## Sunday Times Teaser 3121 – Top Marks

*by Howard Williams*

#### Published Sunday July 17 2022 (link)

A teacher is preparing her end of term class test. After the test she will arrive at each pupil’s score by giving a fixed number of marks for a correct answer, no marks if a question is not attempted, and deducting a mark for each incorrect answer. The computer program she uses to prepare parents’ reports can only accept tests with the number of possible test scores (including negative scores) equal to 100.

She has worked out all possible combinations of the number of questions asked and marks awarded for a correct answer that satisfy this requirement, and has chosen the one that allows the highest possible score for a pupil.

What is that highest possible score?

*by Susan Bricket*

#### Published Sunday July 17 2022 {link)

While waiting for buses, I often look out for interesting number plates on passing cars. From 2001 the UK registration plate format has been 2 letters + a 2-digit number + 3 more letters, the digits being last two of the year of registration with 50 added after six months (for example in 2011, the possible numbers were 11 and 61). I spotted one recently with its five letters in alphabetical order, all different and with no vowels. Looking more closely, I saw that if their numerical positions in the alphabet (A = 1, B = 2 etc.) were substituted for the 5 letters, their sum plus 1 was the 2-digit number and the sum of their reciprocals was equal to 1.

Send the 7-character registration.

*by Victor Bryant*

#### Published Sunday July 03 2022 (link)

My grandson is studying “History since the Battle of Hastings”. I made him a game, which consisted of a row of nine cards, each with a different non-zero digit on it. Throw a standard die, note the number of spots displayed, count that number of places along the row and pause there. Throw the die again, move the corresponding number of places further along and pause again. Repeat this until you come off the end of the row, noting the digit or digits you paused on and put these together in the same order, to produce a number.

Keeping the cards in the same order I asked my grandson to try to produce a square or cube or higher power. He eventually discovered that the lowest possible such number was equal to the number of one of the years that he had been studying.

What is the order of the nine digits along the row?

## Sunday Times Teaser 3118 – Product Dates

*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?

## Sunday Times Teaser 3112 – PIN

*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?