*by Stephen Hogg*

#### Published Sunday February 21 2021 (link)

Our holiday rep, Nero, explained that in Carregnos an eight-digit total of car registrations results from combinations of three Greek capital letters after four numerals (eg 1234 ΩΘΦ), because some letters of the 24-letter alphabet and some numerals (including zero) are not permitted.

For his own “cherished” registration the number tetrad is the rank order of the letter triad within a list of all permitted letter triads ordered alphabetically. Furthermore, all permitted numeral tetrads can form such “cherished” registrations, but fewer than half of the permitted letter triads can.

Nero asked me to guess the numbers of permitted letters and numerals. He told me that I was right and wrong respectively, but then I deduced the permitted numerals.

List these numerals in ascending order

*by Howard Williams*

#### Published Sunday February 14 2021 (link)

I gave Robbie three different, single digit, positive whole numbers and asked him to add up all the different three-digit permutations he could make from them. As a check for him, I said that there should be three threes in his total. I then added two more digits to the number to make it five digits long, all being different, and asked Robbie’s mother to add up all the possible five-digit permutations of these digits. Again, as a check, I told her that the total should include five sixes.

Given the above, the product of the five numbers was as small as possible.

What, in ascending order, are the five numbers?

*by Danny Roth*

#### Published Sunday February 07 2021 (link)

George and Martha run a business in which there are 22 departments. Each department has a perfect-square three-digit extension number, ie, everything from 100 (10²) to 961 (31²), and all five of their daughters are employees in separate departments.

Andrea, Bertha, Caroline, Dorothy and Elizabeth have extensions in numerical order, with Andrea having the lowest number and Elizabeth the highest. George commented that, if you look at those of Andrea, Bertha and Dorothy, all nine positive digits appear once. Martha added that the total of the five extensions was also a perfect square.

What is Caroline’s extension?

*by Stephen Hogg*

#### Published Sunday January 31 2021 (link)

Awaiting guests on poker night, Tel placed (using only clubs, face-up, in order left-to-right) the Ace (=1) to 9 (representing numerals), then interspersed the Ten, Jack, Queen and King (representing -, +, x and ÷ respectively) in some order, but none together, among these.

This “arithmetic expression” included a value over 1000 and more even than odd numbers. Applying * BEDMAS* rules, as follows, gave a whole-number answer. “No

**rackets or pow**

*B***rs, so traverse the expression, left-to-right, doing each**

*E***ivision or**

*D***ultiplication, as encountered, then, again left-to-right, doing each**

*M***ddition or**

*A***ubtraction, as encountered.”**

*S*Tel’s auntie switched the King and Queen. A higher whole-number answer arose. Tel displayed the higher answer as a five-card “flush” in spades (the Jack for + followed by four numeral cards).

Give each answer.

*by Colin Vout*

#### Published Sunday January 24 2021 (link)

The kids had used all the blocks each of them owned (fewer than 100 each) to build triangular peaks — one block on the top row, two on the next row, and so on.

“My red one’s nicer!” “My blue one’s taller!”

“Why don’t you both work together to make one bigger still?” I said. I could see they could use all these blocks to make another triangle.

This kept them quiet until I heard, “I bet Dad could buy some yellow blocks to build a triangle bigger than either of ours was, or a red and yellow triangle, or a yellow and blue triangle, with no blocks ever left over.”

This kept me quiet, until I worked out that I could.

How many red, blue and yellow blocks would there be?

*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?