### by Angela Newing

#### Published Sunday January 30 2022 (link)

Ian was trying to entertain John and Ken, his 2 young nephews, in the local park, which had a 9-hole crazy golf course. He decided to play a round with each of them separately, and gave them money for each hole he lost on a rising scale. He would pay £1 if he lost the first hole (or the first hole after winning one), then £2 if he lost the next consecutive hole, and £3 for the third, and so on.

In the event, Ian won only 5 holes in total between the two rounds, including the first hole against John and the last hole against Ken. There were no ties. At the reckoning after both rounds, both boys received equal amounts of money.

How much did it cost Uncle Ian?

### by Victor Bryant

#### From The Sunday Times, 18th November 1979 (link)

Problems concerning ages have always proved fruitful and entertaining exercises to both mathematicians and non-mathematicians. Trial and error methods, calculators and normal or esoteric mathematical techniques can all be deployed to find the correct solution. The most elegant or the most economical method is naturally the most commendable, but the correct solution, however obtained, is the desideratum.

Our problem concerns six men whose ages are within the range 21 to 89 and any two of them differ by at least 9. If we take the two digits comprising each of the ages of three of the men, and reverse them, we obtain the ages of the other three men.

What is more, if we take the sum of the ages of the first group, we find that it equals the sum of the ages of the second group of three.

Also the sum of the squares of the three ages of the first group equals the sum of the squares of the ages of the second group of three.

Finally, one of the ages in each group is exactly twice an age in the other group.

What are the ages of the six men (in increasing order)?

### by Colin Vout

#### Published Sunday January 23 2022 (link)

Skaredahora used a scale with seven notes, labelled J to P, for an idiosyncratic composition. It had a sequence of chords, each comprising exactly three different notes (the order being immaterial). The sequence started and ended with JNP. Each change from one chord to the next involved two notes increasing by one step in the repeating scale, and the other decreasing by two steps (eg, JNP changing to KLJ).

Every chord (eventually) reachable from JNP was used at least once; every allowable change from one of these chords to another was used exactly once. The number of chord changes between the first pair of JNPs was more than that between the last such pair, but, given that, was the smallest it could be.

With notes in alphabetical order, what was the seventh chord in the sequence?

### by Stephen Hogg

#### Published Sunday January 16 2022 (link)

“Diddums” was Didi’s dada’s name for numbers where the number of positive DIvisors (including 1 and the number), the number of Digits, and the Digit sUM are equal.

Didi enjoyed mental arithmetic, doing it speedily and accurately. For numbers of digits from 1 to 9 she listed in ascending order all possible numbers whose digit sum equalled the number of digits. Working through her list in order, she quickly found the first two diddums (1 and 11) and the next three diddums. After many further calculations, Didi confirmed that the sixth diddum (which is even) was listed.

“Now I’m bored,” she moaned. “Diddums!” said Didi’s dada.

What is the sixth diddum’s digit sum?

### by Danny Roth

#### Published Sunday January 09 2022 (link)

$\begin{array}{|c|c|c|c|c|c|}\hline Year & 2018 & 2019 & 2020 & 2021 & 2022 \\ \hline Pass\;Rate\;(\%) & 30 & 32 & 35 & 36 & X \\ \hline \end{array}$

George and Martha annually examine around 500 candidates (give or take 5%). It is board policy to pass about a third of the entrants but the precise recent percentage pass rates were as above.

The number of entrants in each year up to 2021 was different as was the number of successful candidates. George told Martha of the number of entries for 2022 (different again) and Martha calculated that, if X were to be once again a whole number (also different again but within the above range), the total number of successful candidates over the five-year period would be a perfect square.

How many entries are there for 2022 and how many successful candidates did Martha calculate for 2022?

### by Victor Bryant

#### Published Sunday January 02 2022 (link)

My grandson and I play a simple card game. We have a set of fifteen cards on each of which is one of the words ACE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, SHOUT and SNAP. We shuffle the cards and then display them one at a time. Whenever two consecutive cards have a letter of the alphabet occurring on both we race to shout “Snap!”. In a recent game there were no snaps. I counted the numbers of cards between the “picture-cards” (J/Q/K) and there was the same number between the first and second picture-cards occurring as between the second and third. Also, the odd-numbered cards (3 to 9) appeared in increasing order during the game.

In order, what were the first six cards?

### by Nick MacKinnon

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

A Christmas Carol was published on 19/12/1843, when Bob Cratchit was in his forties, almost seven years after Jacob Marley’s death on Christmas Eve. On Marley’s last Christmas Day, a working day for them all as always, Scrooge said to Cratchit and Marley, “We three will work the same dates next year as this year, except that I’ll cover Cratchit’s birthday so he can have the day off, since Marley never works on his.” On Boxing Day, Scrooge decided to allocate extra work dates for next year to anyone whose number of work dates was going to be below the average of the three of them, bringing them up to exactly that average. Daily, up to and including New Year’s Eve, Scrooge repeated this levelling-up to the new average, never needing fractions of a day.

What was Bob Cratchit’s full date of birth?

### by Richard England

#### From Issue #2093, 2nd August 1997 (link)

Harry, Tom and I were trying to find sets of three two-digit prime numbers such that if we added any two numbers of the set together the answer was double a prime. Within each set the three prime numbers had to be different; but we each found that, having created a set, we could then create a second set with the same properties by changing just one of its primes.

One of my two sets was the same as one of Harry’s sets and the other was the same as one of Tom’s; their other sets were different.

(1) What were my two sets?
(2) Which set might we have found whose three primes do not appear in any other set?

### by Howard Williams

#### Published Sunday December 19 2021 (link)

My two daughters were born on the same day but seven years apart. Every birthday, with only one year’s exception, I have given them both five pounds for each year of their age. They are now grown-up, but I have continued to do this on their birthdays, except for the one year when I couldn’t afford to give either of them anything. Averaged out over all of their birthdays, including the one for which they received nothing, my elder daughter has now received 21 per cent more per birthday than her younger sister.

How much in total have I given to my daughters as birthday presents?

### by Andrew Skidmore

#### Published Sunday December 12 2021 (link)

Anton and Boris live next to a railway line. One morning a goods train passed Anton’s house travelling south just as a slower train passed Boris’s house travelling north. The goods train passed Boris’s house at the same time as a passenger train, heading north at a speed that was half as fast again as the goods train. Similarly, as the slower train passed Anton’s house it passed a passenger train; this was heading south at a speed that was three times as great as that of the slower train.

The passenger trains then passed each other at a point 25 kilometres from Anton’s house before simultaneously passing the two houses.

All four trains travelled along the same route and kept to their own constant speeds.

How far apart do Anton and Boris live?