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Sep 17 21

Sunday Times Teaser 3078 – Digital Daisy-Chains

by Brian Gladman

by Victor Bryant

Published Sunday September 19 2021 (link)

The number 798 is a “digital daisy-chain”; ie, if you spell out each of its digits as a word, then the last letter of each digit is the first letter of the next. Furthermore, the number 182 is a “looped” digital daisy-chain because, in addition, the last letter of its last digit is the first letter of its first digit.

I have written down a large looped digital daisy-chain (with fewer than a thousand digits!). The total of its digits is itself a digital daisy-chain.

What is that total?

Sep 10 21

Sunday Times Teaser 3077 – Shuffling Series Schedules

by Brian Gladman

by Colin Vout

Published Sunday September 12 2021 (link)

A TV company planned a set of programmes to fill a weekly slot (one programme per week for many weeks) with six consecutive series of three different types (Arts, Biography and Comedy). None of the series was followed by another of the same type (eg, there could be an Arts series for three weeks then a Comedy series for four weeks and so on). Then it decided to change the order of the series within the same overall slot, but to minimise disruption it would not alter the gaps between series of the same type. It did this by scheduling each of the three Arts series 6 weeks earlier than first planned, each of the two Biography series 20 weeks later than first planned, and the Comedy series 21 weeks earlier than first planned.

How many programmes are there in each of the six series (listed in their final order)?

Sep 4 21

Sunday Times Teaser 3076 – Bee Lines

by Brian Gladman

by Nick MacKinnon

Published Sunday September 05 2021 (link)

Three bees are trapped inside three empty cuboidal boxes of different sizes, none of whose faces are squares. The lengths of the edges of each box in centimetres are whole numbers, and the volume of each box is no more than a litre. Starting at a corner, each bee moves only in straight lines, from corner to corner, until it has moved along every edge of its box. The only points a bee visits more than once are corners of its box, and the total distance moved by each bee is a whole number of centimetres. Given the above, the sum of these three distances is as small as it could be.

What is the sum of the distances that the bees moved?

Aug 28 21

Sunday Times Teaser 3075 – Prime Cuts for Dinner

by Brian Gladman

by Peter Good

Published Sunday August 29 2021 (link)

Tickets to the club dinner were sequentially numbered 1, 2, …, etc. and every ticket was sold. The number of guests for dinner was the highest common factor of three different two-figure numbers and the lowest common multiple of three different two-figure numbers. There were several dinner tables, each with the same number of seats, couples being seated with friends. The guests on each table added their ticket numbers together and obtained one of two prime numbers, both less than 150, but if I told you the larger prime number you would not be able to determine the other.

What was the larger of the two prime numbers?

Aug 20 21

Sunday Times Teaser 3074 – Timely Overthrows

by Brian Gladman

by Howard Williams

Published Sunday August 22 2021 (link)

Without changing their size, Judith sews together one-foot squares of different colours that her mother has knitted, to make rectangular throws. These are usually all of the same dimensions using fewer than a hundred squares. She has observed that it takes her mother 20 per cent longer to knit each square than it takes her to sew two single squares together.

As a one-off she has completed a square throw whose sides have the same number of squares as the longer side of her usual rectangular throws. The average time it took per square foot, both knitting and sewing, to complete the square throw was 2 per cent longer than that of the rectangular throws.

What are the dimensions in feet of the rectangular throws?

Aug 14 21

Sunday Times Teaser 3073 – Snookered

by Brian Gladman

by Nick MacKinnon

Published Sunday August 15 2021 (link)

The playing surface of a snooker table is a twelve-foot by six-foot rectangle. A ball is placed at P on the bottom cushion (which is six feet wide) and hit so it bounces off the left cushion, right cushion and into the top-left pocket.

Now the ball is replaced at P and hit so it bounces off the left cushion, top cushion and into the bottom right pocket, after travelling 30% further than the first shot took. The ball always comes away from the cushion at the same angle that it hits the cushion.

How far did the ball travel on the second shot?

Aug 6 21

Sunday Times Teaser 3072 – Dial M for Marriage

by Brian Gladman

by Danny Roth

Published Sunday August 08 2021 (link)

George and Martha work in a town where phone numbers have seven digits. “That’s rare!” commented George. “If you look at your work number and mine, both exhibit only four digits of the possible ten (0-9 inclusive), each appearing at least once. Furthermore, the number of possible phone numbers with that property has just four single-digit prime number factors (each raised to a power where necessary) and those four numbers are the ones in our phone numbers.”

“And that is not all!” added Martha. “Both numbers have their highest digits first, working their way down to the lowest, and both give perfect numbers [1] when their digits are summed.”

What are two phone numbers?

[1] A perfect number equals the sum of its factors, eg, 6 = 1 + 2 + 3

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?