*by Miss W. M. Jeffree*

#### Published 23rd December 1956 (link)

Each square is to be occupied by either a zero or a cross, which stands for one particular digit which you may discover for yourself.

No row or column begins with a zero.

The following clues give the prime factors of the various rows and columns, each letter representing a different prime:

Rows:

(i) | a b c d |

(ii) | a² b² e f |

(iii) | a b² g h |

(iv) | a b i j² k |

(v) | a² b e j l |

(vi) | a b² i k m |

Columns:

(i) | a b² i j k m |

(ii) | a² b e i j² k |

(iii) | a b² m n p |

(iv) | a² b c d e |

(v) | a b i j k l |

*by Stephen Hogg*

#### Published Sunday November 28 2021 (link)

My four toilet rolls had zigzag-cut remnants (not unlike that shown). Curiously, each polygonal remnant had a different two-figure prime number of sides, each with a digit sum itself prime.

Calculating the average number of sides for the four remnants and the average of their squares, I found that the average of the squares minus the square of the average had a value whose square root (the “standard deviation”) was a whole number.

I also noticed that a regular convex polygon with a number of sides equal to the average number of sides would have an odd whole-number internal angle (in degrees).

Give the “standard deviation”.

## Sunday Times Teaser 3087 – Hexagonia

*by Nick MacKinnon*

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

Hexagonia is a hexagonal republic and is divided into 24 numbered counties, as shown. The counties are to be reorganised into four departments, each composed of six edge-connected counties. No two departments will have the same shape or be reflections of each other, and the president’s residence in Department A will be built on an axis of symmetry of the department. Every sum of the county numbers in a department will be, in the prime minister’s honour, a prime number, and her mansion will be built in the middle of a county in Department B, on an axis of symmetry of the department, and as far as possible from the president’s residence.

In what county will the Prime Minister’s mansion be built?

*by Howard Williams*

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

I have two identical four-sided dice (regular tetrahedra). Each die has the numbers 1 to 4 on the faces. Nia and Rhys play a game in which each of them takes turns throwing the two dice, scoring the sum of the two hidden numbers. After each has thrown both dice three times, if only one of them scores an agreed total or over, then he or she is the winner. Otherwise the game is drawn.

After Nia has thrown once, and Rhys twice, they both have a chance of winning. If Rhys had scored less on his first throws and Nia had scored double her first throw total, then Nia would have had a 35 times greater chance of winning.

What are Nia’s and Rhys’ scores and the agreed winning total?

*by Victor Bryant*

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

I wrote down a number which happened to be a multiple of my lucky number. Then I multiplied the written number by my lucky number to give a second number, which I also wrote down. Then I multiplied the second number by my lucky number again to give a third, which I also wrote down. Overall, the three numbers written down used each of the digits 0 to 9 exactly once.

What were the three numbers?

## Sunday Times Teaser 3084 – Face Value

*by Susan Bricket*

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

Plato: I have written a different whole number (chosen from 1 to 9 inclusive) on each of the faces of one of my regular solids and have labelled each vertex with the product of the numbers on its adjacent faces. If I tell you the sum of those eight vertex labels, you can’t deduce my numbers, but if I rearrange the numbers on the faces and tell you the new sum, then you can deduce the numbers.

Eudoxus: Tell me the new sum then.

Plato: No, but I’ll tell you it’s a 10th power.

Eudoxus: Aha! I know your numbers now.

Plato: Yes, that’s right. But if I now told you the original sum, you couldn’t work out which numbers were originally opposite each other.

What was the original sum?

*by Colin Vout*

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

Inside a shed I saw where the council was preparing signposts for seven neighbouring villages. Between any two villages there is at most one direct road, and the roads don’t meet except at village centres, where the signposts are to stand. The arms were all affixed, and labelled except for one name to be completed on each. The names in clockwise order on each were as follows:

Barton, Aston, ?;

Barton, Grafton, ?;

Carlton, Forton, Eaton, ?;

Grafton, Aston, ?;

Dalton, Grafton, Eaton, ?;

Barton, Forton, Carlton, ?;

Dalton, Aston, ?

Starting at Dalton, I walked along roads, taking the road furthest rightwards at each village and returning to Dalton. I chose the first road so that I visited as many other villages as possible with this method.

In order, what were the other villages I visited?

## New Scientist Enigma 871 – Pandigitals

by Colin Singleton

From Issue #2026, 20th April 1996 [link]

Uncle Joe had written the digits 0 to 9 on ten cards, some red, the others blue, one digit per card. “Now boys,” he said, “you have to arrange the red cards on the table to form a number, and divide the blue cards into two groups to form two separate numbers. The ‘red’ number must be the product of the ‘blue’ numbers.”

“Like this?” said Tom. “Or this?” said Dick. “Or this?” said Harry.

“All different, and all correct!” replied Uncle Joe.

Only Tom had included a single-digit number in his arrangement. What was Harry’s ‘red’ number?

## A Holiday Brain Teaser

by A. D. Denton

From The Sunday Times, 4th August 1957 [link]

The series below is peculiar in that if any two of the numbers are multiplied together and added to unity, the result is a perfect square:

1, 3, 8

Let A[4], be a fourth number in this series. There is a further series with the same characteristic:

8, B[2], 190, B[4]

in which not only is B[1], the same as A[3], but B[2] is also the same as A[4].

What is B[4]?

(Note: zero is excluded)

## Sunday Times Teaser 3802 – In the Swim

*by Nick MacKinnon*

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

Expert mathematicians Al Gritham, Ian Tadger, Carrie Over and Tessa Mole-Poynte have a swimming race for a whole number of lengths. At some point during each length (so not at the ends) they calculate that they have completed a fraction of the distance, and at the finish they compare their fractions (all in the lowest terms). Al says, “My fractions have prime numerators, the set of numerators and denominators has no repeats, their odd common denominator has two digits, and the sum of my fractions is a whole number.” Tessa says, “Everybody’s fractions have all the properties of Al’s, but one of mine is closer to an end of the pool than anybody else’s.”

What were Tessa’s fractions?