*by Victor Bryant*

#### Published Sunday January 17 2021 (link)

I wrote an odd digit in each of the sixteen cells of a four-by-four grid, with no repeated digit in any row or column, and with each odd digit appearing three or more times overall. Then I could read four four-figure numbers across the grid and four four-figure numbers down. I calculated the average of the four across numbers and the larger average of the four down numbers. Each was a whole number consisting entirely of odd digits, and each used an odd number of different odd digits.

What were those two averages?

*by R. Postill*

#### From The Sunday Times, 3rd March 1974

My typewriter had the standard keyboard:

row 1: QWERTYUIOP

row 2: ASDFGHJKL

row 3: ZXCVBNM

until I was persuaded by a time-and-motion expert to have it ‘improved’. When it came back I found that none of the letters was in its original row, though the number of letters per row remaining unchanged. The expert assured me that, once I got used to the new system, it would save hours.

I tested it on various words connected with my occupation — I am a licensed victualler — with the following results. The figures in parentheses indicate how many rows I had to use to produce the word:

BEER (1)

STOUT (1)

SHERRY (2)

WHISKY (3)

HOCK (2)

LAGER (2)

VODKA (2)

CAMPARI (2)

CIDER (3)

FLAGON (2)

SQUASH (2, but would have been 1 but for a single letter)

Despite feeling a trifle MUZZY (a word which I was able to type using two rows) I persevered, next essaying CHAMBERTIN.

Which rows, in order, did I use?

*by Colin Vout*

#### Published Sunday January 10 2021 (link)

The modernist music of Skaredahora eschewed traditional scales; instead he built scales up from strict mathematical rules.

The familiar major scale uses 7 notes chosen from the 12 pitches forming an octave. The notes are in (1) or out of (0) the scale in the pattern 101011010101, which then repeats. Six of these notes have another note exactly 7 steps above (maybe in the next repeat).

He wanted a different scale using 6 notes from the 12 pitches, with exactly two notes having another note 1 above, and one having another 5 above. Some notes could be involved in these pairings more than once.

His favourite scale was the one satisfying these rules that came first numerically when written out with 0s & 1s, starting with a 1.

What was Skaredahora’s favourite scale?

*by Mark Bryant*

#### From New Scientist #1752, 19th January 1991

In Susan Denham’s Enigma last week she related details of an unusual frame of snooker (and she reminded readers of the basic rules). Reading that teaser reminded me of a frame which I once saw between the two great players Dean and Verger.

Dean had a break; Verger followed this with a break of 1 point fewer; Dean followed this with a break of 1 point fewer; Verger followed this with a break of 1 point fewer; and so on, up to and including the final clearance.

In any one break no colour (apart from the red) was potted more than once. Dean never potted the blue. Each player potted the black twice as many times as his opponent potted the yellow. The brown was potted more times than the green.

How many times (in total) did the winner pot each of the colours? (Red, Yellow, Green, Brown, Blue, Pink, Black).

*by Stephen Hogg*

#### Published Sunday January 03 2021 (link)

For Christmas 1966 I got 200 Montini building blocks; a World Cup Subbuteo set; and a Johnny Seven multi-gun. I built a battleship on the “Wembley pitch” using every block, then launched seven missiles at it from the gun. The best game ever!

Each missile blasted a different prime number of blocks off the “pitch” (fewer than remained). After each shot, in order, the number of blocks left on the “pitch” was: a prime; a square; a cube; a square greater than 1 times a prime; a cube greater than 1 times a prime; none of the aforementioned; and a prime.

The above would still be valid if the numbers blasted off by the sixth and seventh shots were swapped.

How many blocks remained on the “pitch” after the seventh shot?

*by Susan Denham*

#### From New Scientist #1751, 12th January 1991

In snooker there are 15 red balls worth one point each. If a player pots a red it stays in the hole and he (or she) is allowed to try to pot one of the colours yellow, green, brown, blue, pink or black (worth 2-7 points in that order). If a colour is potted it is brought out again and the player can try for another red, and so on. This continues until all the reds have gone. Then the remaining six colours are potted in ascending order.

The total points achieved in one such run is called a “break”. For example:

red + pink + red + black + red

would be a break of 16.

Having completed the break, the player sits down and lets the other player try for a red and continue the break, and so on. At the end the winning player is the one with the higher grand total of points. A player may choose not to pot the final black if the result is already determined without it. No other rules concern today’s story.

Stephens and Hendry play a frame of snooker. Stephens starts with a break of 3, Hendry follows with a break of 4, Stephens with a break of 5, and so on, and this pattern continues to the end. As usual in quality snooker, the black was potted more times than the yellow, and the pink was potted more times that the blue.

How many times did Stephens pot the brown? And how many times did Hendry pot the brown?

## Sunday Times Teaser 3040 – Moving Digit

*by Andrew Skidmore*

#### Published Sunday December 27 2020 (link)

Jonny has opened a new bank account and has set up a telephone PIN. His sort code is comprised of the set of three two-figure numbers with the smallest sum which give his PIN as their product. He was surprised to find that the PIN was also the result of dividing his eight-figure account number by one of the three two-figure numbers in the sort code.

The PIN has an unusual feature which Jonny describes as a moving digit. If the number is divided by its first digit then the number which results has the same digits in the same order except that first digit is now at the end.

The account number does not contain the digit which moved.

What is the account number?

*by Howard Williams*

*by Howard Williams*

#### Published Sunday December 20 2020 (links)

James is laying foot-square stones in a rectangular block whose short side is less than 25 feet. He divides this area into three rectangles by drawing two parallel lines between the longest sides and into each of these three areas he lays a similar pattern.

The pattern consists of a band or bands of red stones laid concentrically around the outside of the rectangles with the centre filled with white stones. The number of red stone bands is different in each of the rectangles but in each of them the number of white stones used equals the number of outer {1] red stones used.

The total number of stones required for each colour is a triangular number (ie, one of the form 1+2+3+…).

What is the total area in square feet of the block?

[1] this word is unnecessary (it also creates an ambiguity).

*by Danny Roth*

*by Danny Roth*

#### Published Sunday December 13 2020 (link)

George and Martha were participating in the local village raffle. 1000 tickets were sold, numbered normally from 1 to 1000, and they bought five each. George noticed that the lowest-numbered of his tickets was a single digit, then each subsequent number was the same multiple of the previous number, eg, 7 21 63 189 567. Martha’s lowest number was also a single digit, but her numbers proceeded with a constant difference, eg, 6 23 40 57 74. Each added together all their numbers and found the same sum. Furthermore, the total of all the digits in their ten numbers was a perfect square.

What was the highest numbered of the ten tickets?

*by John Owen*

#### Published Sunday December 06 (link)

Last year I was given a mathematical Advent calendar with 24 doors arranged in four rows and six columns, and I opened one door each day, starting on December 1. Behind each door is an illustrated prime number, and the numbers increase each day. The numbers have been arranged so that once all the doors have been opened, the sum of the numbers in each row is the same, and likewise for the six columns. Given the above, the sum of all the prime numbers is as small as it can be.

On the 24th, I opened the last door to find the number 107.

In order, what numbers did I find on the 20th, 21st, 22nd and 23rd?