### by Stephen Hogg

#### Published Sunday August 14 2022 (link)

At Teaser Tor trig. point I found a geocaching box. The three-figure compass bearings (bearing 000=north, 090=east, etc.) from there to the church spires at Ayton, Beeton and Seaton were needed to decode the clue to the next location.

Each spire lay in a different compass quadrant (eg 000 to 090 is the North-East quadrant). Curiously, each of the numerals 1 to 9 occurred in these bearings and none of the bearings were prime values.

Given the above, if you chose one village at random to be told only its church spire’s bearing, it might be that you could not calculate the other two bearings with certainty, but it would be more likely you could.

Give the three bearings, in ascending order.

### by Colin Vout

#### Published Sunday August 07 2022 (link)

A gardener was laying out the border of a new lawn; he had placed a set of straight lawn edging strips, of lengths 16, 8, 7, 7, 7, 5, 4, 4, 4 & 4 feet, which joined at right angles to form a simple circuit. His neighbour called over the fence, “Nice day for a bit of garden work, eh? Is that really the shape you’ve decided on? If you took that one joined to its two neighbours, and turned them together through 180°, you could have a different shape. Same with that one over there, or this one over here — oh, look, or that other one.” The gardener wished that one of his neighbours would turn through 180°.

What is the area of the planned lawn, in square feet?

### by Nick MacKinnon

#### Published Sunday July 31 2022 (link)

A factory makes six types of cylindrical pipe, A to F in decreasing size, whose diameters in centimetres are whole numbers, with type A 50 per cent wider than type B. The pipes are stacked in the yard as a touching row of As with an alternating row of touching Bs and Cs in the next layer, with each B touching two As. Type Ds fill the gap between the As and the ground; Es fill the gap between As and the Bs; and Fs fill the gap between As, Ds and the ground. Finally another row of As is put on top of the stack, giving a height of less than 5 metres.

What is the final height of the stack in centimetres?

### by Angela Newing

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

Five witnesses were interviewed following a robbery at the bank in the High Street. Each was asked to give a description of the robber and his actions. The details given were: height, hair colour, eye colour, weapon carried, escape method.

$\begin{array}{|c|c|c|c|c|c|}\hline \mathbf{Witness} & \mathbf{Height} & \mathbf{Hair\;Colour} & \mathbf{Eye\;colour} & \mathbf{Weapon}& \mathbf{Escape}\\ \hline \mathbf{one} & short & fair & brown & cricket\;bat & motorbike \\ \hline \mathbf{two} & tall & fair & grey & gun & car \\ \hline \mathbf{three} & tall & dark & brown & crowbar & motorbike \\ \hline \mathbf{four} & short & ginger & blue & knife & car \\ \hline \mathbf{five} & tall & dark & blue & stick & pusbike \\ \hline \end{array}$

When the police caught up with the perpetrator, they found that each of the five witnesses had been correct in exactly two of these characteristics.

What was the robber carrying, and how did he get away?

### 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?

### 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?