*by Peter Good*

#### Published Sunday May 28 2023 (link)

In their art class, Jack and Jill each had a box of spherical glass marbles. Each box contained marbles of at least three different sizes, and the radius of every marble was a whole number of cm. Jack placed three marbles of equal size from his box onto a desk, and placed each of the others in turn on top so that all four marbles touched each other and formed a triangular pyramid. He worked out the height of each pyramid (from desk to top of top marble) and obtained a different whole number of cm each time. Jill was also able to do this with her marbles but they were all of different sizes to Jack’s. None of the pyramids was higher than 30cm.

List all of the different marble radii in ascending order.

*by Howard Williams*

#### Published Sunday May 21 2023 (link)

My new craft project involves card folding and requires a certain amount of precision and dexterity. For the penultimate stage a rectangular piece of card, with sides a whole number of centimetres (each less than 50cm), is carefully folded so that one corner coincides with that diagonally opposite to it. The resulting five-sided polygon also has sides of integer lengths in cm. The perimeter of the polygon is five twenty-eighths smaller than that of the perimeter of the original rectangular card.

As a final check I need to find the new area of the card.

What, in square centimetres, is the area of the polygon?

## Sunday Times Teaser 3164 – Touching Base

*by Andrew Skidmore*

#### Published Sunday May 14 2023 (link)

Liam has a pack of twenty cards; each card has a number printed on it as a word rather than figures.

There are two cards with **ONE**, two cards with **TWO** etc, up to two cards with **TEN**. He has taken four of the cards to construct an addition sum. The largest value card he has used is the sum of the values of the other cards, eg, **ONE + ONE + TWO = FOUR**.

If I now work in the base equal to that sum (ie, base 4 in the example), and consistently replace letters with single digits, this gives a correct sum. All of the possible digits in that base are used.

Leaving the answer in the working base, what is the total of my addition sum?

## New Scientist Enigma 835 -Treble Top

*by Mark Bryant*

#### From Issue #1990, 12th August 1995

Whenever I play darts I keep track of my score by writing down how many points I scored for each go (consisting of three darts) followed by the number of that go. For example, if I scored 27, 154 and 84 on my first three visits I would have written 2711542843.

After a recent game I noticed that if the digits were consistently replaced by letters, with different letters for different digits, then it read:

**TREBLETOPSATDARTS**

My total score was a prime number.

Please find the value of **PLEASE**.

*by Victor Bryant*

#### Published Sunday May 07 2023 (link)

To enable me to spell out ONE, TWO, up to NINE one or more times, I bought large quantities of the letters E, F, G, H, I, N etc. Then in a box labelled “ONE” I put equal numbers of Os, Ns and Es; in a second box labelled “TWO” I put equal numbers of Ts, Ws and Os; in box “THREE” I put equal numbers of Ts, Hs and Rs, together with double that number of Es; etc. In this way I made nine boxes from which my grandson could take out complete sets to spell out the relevant digit. In total there was a prime number of each of the letters, with equal numbers of Ns and Vs, but more Ts. Furthermore, the grand total number of letters in the boxes was a two-figure prime.

In the order ONE to NINE, how many sets were in each box?

*by Stephen Hogg*

*by Stephen Hogg*

#### Published Sunday April 30 2023 (link)

Housing its priceless Ming collection, the museum’s main hall had eight flashing, motion-sensor alarms. Each alarm’s flash ON and OFF times were settable from 1s to 4s in 0.1s steps. This allowed the flash “duty cycle” (the percentage of one ON+OFF period spent ON) to be altered. To conserve power, all duty cycles were set under 50%. All eight duty cycles were whole numbers, not necessarily all different.

Four alarms were set with consecutive incremental ON times (eg 2.0, 2.1, 2.2, 2.3). One pair of these had equal OFF times, as did the other pair (but a different OFF time from the first pair). Curiously, these two statements also apply to the other four alarms if the words ON and OFF are exchanged.

Give the lowest and highest duty cycles for each set of four alarms.

*Colin Vout*

#### Published Sunday April 23 2023 (link)

I arrived very late to watch the croquet game. Someone had listed the times when the four balls (blue, black, red and yellow) had run through hoops 1 to 12; none had yet hit the central peg to finish. Blue had run each hoop earlier than black, and every hoop had been run by colours in a different order. The only change in running order from one hoop to the next-numbered hoop was that two colours swapped positions. These swapping pairs (to obtain the order for the next-numbered hoop) were in non-adjacent positions for hoops 5 and 10, but no others. For one particular colour, all the hoops where it passed through earlier than the yellow were before all those where it passed through later than the yellow.

In what order had the balls run the twelfth hoop?

*by Edmund Marshall*

#### Published Sunday April 16 2023 (link)

At Church on Sunday, the hymn board showed just four different hymn numbers, each having the same three, different, non-zero digits, but in a different order. I noticed that the first hymn number was a perfect square, and that there was at least one other perfect square, formed from the same three digits, which did not appear on the board. The sum of the four hymn numbers was a prime number. If I told you the first digit of that prime number, you would be able to tell me the first hymn number on the board.

What was the first hymn number on the board?

## Sunday Times Teaser 3159 – King Coin

*by Mark Valentine*

#### Published Sunday April 09 2023 (link)

The new King’s currency has 64 Minims (circular coins) per Suttas (notes). Each coin’s value is proportional to its face area, and is a whole number of cm in diameter, starting with 1 cm for 1 Minim.

The King only wanted denominations such that his citizens can pay any amount below 1 Suttas using no more than a certain number of coins for the transaction. This number is the smallest possible, given the above conditions. His mint suggested that if just two values could require an extra coin, they could reduce the number of denominations needed. The King agreed and placed one of each minted denomination flat and face up in a rectangular display, with each coin’s edge resting along the display’s base. The order of the coins minimised the width of the display, with the smallest coin to the right of the centre.

What are the diameters in cm, from left to right?

## Sunday Times Teaser 3158 – Digital Trio

*by Danny Roth*

#### Published Sunday April 02 2023 (link)

“I have a couple of subtraction problems for you”, George told Martha.

Look: N1 – N2 = N3 and N3 – N4 = N5.

Can you solve them if I tell you that N1, N3 and N5 are all three-digit whole numbers whose sum is less than 2000, the same three non-zero digits appearing in all three numbers but no digit being repeated within any of those numbers? N2 and N4 are both two-digit whole numbers using two of the three digits mentioned above, and the first digit of N1 is not equal to the first digit of N2.

What is N1?