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Aug 1 21

Sunday Times Teaser 3071 – Three-Cornered Problem

by Brian Gladman

by Oliver Tailby

Published Sunday August 01 2021 (link)

I have a set of equal-sized equilateral triangles, white on one face and black on the back. On the white side of each triangle I have written a number in each corner. Overall the numbers run from 1 to my age (which is less than thirty). If you picture any such triangle then it occurs exactly once in my set (for example, there is just one triangle containing the numbers 1, 1 and 2; but there are two triangles containing the numbers 1, 2 and 3).

The number of triangles that contain at least one even number is even.

The number of triangles that contain at least one odd number is odd.

The number of triangles that contain at least one multiple of four is a multiple of four.

How old am I?

Jul 27 21

New Scientist Enigma 886 – Set Square

by Brian Gladman

by Susan Denham

From Issue #2041, 3rd August 1996 (link)

Last week I watched a thrilling five-set tennis match between the two top players, Pampas and Grassy. Pampas won the first set easily and the second in a tie-break. He then lost the next two sets and towards the end of the final set the scoreboard showing the games won looked like this:

\[\begin{array}{|l|c|c|c|c|c|}\hline \mathbf{Pampas} & \mathbf{6} & \mathbf{7} & \mathbf{5} & \mathbf{0}& \mathbf{7} \\
\hline \mathbf{Grassy} & \mathbf{7} & \mathbf{2} & \mathbf{3} & \mathbf{6}& \mathbf{1} \\
\hline \end{array}\]

Pampas then went on to win the next two games (and hence the match).

I remember on a previous occasion when they met the match also went to five sets. Towards the end of the match I looked at the scoreboard and each of the two rows of games won formed a five-figure perfect square. On that occasion Grassy then went on to win in two more games.

What did the score-board look like at the very end of that match?

Jul 23 21

Sunday Times Teaser 3070 – Film Binge

by Brian Gladman

by Colin Vout

Published Sunday July 25 2021 (link)

I’m going to have a day binge-watching films. My shortlist is:

Quaver, starring Amerton, Dunino, Emstrey, Fairwater and Joyford;

Rathripe, starring Amerton, Codrington, Fairwater,Glanton and Ingleby;

Statecraft, starring Amerton, Codrington, Dunino, Hendy and Ingleby;

Transponder, starring Codrington, Dunino, Emstrey, Hendy and Ingleby;

Underpassion, starring Blankney, Emstrey, Fairwater, Hendy and Ingleby;

Vexatious, starring Amerton, Blankney, Dunino, Emstrey and Joyford;

Wergild, starring Blankney, Fairwater, Glanton, Hendy and Joyford;

X-axis, starring Blankney, Codrington, Fairwater, Glanton and Ingleby;

Yarborough, starring Blankney, Dunino, Glanton, Hendy and Joyford;

Zeuxis, starring Amerton, Codrington, Emstrey, Glanton and Joyford.

I dislike Ingleby and Joyford, so I don’t want either of them in more than two films; but I want to see each of the others in at least two films.

Which are the films I should watch?

Jul 17 21

Sunday Times Teaser 3069 – Fit for Purpose

by Brian Gladman

by Peter Good

Published Sunday July 18 2021 (link)

George and Martha bought a new toy for their son Clark. It consisted of a rectangular plastic tray with dimensions 15x16cm and eight plastic rectangles with dimensions 1x2cm, 2x3cm, 3x4cm, 4x5cm, 5x6cm, 6x7cm, 7x8cm and 8x9cm. The rectangles had to be placed inside the tray without any gaps or overlaps. Clark found every possible solution and he noticed that the number of different solutions which could not be rotated or reflected to look like any of the others was the same as his age in years.

How old was Clark?

