Skip to content
Dec 30 20

New Scientist Enigma 597 – Break-Uprithmetic

by Brian Gladman

by Susan Denham

From New Scientist #1751, 12th January 1991

In snooker there are 15 red balls worth one point each. If a player pots a red it stays in the hole and he (or she) is allowed to try to pot one of the colours yellow, green, brown, blue, pink or black (worth 2-7 points in that order). If a colour is potted it is brought out again and the player can try for another red, and so on. This continues until all the reds have gone. Then the remaining six colours are potted in ascending order.

The total points achieved in one such run is called a “break”. For example:

red + pink + red + black + red

would be a break of 16.

Having completed the break, the player sits down and lets the other player try for a red and continue the break, and so on. At the end the winning player is the one with the higher grand total of points. A player may choose not to pot the final black if the result is already determined without it. No other rules concern today’s story.

Stephens and Hendry play a frame of snooker. Stephens starts with a break of 3, Hendry follows with a break of 4, Stephens with a break of 5, and so on, and this pattern continues to the end. As usual in quality snooker, the black was potted more times than the yellow, and the pink was potted more times that the blue.

How many times did Stephens pot the brown? And how many times did Hendry pot the brown?

Dec 24 20

Sunday Times Teaser 3040 – Moving Digit

by Brian Gladman

by Andrew Skidmore

Published Sunday December 27 2020 (link)

Jonny has opened a new bank account and has set up a telephone PIN. His sort code is comprised of the set of three two-figure numbers with the smallest sum which give his PIN as their product. He was surprised to find that the PIN was also the result of dividing his eight-figure account number by one of the three two-figure numbers in the sort code.

The PIN has an unusual feature which Jonny describes as a moving digit. If the number is divided by its first digit then the number which results has the same digits in the same order except that first digit is now at the end.

The account number does not contain the digit which moved.

What is the account number?

Dec 18 20

Sunday Times Teaser 3039 – Three Patterned Paving

by Brian Gladman

by Howard Williams

Published Sunday December 20 2020 (links)

James is laying foot-square stones in a rectangular block whose short side is less than 25 feet. He divides this area into three rectangles by drawing two parallel lines between the longest sides and into each of these three areas he lays a similar pattern.

The pattern consists of a band or bands of red stones laid concentrically around the outside of the rectangles with the centre filled with white stones. The number of red stone bands is different in each of the rectangles but in each of them the number of white stones used equals the number of outer {1] red stones used.

The total number of stones required for each colour is a triangular number (ie, one of the form 1+2+3+…).

What is the total area in square feet of the block?

[1] this word is unnecessary (it also creates an ambiguity).

Dec 11 20

Sunday Times Teaser 3038 – Progressive Raffle

by Brian Gladman

by Danny Roth

Published Sunday December 13 2020 (link)

George and Martha were participating in the local village raffle. 1000 tickets were sold, numbered normally from 1 to 1000, and they bought five each. George noticed that the lowest-numbered of his tickets was a single digit, then each subsequent number was the same multiple of the previous number, eg, 7 21 63 189 567. Martha’s lowest number was also a single digit, but her numbers proceeded with a constant difference, eg, 6 23 40 57 74. Each added together all their numbers and found the same sum. Furthermore, the total of all the digits in their ten numbers was a perfect square.

What was the highest numbered of the ten tickets?

Dec 4 20

Sunday Times Teaser 3037 – Prime Advent Calendar

by Brian Gladman

by John Owen

Published Sunday December 06 (link)

Last year I was given a mathematical Advent calendar with 24 doors arranged in four rows and six columns, and I opened one door each day, starting on December 1. Behind each door is an illustrated prime number, and the numbers increase each day. The numbers have been arranged so that once all the doors have been opened, the sum of the numbers in each row is the same, and likewise for the six columns. Given the above, the sum of all the prime numbers is as small as it can be.

On the 24th, I opened the last door to find the number 107.

In order, what numbers did I find on the 20th, 21st, 22nd and 23rd?

