 2022 (link)

I’m planting four differently-coloured plots in a line. Mrs Hyde’s design principles classify eight colours into five categories. Adjacent plots mustn’t have the same category, or even certain pairs of categories.

“Every scheme you’ve suggested has at least one adjacent pair of categories that breaks my rules”, said Mrs Hyde. “White, Green, Yellow, White has Honest and Social adjacent; Blue, Orange, Pink, Violet has Ardent and Social adjacent; Green, Violet, Blue, Orange has Ardent and Honest adjacent; Violet, White, Orange, Red has Droll and Honest adjacent; and White, Violet, Green, Blue has Jolly and Social adjacent. However, if you change one colour in one of your suggestions then it will work.”

What are the four colours in order in the changed suggestion that would be allowable?

*by Bill Kinally*

#### Published Sunday December 04 2022 (link)

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

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

What was the larger extension number he dialled?