*by Colin Vout*

#### Published Friday April 19 2024 (link)

In retrospect it was inadvisable to ask an overenthusiastic mathematician to overhaul our local tram routes. They allocated a positive whole number less than 50 to each district’s tram stop. To find the number of the tram going from one district to another you would “simply” (their word, not mine) find the largest prime divisor of the difference between the two districts’ numbers; if this was at least 5 it was the unique route number, and if not there was no direct route.

The routes, each in increasing order of the stop numbers, were: Atworth, Bratton, Codford; Atworth, Downlands, Enford; Bratton, Figheldean, Enford; Downlands, Figheldean, Codford; Codford, Enford.

What were the route numbers, in the order quoted above?

*by Victor Bryant*

#### Published Friday April 12 2024 (link)

I started with a six-by-six grid with a 0 or 1 in each of its 36 squares: they were placed in chequerboard style with odd rows 101010 and even rows 010101. Then I swapped over two of the digits that were vertically adjacent. Then in three places I swapped a pair of horizontally adjacent digits.

In the resulting grid I read each of the six rows as a binary number (sometimes with leading zeros) and I found that three of them were primes and the other three were the product of two different primes. The six numbers were all different and were in decreasing order from the top row to the bottom.

What (in decimal form) were the six decreasing numbers?

*by Howard Williams*

#### Published Friday April 05 2024 (link)

I have three daughters and a grandson. They are all of different ages, with the eldest being younger than 35 years old, and my grandson being the youngest.

Three years ago the square of the age of my eldest daughter was equal to the sum of the squares of the ages of the other three. In another three years’ time the sum of the square of the age of my eldest daughter plus the square of the age of my grandson will be equal to the sum of the squares of the ages of my other two daughters.

In ascending order, what are their ages?

## Sunday Times Teaser 3210 – Random Rabbit

*by Mark Valentine*

#### Published Friday March 29 2024 (link)

Every year Easter Bunnies must pass a series of demanding tests, the most important being the double jump. Rabbits perform a single jump comprising two hops, with the total distance scoring. For both hops, Max knows he has an equal chance of jumping any distance between two limits. For the first hop, the limits are 80 and 100cm. For his weaker second hop, these limits decrease but keep the same proportion.

However, the instructors have increased the required standard from 152 to 163cm. Max is worried; he can still pass, but his chance is half what it was before.

What, as a percentage, is his chance of passing this year?

*by J S Rowley*

#### Published 28th January 1968 (link)

Some years ago the Bell family were holding their usual annual special birthday party. Four members of the family, of four different generations, had birthdays on the same day of the year. They were old Adam, his son Enoch, Enoch’s son Joseph and Joseph’s son David. On this occasion David remarked that the sum of any three of their four ages was a perfect square

Some years later old Adam died on his birthday, but it so happened that on the very same day David’s son Samuel was born, and the annual party was continued in subsequent years.

In 1967 at the usual party Samuel made exactly the same remark that David had made, on the previous occasion.

In what year did Adam die and how old was he then?

(Perhaps I should mention that no one survived to be 100!).

## Sunday Times Teaser 3209 – All in Order

*by Victor Bryant*

#### Published Friday March 22 2024 (link)

Audley’s age is a two-figure number. He has that number of cards and on them he has spelt out the consecutive numbers from one up to and including his age, (“one”, “two”, etc) with one number on each card. Then he has arranged the cards in a row in alphabetical order. It turns out that two of the numbers are in the “correct” place; ie in the same place as if he had arranged the cards in numerical order).

If he had done all this a year ago, or if he repeated this whole exercise in a year’s time, there would be no card in the correct place.

How old is he?

**by Colin Vout**

**by Colin Vout**

#### Published Friday March 15 2024 (link)

For Skaredahora’s quartet four players read the same musical score, but from different compass directions. There are symbols of three types, indicating different whole number beat durations, on a square 17×17 grid. Each player reads the beat position in their left-to-right direction, and pitch in their bottom-to-top.

Each player plays four notes; South reads a plus at (beat,pitch) position (3,12), a circle at (14,1), a cross at (16,3), and a plus at beat 9. For example, if a cross indicates three beats, South plays a note of pitch 3 at beat 16, which is still sounding at beats 17 and 18, while East plays a note of pitch 2 sounding at beats 3, 4 and 5.

No player sounds more than one note at the same time. All possible pitch differences between notes sounding simultaneously from different players occur, except zero and exactly one other value.

Which non-zero pitch difference never occurs? What pitch does South play at beat 9?

*by Rachel Blunt*

#### Published 7th March 1982 (see link)

My mathematics class consists of six boys and six girls. In their annual examination each was awarded integral an mark out of 100.

Disappointingly no boy received a distinction (over 80) but all the boys managed over 40. The lowest mark in the class was 36.

Upon listing the boys’ marks I noticed that all their marks were different prime numbers and that their average was an even number. Further three of the boys’ marks formed an arithmetic progression, and the other three another arithmetic progression.

Turning, my, attention to the girls I found that their marks were all different. There was little overall difference in the performance of the

sexes, the total of the girls’ marks being just one more than the total of the boys’. Three of the girls’ marks formed one geometric progression, and

the other three formed another geometric progression with the same ratio as the first one.

Finally when listing the results in numerical order I was pleased to see that Annie (who did so badly last year) had come seventh in the class.

What were the top six marks (in descending order)?

*by Andrew Skidmore*

#### Published Friday March 08 2024 (link)

Fabulé’s next creation will be a set of equal-sized silver regular dodecahedra, but some of the faces will be gold-plated. He is undecided whether to go ahead with either a “Charm” set or a “Partial” set.

“Charm” is composed of dodecahedra with at least one gold-plated face but with no gold-plated face having a common side with more than one other gold-plated face. “Partial” is composed of dodecahedra with exactly six gold-plated faces. All the items in each set are distinguishable.

What is the maximum number of dodecahedra possible in (a) “Charm” (b) “Partial”?

*by Peter Good*

#### Published Friday March 01 2024 (link)

Two buildings on level ground with vertical walls were in need of support so engineers placed a steel girder at the foot of the wall of one building and leaned it against the wall of the other one. They placed a shorter girder at the foot of the wall of the second building and leaned it against the wall of the first one. The two girders were then welded together for strength.

The lengths of the girders, the heights of their tops above the ground, the distance between their tops and the distance between the two buildings were all whole numbers of feet. The weld was less than ten feet above the ground and the shorter girder was a foot longer than the distance between the buildings.

What were the lengths of the two girders?