### by Edmund Marshall

#### Published Sunday June 26 2022 (link)

If a date during the current millennium is day D in month M during year (2000+N), it is said to be a product date if the product of D and M equals N (for example 11 February 2022). My daughter and I have been investigating the numbers of days from one product date to the next product date. I was able to establish the longest such interval L, while my daughter worked out the shortest such interval S. We were surprised to find that L is a whole number multiple of S.

What is that multiple?

### by Colin Vout

#### Published Sunday June 19 2022 (link)

A square, a triangle and a circle went into a bar. The barman said, “Are you numbers over 18?” They replied, “Yes, but we’re under a million.” The square boasted, “I’m interesting, because I’m the square of a certain integer.” The triangle said, “I’m more interesting; I’m a triangular number, the sum of all the integers up to that same integer.” The circle said, “I’m most interesting; I’m the sum of you other two.” “Well, are you actually a circular number?” “Certainly, in base 1501, because there my square ends in my number exactly. Now, shall we get the drinks in?” The square considered a while, and said, “All right, then. You(’)r(e) round!”

In base 10, what is the circular number?

### by Nick MacKinnon

#### Published Sunday June 12 2022 (link)

In an election for golf-club president, voters ranked all four candidates, with no voters agreeing on the rankings. Three election methods were considered.

Under First-past-the-post, since the first-preferences order was A, B, C, D, the president would have been A.

Under Alternative Vote, since A had no majority of first preferences, D was eliminated, with his 2nd and 3rd preferences becoming 1st or 2nd preferences for others. There was still no majority of 1st preferences, and B was eliminated, with his 2nd preferences becoming 1st preferences for others. C now had a majority of 1st preferences, and would have been president.

Under a Borda points system, candidates were given 4, 3, 2, or 1 points for each 1st, 2nd, 3rd or 4th preference respectively. D and C were equal on points, followed by B then A.

How many Borda points did each candidate receive?

### by Stephen Hogg

#### Published Sunday June 05 2022 (link)

On Whit Monday, Zak began self-isolating upstairs. At lunchtime Kaz shouted up, “What’s a Geometric Mean?” “It’s the Nth root of the product of N values,” Zak replied.

On TV, Teaseside hospital’s “geovid” admissions for the seven days prior were listed alongside their Geometric Mean. Kaz stated that chronologically the numbers comprised a decreasing set of two-figure values, Friday’s value equalling the Geometric Mean. She added that, curiously, there was a value double the Geometric Mean, but not triple, whereas the Geometric Mean was triple a data value, but not double a data value. She then told Zak just the Geometric Mean.

Zak worked out the unique data set.

Give the seven numbers in chronological order.

### by Danny Roth

#### Published Sunday May 29 2022 (link)

George and Martha have recently taken a great-grandchild to a toddler’s birthday party. The youngsters like to traipse around over a pen with a large number of brightly coloured plastic balls. Actually there were 200 in total, some of red, yellow, blue and green. There were at least 30 but fewer than 70 of each colour, with the following properties:

Red – perfect square

Yellow – prime number

Blue – palindromic number

Green – divisible by three single-digit prime numbers

George told Martha the above information and the number of red balls. Martha was then able to work out the numbers of each of the others.

How many of each colour were there?

### by Peter Good

#### Published Sunday May 22 2022 (link)

A plumber was trying to empty a tank containing 100 litres of water using three buckets, each marked with a different whole number of litres capacity between 10 and 20 litres. He calculated that he could exactly empty the tank, but only by using all three buckets and completely filling each bucket a different number of times. He filled and emptied each bucket the calculated number of times but the tank still contained 6 litres of water, because the smallest bucket had a dent that reduced its capacity by 3 litres.

What were the marked capacities of the three buckets?

### by Andrew Skidmore

#### Published Sunday May 15 2022 (link)

Callum has opened a new current account and has been given a telephone PIN that is composed of non-zero digits (fewer than six). He has written down the five possible rearrangements of his PIN. None of these five numbers are prime; they can all be expressed as the product of a certain number of different primes.

The PIN itself is not prime; it can also be expressed as the product of different primes but the number of primes is different in this case. The sum of the digits of his PIN is a square.

What is the PIN?

### by Howard Williams

#### Published Sunday May 08 2022 (link)

I have an analogue wall clock with a second hand and also a separate 24-hour hh:mm:ss digital clock. The wall clock loses a whole number of seconds over a two-digit period of seconds. The digital clock gains at a rate 2½% greater than the wall clock loses. After resetting both clocks to the correct time, I noticed that they both displayed the same but wrong time later in the same week, and one hour earlier than the time of setting.

I can reset one of the clocks at an exact hour so that it will show the correct time when the televised rugby kicks off at 19:15:00 on the 31st.

What is the latest time (hour and date) when I can do this?

### by Victor Bryant

#### Published Sunday May 01 2022 (link)

I have written down three 3-figure numbers in decreasing order and added them up to give their 4-figure sum, which is a perfect square: the digit 0 occurred nowhere in my sum.

Now I have attempted to replace digits consistently by letters and I have written the sum as

CUP + AND + LIP = SLIP

However, there’s “many a slip twixt cup and lip” and unfortunately one of those thirteen letters is incorrect. If you knew which letter was incorrect then you should be able to work out the three 3-figure numbers.

What are they?

### by Colin Vout

#### Published Sunday April 24 2022 (link)

In theoretical golf, you have a set of “clubs” each of which hits a ball an exact number of yards forwards or backwards. You own the following set: 3, 8, 17, 19 and 35. For example, if a hole is 31 yards long, you can reach it in three strokes, with two forward hits of the 17 and a backward hit of the 3. In the next competition, you are only allowed to use three clubs, and the course consists of three holes whose lengths are 101, 151 and 197 yards. In order to get round the course in the fewest possible strokes, you must make a wise choice of clubs.

Which three clubs (in ascending order) should you choose, and what will the individual hole scores be (in order)?