### by Howard Williams

#### Published Sunday April 04 2021 (link)

My kitchen floor is tiled with identically-sized equilateral triangle tiles while the floor of the bathroom is tiled with identically-sized regular hexagon tiles, the tiles being less than 1m across. In both cases the gaps between tiles are negligible. After much experimenting I found that a circular disc dropped at random onto either the kitchen or bathroom floor had exactly the same (non-zero) chance of landing on just one tile.

The length of each side of the triangular tiles and the length of each side of the hexagon tiles are both even triangular numbers of mm (ie, of the form 1+2+3+…).

What are the lengths of the sides of the triangular and hexagonal tiles?

### by Danny Roth

#### Published Sunday March 28 2021 (link)

George and Martha have a telephone number consisting of nine digits; there is no zero and the others appear once each. The total of the digits is obviously 45, so that the number is divisible by nine. Martha noticed that, if she removed the last (ie, the least significant) digit, an eight-digit number would remain, divisible by eight. George added that you could continue this process, removing the least significant digit each time to be left with an n-digit number divisible by n right down to the end.

What is their telephone number?

### by Stephen Hogg

#### Published Sunday March 21 2021 (link)

“Argent bend sinister abased sable in dexter chief a hog enraged proper” blazons our shield (shaped as a square atop a semi-circle, with a 45° diagonal black band meeting the top corner). We’ve three shields. For the first, in centimetres, the top width (L) is an odd perfect cube and the vertical edge height of the band (v) is an odd two-figure product of two different primes. The others have, in inches, whole-number L (under two feet) and v values (all different). For each shield, the two white zones have almost identical areas. All three v/L values, in percent, round to the same prime number.

Give the shortest top width, in inches.

#### by Colin Vout

Published Sunday March 14 2021 (link)

Here is the shipping forecast for the regions surrounding our island.

First, the coastal regions.

$\begin{array}{|l|c|c|c|c|c|}\hline \mathbf{Region} & \mathbf{Direction} & \mathbf{Strength} & \mathbf{Sea\;State} & \mathbf{Weather}& \mathbf{Visibility} \\ \hline \mathbf{Hegroom} & E & 6 & rough & drizzle & moderate \\ \hline \mathbf{Forkpoynt} & E & 7 & rough & rain & good \\ \hline \mathbf{Angler} & NE & 7 & rough & drizzle & moderate \\ \hline \mathbf{Dace} & NE & 7 & smooth & drizzle & moderate \\ \hline \end{array}$

Now, the offshore regions.

$\begin{array}{|l|c|c|c|c|c|}\hline \mathbf{Region} & \mathbf{Direction} & \mathbf{Strength} & \mathbf{Sea\;State} & \mathbf{Weather}& \mathbf{Visibility} \\ \hline \mathbf{Back} & E & gale\;8 & rough & rain & moderate \\ \hline \mathbf{Greigh} & E & 7 & smooth & drizzle & poor \\ \hline \mathbf{Intarsia} & SE & 7 & rough & drizzle & moderate \\ \hline \mathbf{Catter} & E & gale\;8 & high & drizzle & moderate \\ \hline \mathbf{Eighties} & E & 7 & rough & fair & good \\ \hline \end{array}$

In this forecast, no element jumps from one extreme to another between adjacent regions. The extremes are:

$\begin{array}{|l|c|c|} \hline Wind\;Direction & SE & NE \\ \hline Wind\;Strength & 6 & gale\; 8 \\ \hline Sea State & smooth & high \\ \hline Weather & fair & rain \\ \hline Visibility & good & poor \\ \hline \end{array}$

Each region adjoins four others (meeting at just one point doesn’t count).

Which of the regions does Angler touch?

### by Victor Bryant

#### Published Sunday March 07 2021 (link)

Given any number, one can calculate how close it is to a perfect square or how close it is to a power of 2. For example, the number 7 is twice as far from its nearest perfect square as it is from its nearest power of 2. On the other hand, the number 40 is twice as far from its nearest power of 2 as it is from its nearest square.

I have quite easily found a larger number (odd and less than a million!) for which one of these distances is twice the other.

What is my number?

### by J E Kessel and J R Partridge

#### From The Sunday Times, 17th November 1974

You see, Inspector, the combination of my safe is a six-figure number. In case anyone needed to get into it while I was away, I gave each of my clerks (Atkins, Browning and Clark) one of the two-figure numbers which make up the combination. I also told each the position in the combination of the number of another clerk, but not the number itself.

Browning must have overheard me telling a friend that it is a coincidence that two of these numbers are squares and if you put them together you get a four-figure number that equals the other clerk’s number squared. I remember I also said something about whether or not the combination is divisible by this clerk’s number.

When he was caught, Browning said, “I can’t understand why the alarm went off; I know Clark’s is the first number”. I later realised that what I’d told my friend about whether or not that other number was a factor was wrong, which was lucky for me as Browning had got his own number in the right place.

What was the combination?

### by Zoe Mensch

#### From New Scientist #3323, 27th February 2021

My nine-digit passport number has some remarkable properties. Not only does it use all the digits from 1 to 9, but if I label the number ABCDEFGHI, then:

A is divisible by 1
AB is divisible by 2
ABC is divisible by 3
ABCD is divisible by 4
ABCDE is divisible by 5
ABCDEF is divisible by 6
ABCDEFG is divisible by 7
ABCDEFGH is divisible by 8
ABCDEFGHI is divisible by 9

You could program a computer to find this passport number, but there are shortcuts to figuring it out with a pen and paper. For example, a number is only divisible by 3 if its digits add up to a multiple of 3 (e.g.: 372 is divisible by 3 because 3+7+2 = 12). And a number is only divisible by 4 if its last two digits form a number that is also a multiple of 4 (hence 9324 is divisible by 4 because 24 also is).

What is my passport number?

### by Andrew Skidmore

#### Published Sunday February 28 2021 (link)

Jed farms a large, flat, square area of land. He has planted trees at the corners of the plot and all the way round the perimeter; they are an equal whole number of yards apart.

The number of trees is in fact equal to the total number of acres (1 acre is 4840 square yards). If I told you an even digit in the number of trees you should be able to work out how far apart the trees are.

How many yards are there between adjacent trees?

### by Stephen Hogg

#### Published Sunday February 21 2021 (link)

Our holiday rep, Nero, explained that in Carregnos an eight-digit total of car registrations results from combinations of three Greek capital letters after four numerals (eg 1234 ΩΘΦ), because some letters of the 24-letter alphabet and some numerals (including zero) are not permitted.

For his own “cherished” registration the number tetrad is the rank order of the letter triad within a list of all permitted letter triads ordered alphabetically. Furthermore, all permitted numeral tetrads can form such “cherished” registrations, but fewer than half of the permitted letter triads can.

Nero asked me to guess the numbers of permitted letters and numerals. He told me that I was right and wrong respectively, but then I deduced the permitted numerals.

List these numerals in ascending order

### by Howard Williams

#### Published Sunday February 14 2021 (link)

I gave Robbie three different, single digit, positive whole numbers and asked him to add up all the different three-digit permutations he could make from them. As a check for him, I said that there should be three threes in his total. I then added two more digits to the number to make it five digits long, all being different, and asked Robbie’s mother to add up all the possible five-digit permutations of these digits. Again, as a check, I told her that the total should include five sixes.

Given the above, the product of the five numbers was as small as possible.

What, in ascending order, are the five numbers?