*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?

*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?

# 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?

*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?

*by Susan Bricket*

#### Published Sunday November 08 2020 (link)

In the bears’ villa there are three floors, each with 14 rooms. The one switch in each room bizarrely toggles (on <—> off) not only the single light in the room but also precisely two other lights on the same floor; moreover, whenever A toggles B, then B toggles A.

As Goldilocks moved from room to room testing various combinations of switches, she discovered that on each floor there were at least two separate circuits and no two circuits on a floor had the same number of lights. Furthermore, she found a combination of 30 switches that turned all 42 lights from “off” to “on”, and on one floor she was able turn each light on by itself.

(a) How many circuits are there?

(b) How many lights are in the longest circuit?

## Sunday Times Teaser 3032 – Darts Display

*by Andrew Skidmore*

#### Published Sunday November 01 2020 (link)

I noticed a dartboard in a sports shop window recently. Three sets of darts were positioned on the board. Each set was grouped as if the darts had been thrown into adjacent numbers (eg, 5, 20, 1) with one dart from each set in a treble. There were no darts in any of the doubles or bulls.

The darts were in nine different numbers but the score for the three sets was the same. If I told you whether the score was odd or even you should be able to work out the score. The clockwise order of numbers on a dartboard is:

20 1 18 4 13 6 10 15 2 17 3 19 7 16 8 11 14 9 12 5

What was the score that all three sets of darts made? [i.e. that made by each set of three darts]

**by Keith Austin**

**by Keith Austin**

#### From Issue #1734, 15th September 1990

We spent the day camped at the entrance to Apache Pass waiting for the cover of darkness. I passed the time playing cards with a grizzly old prospector, using a pack given him by Billy the Kid.

He took the spades from ace to ten and laid them in a row so:

6 2 9 7 3 10 8 1 4 5

where 1 is the ace.

“That’s the order Wyatt Earp dealt me just before the gunfight at the OK Corral. You can exchange any to cards provided the numbers on them are consecutive but the cards are not next to each other. For example, you can exchange the 7 and the 8, or the 2 and the 3, but not the 4 and the 5”.

“So I can exchange the 7 and 8 and get:

6 2 9 8 3 10 7 1 4 5

and then exchange the 3 and 4 and get:

6 2 9 8 4 10 7 1 3 5

and then exchange the 4 and 5 and get:

6 2 9 8 5 10 7 1 3 4

and so on?” I asked.

“You’ve got it. Now look at this old treasure map I got from Geronimo. It has 6 rows of numbers on it:

A: 7 2 10 9 4 5 3 1 8 6

B: 5 9 2 4 8 1 3 10 7 6

C: 7 4 5 2 1 10 8 3 6 9

D: 4 5 7 8 1 3 2 10 6 9

E: 10 3 4 2 1 9 6 5 7 8

F: 6 10 8 5 3 1 2 4 7 9

“Geronimo said the to discover the treasure it was necessary to find which of these six rows can be obtained from Wyatt’s row by the allowed

operations.”

Find all those of the six rows which can be obtained from Wyatt’s row.

*by Howard Williams*

#### Published Sunday October 25 2020 (link)

Jenny is using her calculator, which accepts the input of numbers of up to ten digits in length, to prepare her lesson plan on large numbers. She can’t understand why the results being shown are smaller than she expected until she realizes that she has entered a number incorrectly.

She has entered the number with its first digit being incorrectly entered as its last digit. The number has been entered with its second digit first, its third digit second etc. and what should have been the first digit entered last. The number she actually entered into her calculator was 25/43rds of what it should have been.

What is the correct number?

*by Susan Denham*

#### From Issue #1732, 1st September 1990

In the two division sums below, the six-figure dividend is the same but the divisor and the answer have been interchanged (that is, the first is x ÷ y = z and the second is x ÷ z = y). No digit occurs in both the divisor and the answer, and no digit in the dividend is used in either the divisor or the answer.

What is the dividend?

# Sunday Times Teaser 3030 – Pot Success

*by John Owen*

#### Sunday October 18 2020 (link)

In snooker, pot success (PS) is the percentage of how many pot attempts have been successful in that match (eg, 19 pots from 40 attempts gives a PS of 47.5). In a recent match, my PS was a whole number at one point. I then potted several balls in a row to finish a frame, after which my improved PS was still a whole number. At the beginning of the next frame, I potted the same number of balls in a row, and my PS was still a whole number. I missed the next pot, my last shot in the match, and, remarkably, my PS was still a whole number.

If I told you how many balls I potted during the match, you would be able to work out those various whole-number PS values.

How many balls did I pot?