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)

### 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?

### by Kenneth Armstrong

#### From Issue #1796, 23rd November 1991 [link]

My children had been practising addition, forming additive sequences of numbers, and when the numbers got too large for them, extending the sequence the other way using subtraction and getting some negative numbers. Eventually, they noticed that one of their sequences had numbers in it divisible by 2, 3, 5 and 7 but none divisible by 11. Part of that sequence was:

… –2 3 1 4 …

the rule going from left to right being, of course, that one term plus the following one gives the next.

The children soon saw too that every sequence they wrote down had terms divisible by 2 and 3, and probably 7 also.

What I want you to find is the two sequences like this, each having no terms divisible by 5, 11 or 13. In each sequence the four consecutive terms we want should have:

just the first term negative;
the third term 3;
and the fourth term less than 100

(In fact, one of the sequences will have terms divisible by 17 and none divisible by 19, while the other will have terms divisible by 19 and none divisible by 17, but you don’t need these facts to find them).

Please send in the four terms of the two sequences.

### by Howard Williams

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

I have four different two-digit numbers, each having at least one digit which is a three. When I multiply any three of these numbers together I get a product that, with the inclusion of a leading zero, is one or more repetitions of the repetend of the reciprocal of the fourth two-digit number. A repetend is the repeating or recurring decimal of a number. For example 1 divided by 27 is 0.037037……, giving a repetend of 037; in that case, the product would be 37 or 37037 or 37037037 etc.

What, in ascending order, are the four two-digit numbers?

### by Andrew Skidmore

#### Published Sunday October 03 2021

The raffle tickets at the Mathematical Society Dinner were numbered from 1 to 1000. There were four winning tickets and together they used each of the digits from 0 to 9 once only. The winning numbers could be classified uniquely as one square, one cube, one prime and one triangular number. For example, 36 is a triangular number as 1+2+3+4+5+6+7+8 = 36, but it cannot be a winner as 36 is also a square. The tickets were all sold in strips of five, and two of the winning numbers were from consecutive strips. The first prize was won by the holder of the smallest-numbered winning ticket, which was not a cube.

List the four winning numbers in ascending order.

### by Angela Newing

#### Published Sunday September 26 2021 (link)

Oak Hall at Woodville University has groups of five study bedrooms per flat and they share a kitchen/diner. In one flat live language students Andy, Bill, Chris, Dave and Ed. Bill, whose home town is Dunstable is reading French. The person in room 5 comes from Colchester and Dave comes from Brighton. The chap reading German has the room with a number one greater than the man from Gloucester. Chris occupies room 3, and Ed is reading Italian. The man in room 2 is reading Spanish, and the man reading English has a room whose number is two different from the student from Reigate.

What is Andy’s subject and where is his home?

### 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?

### 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)?

### 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?

### 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?