Jul 9 21

Sunday Times Teaser 3068 – Valued Playwrights

by Brian Gladman

by Victor Bryant

Published Sunday July 11 2021 (link)

I have given each letter of the alphabet a different whole-number value from 1 to 26. For example, P=4, L=8, A=3 and Y=24. With my numbers I can work out the value of any word by adding up the values of its letters, for example the word PLAY has a value of 39.

It turns out that the playwrights:

BECKETT, FRAYN, PIRANDELLO, RATTIGAN, SHAKESPEARE and SHAW

all have the same prime value.

Also COWARD, PINERO and STOPPARD have prime values.

What are these three prime numbers?

Jul 6 21

New Scientist Enigma 631 – Fidget Digit

by Brian Gladman

by Susan Denham

From Issue #1785, 7th September 1991 (link)

My niece (whose age is a two-figure number) is very good at arithmetic, and to keep her occupied on a recent train journey I asked her to find a number with all its digits different and with the sum of its digits a multiple (more than one times) of her age.

She wrote down a list of lots of number with both those properties. So I then asked her to add 1 to each of her numbers, and to pick out from the new list each number which still had all its digits different and with the sum of its digits equal to a multiple (again, more than one times) of her age next birthday.

There were still quite a few numbers in this new list which had these properties. So I asked her to find one of them which, when multiplied by her age, gave an answer which still had all its digits different — which she did!

How old is she?

Jul 2 21

Sunday Times Teaser 3067 – Reciprocal Arrangement

by Brian Gladman

by Colin Vout

Published Sunday July 04 2021 (link)

Twelve men from our football squad had turned up for training, and I’d promised them a game of six-a-side at the end of the session; so while they were off on a gentle three-mile run I worked out what the two teams would be. They were wearing their squad numbers, which had one or two digits: 2, 3, 4, 5, 6, 7, 8, 9, 15, and three others. It appealed to me when I found that I could divide them into two teams of six, such that the sum of the reciprocals of the squad numbers in each team equalled one exactly.

What were the squad numbers in the team containing number 2?

Jun 25 21

Sunday Times Teaser 3066 – Leaning Tower of Pesos

by Brian Gladman

by Howard Williams

Published Sunday June 27 2021 (link)

I have six old coins worth an odd number of pesos, comprising a mixture of one- and two-peso coins. Both denominations are of the same diameter and made of the same metal, but the two-peso coins are twice the thickness of the one-peso coins.

After making a vertical stack of the coins I then slid each of the top five coins as far as possible to the right, to make the pile lean as much as possible in that direction, without toppling. I managed to get the rightmost edge of the top coin a distance of one and a quarter times its diameter further right than the rightmost edge of the bottom coin.

Starting from the top, what is the value of each of the six coins?

Jun 18 21

Sunday Times Teaser 3065 – Members of the Jury

by Brian Gladman

by Danny Roth

Published Sunday June 20 2021 (link)

A jury had twelve members, all with different ages (at least 20 but not more than 65), except that two were twins with a common age over 40. The average age was a prime number. A counsel objected to one of the members and he was replaced by another with the result that the average age was reduced to another prime number. Between the two juries, there were twelve different ages and they formed a progression with a common difference (eg, 1, 4, 7, 10, 13, etc. or 6, 13, 20, 27, 34, etc.,). None of the individuals had a perfect square age, and the replacement jury still included both twins.

How old were the twins?

Jun 11 21

Sunday Times Teaser 3064 – Turnip Prize

by Brian Gladman

by Victor Bryant

Published Sunday June 13 2021 (link)

The Turnip Prize is awarded to the best piece of work by an artist under fifty. This year’s winning entry consisted of a mobile made up of many different plain white rectangular or square tiles hanging from the ceiling. The sides of the tiles were all whole numbers of centimetres up to and including the artist’s age, and there was precisely one tile of each such possible size (where, for example, a 3-by-2 rectangle would be the same as a 2-by-3 rectangle). Last week one of the tiles fell and smashed and then yesterday another tile fell and smashed. However, the average area of the hanging tiles remained the same throughout.

How old is the artist?