*by John Owen*

#### Published Sunday August 04 2024

Ann, Beth and Chloe play a game with tiles. The prime numbers from 3 to 41 inclusive are written on 12 tiles, and each player receives 4 tiles. An odd starting number is chosen at random, then each player in turn tries, if possible, to increase the total to a prime number, by adding two of their tiles. For example, if the starting number is 5, Ann could play 11 and 31 to make the total 47, then Beth might be able to play 5 and 7 to make 59 and so on.

In a game where the starting number was 3, Ann was first, but couldn’t play. Beth and Chloe both took their turns, but again Ann was unable to play.

In ascending order, which four tiles did Beth and Chloe play?

*by Phil Jackson*

#### Published Friday July 19 2024 (link)

When installing the Chess Club cupboard in its new room, the members carelessly jammed it on the low ceiling and back wall, as shown from the side (not to scale).

Measurements were duly taken and the following were noted. The floor and ceiling were level and the back wall was vertical. The cupboard was a rectangular cuboid. The ceiling height was just 2cm greater than the cupboard height. Also, the depth of the cupboard from front to back was 148cm less than the cupboard height. Finally the point on the ceiling where the cupboard jammed was a distance from the back wall that was 125cm less than the cupboard height.

What is the cupboard height?

## Sunday Times Teaser 3226 – Prime Choice

*by Andrew Skidmore*

#### Published Thursday July 18 2024 (link)

I have written down three numbers, which together use each of the digits from 0 to 9 once.

All the numbers are greater than one hundred and the two smaller numbers are prime. I have also written down the product of the three numbers. If I told you how many digits the product contains you wouldn’t be able to tell me the value of the product. However, if I also told you whether the product is odd or even then you would be able to work out all the three numbers.

What, in increasing order, are my three numbers?

*by Bernardo Recamán*

#### Published Friday July 12 2024 (link)

When Daniela, the youngest of my nieces, was born not so long ago, her brother John remarked that Daniela’s year of birth was very special: no number added to the sum of that number’s digits was equal to her year of birth. The same was true for John’s year of birth.

“Well” intervened Lorena, the eldest of my nieces. “If you want to know, my own year of birth — earlier than yours — also shares that very same property, and so does the year of birth of our father Diego. The same is even true for our grandmother Anna, who, as you know, was born in the 1950s.”

When were Daniela, John, Lorena, Diego and Anna born?

*by Colin Vout*

#### Published Friday July 05 2024 (link)

On a particular stringed instrument, a fingering pattern of 2,2,3,1 means that the pitches of strings 1 to 4 are raised by that many steps respectively from the “open strings” (ie, their pitches when not fingered); this gives a “chord” of pitches C,E,G,A# in some order. The fingering pattern 4,1,0,x (for some whole number x from 0 to 11 inclusive) would give pitches F,A,C,D in some order, if these were all shifted by some fixed amount.

[Pitches range through C,C#,D,D#,E,F,F#,G,G#,A,A#,B, which then repeat.]

What are the pitches of “open strings” 1 to 4, and what is the value of x?

## Sunday Times Teaser 3223 – Shaping Up

*by Peter Good*

#### Published Friday **June** 28 2024 (link)

**June**

Clark wondered how many different shapes he could draw with identical squares joined edge to edge and each shape containing the same number of squares. He only drew five different shapes containing four squares, for example, because he ignored rotations and reflections (see diagram, above). He drew all of the different shapes containing 1, 2, 3, 4, 5 and 6 squares and wrote down the total number of different shapes in each case. He took the six totals in some order, without reordering the digits within any total, and placed them end to end to form a sequence of digits which could also be formed by placing six prime numbers end to end.

In ascending order, what were the six prime numbers?

## Sunday Times Teaser 3222 – Squarism

*by Mark Valentine*

#### Published Friday June 21 2024 (link)

There are 84 marbles, numbered sequentially from 1, in a bag. Oliver (going first) and Pam take turns, each removing two marbles from the bag. The player finds the “most square” integer factorisation of each number. The difference between the two factors for each number is then added to the player’s score (eg removing marbles 1 and 84 scores 5 points since 1=1×1 and 84=7×12), and the game ends when all marbles are removed.

After each of the first four turns (two each), both players’ expected final score was a whole number. For example, if there were 90 points remaining, with 4 turns left for Oliver and 5 for Pam, Oliver would expect to score another 40 points. In addition, each player’s score for their second turn was the same as their first. Also, Pamela scored the lowest possible game total.

What was Oliver’s expected final score after Pamela’s second turn?

## Sunday Times Teaser 3221 – Faulty Towers

*by Victor Bryant*

#### Published Friday June 14 2024 (link)

The famous “Tower of Hanoi” puzzle consists of a number of different-sized rings in a stack on a pole with no ring above a smaller one. There are two other poles on which they can be stacked and the puzzle is to move one ring at a time to another pole (with no ring ever above a smaller one) and to end up with the entire stack moved to another pole.

Unfortunately I have a faulty version of this puzzle in which two of the rings are of equal size. The minimum number of moves required to complete my puzzle (the two equal rings may end up in either order) consists of four different digits in increasing order.

What is the minimum number of moves required?

## Sunday Times Teaser 3220 – Graffiti

*by Angela Newing*

#### Published Friday June 07 2024 (link)

Six pupils were in detention having been caught painting graffiti on the school wall. The teacher decided to show them some examples of abstract art and produced seven different pictures in order **ABCDEFG** for them to study for a few minutes. Teacher took them back, showed them again upside down and in random order, and asked them to write down which of **ABCDEFG** each one was.

The answers were:

Helen |
A B C D E F G |

Ian |
A B C F E D G |

Jean |
A D F G C E B |

Kevin |
A D B F C E G |

Lee |
A F C G E D B |

Mike |
C A F B E D G |

It transpired that each picture had been correctly identified by at least one pupil, and everyone had a different number of correct answers.

What was the correct order for the upside down ones?

## Sunday Times Teaser 3219 – Number Funnel

*by Howard Williams*

#### Published Friday May 31 2024 (link)

Betty, being a keen mathematician, has devised a game to improve her children’s maths additions. She constructed an inverted triangular tiling of hexagonal shapes, with nine hexagons in the top row, eight in the second row etc, reducing to one at the bottom. Then, completing the top row with 0s and 1s, she asks them to complete each row below by adding in each hexagon shape the numbers in the two hexagons immediately above it.

In the last game she noticed that if the top row is considered as a base-2 binary number, then this is exactly four and a half times the bottom total.

What is the bottom total?