Nov 27 20

Sunday Times Teaser 3036 – That Old Chestnut

by Brian Gladman

by Colin Vout

Published Sunday November 29 2020 (link)

Clearing out an old drawer I found a wrinkled conker. It was my magnificent old 6709-er, a title earned by being the only survivor of a competition that I had had with friends. The competition had started with five conkers, veterans of many campaigns; each had begun at a different value between 1300 and 1400.

We used the rule that if an m-er beat an n-er in an encounter (by destroying it, of course!) the m-er would become an m+n+1-er; in effect, at any time the value of a conker was the number of destroyed conkers in all confrontations in its “ancestry”.

I recall that at the beginning of, and throughout, the competition, the value of every surviving conker was a prime number.

What were the values of the five conkers at the start?

Nov 21 20

Sunday Times Teaser 3035 – Friend of the Devil

by Brian Gladman

by Stephen Hogg

Published Sunday November 22 2020 (link)

My friend, “Skeleton” Rose, rambled on with me and my uncle (“The Devil” and “Candyman”) about Mr Charlie, who gave, between us, three identical boxes of rainbow drops.

Each identical box’s card template had a white, regular convex polygonal base section with under ten sides, from each of which a similar black triangular star point extended. All these dark star points folded up to an apex, making an enclosed box.

The number of sweets per box equalled the single-figure sum of its own digits times the sum of the star points and the box’s faces and edges. If I told you how many of the “star point”, “face” and “edge” numbers were exactly divisible by the digit sum, you would know this number of sweets.

How many sweets were there in total?

Nov 18 20

New Scientist Enigma 587 – Find the Fractions

by Brian Gladman

by John Magill

From New Scientist #1740, 27th October 1990

\[\Large\begin{array} {|c|c|c|}
\hline \frac{29}{e} & \frac{14}{c} & \frac{11}{e} \\
\hline\frac{7}{d} & \frac{8}{c} & \frac{5}{a} \\
\hline\frac{53}{e} & \frac{2}{c} & \frac{7}{b} \\
\hline \end{array}\]

In the figure each letter represents a number, the same letter standing for the same number wherever it appears. The nine entries are vulgar fractions in reduced form, that is, in each fraction the numerator and denominator have no common factor apart from 1. When completed, the figure is a magic square where each of the rows, columns and diagonals sum to the same value, a value greater than a half and less than 1.

What is the magic square revealed by substituting, for each letter, its number?

Nov 16 20

by Brian Gladman

New Scientist Enigma 586 – The Paint Runs Down

by Keith Austin

From New Scientist #1739, 20th October 1990

In the diagram the lines indicate pipes for the paint to flow down. At each of A, B and C either red or blue paint is poured into the top of the pipe. The circles are paint processors. For the circles marked with a star, if all the paint flowing down into the circle is red then blue flows out of the bottom of the circle, otherwise red flows out. For the other circles, if all the paint flowing into the circle is blue, then blue flows out of the bottom otherwise red flows out. For example, if red is poured into the top of A and C and blue into B, then blue flows out of the pipe 5 and red out of pipes 1, 2, 3, 4, 6 and 7 at the bottom.

Some of the seven numbered bottom pipes are to be directed into a storage drum in such a way that the following condition holds whatever the choice of three paints put in A, B, C:

If red is poured into A then at least one pipe into the drum brings blue;
if blue is poured into A then all the pipes into the drum bring red.

Which of the numbered pipes should go into the drum?

Nov 14 20

Sunday Times Teaser 3034 – Reservoir Development

by Brian Gladman

by Graham Smithers

Published Sunday November 15 2020 (link)

A straight track from an observation post, O, touches a circular reservoir at a boat yard, Y, and a straight road from O meets the reservoir at the nearest point, A, with OA then extended by a bridge across the reservoir’s diameter to a disembarking point, B. Distances OY, OA and AB are whole numbers of metres, with the latter two distances being square numbers.

Following development, a larger circular reservoir is constructed on the other side of the track, again touching OY at Y, with the corresponding new road and bridge having all the same properties as before. For both reservoirs, the roads are shorter than 500m, and shorter than their associated bridges. The larger bridge is 3969m long.

What is the length of the smaller bridge?