*by Susan Bricket*

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

In this game, two players take turns drawing a single horizontal or vertical line between two unjoined adjacent dots in a rectangular grid. A player who completes the fourth side of a 1×1 box earns one point and takes another turn. The winner is the one with the most points at the end. A recent game with 4×4 dots (as above) between two beginners, reached a critical position when no box had more than two sides already drawn and the next player was forced to complete the third side of some box. It turned out that this was the shortest possible 4×4 game to reach a critical position. Their next 4×4 game reached a critical position in the largest possible number of turns.

How many turns were in each game?

**by Victor Bryant**

**by Victor Bryant**

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

My bedside clock displays the time and date using eight digits; for example at 9.43am on 28th June the display would be (see image, above).

Each digit in the electronic display lights up some (or all) of seven light segments, the above display lighting up a total of 45 segments.

On one occasion recently the displayed digits were all different and the total number of lit segments was prime. The same was true exactly one day later. Then, just one minute after the second occasion, the number of lit segments was the average of the numbers of lit segments on those two previous occasions.

What was that third display?

*by Michael Fletcher*

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

Father Christmas was rewarding his helpers (sprites, elves and fairies). He had 333 helpers, of whom 111 were elves.

He had 28 chocolate bars to give away in proportion to the size of the groups. If the groups’ shares were 9.5, 9.3 and 9.2 then they would actually receive 10, 9 and 9, because the bars can’t be split and the extra bar goes to the group with the highest remainder (for two extra bars, one each would go to the two groups with the highest remainders). In fact the sprites, elves and fairies received 7, 9 and 12 bars respectively. “I’ve got another chocolate bar,” said Father Christmas, “so now the sprites, elves and fairies will receive 6, 10 and 13 bars respectively.” “What’s happened to our chocolate bar?” asked the sprites.

How many sprites are there?

*by Danny Roth*

*by Danny Roth*

#### Published Sunday June 25 2023 (link)

George and Martha use a local underground station, which has up and down escalators and an adjacent 168-step staircase. Martha always uses the escalator. George, being the fitness fanatic, always uses the stairs. In olden times, he could keep up with her while climbing or descending at seven steps per second. But now he can only ascend at N steps per second and descend at N + 1 steps per second, N being a whole number. They start together at the bottom and go up and down continually, losing negligible time in turning. Interestingly, at some point after they have both been all the way up and down, but before ten minutes have elapsed, Martha overtakes George precisely half-way up or half-way down.

After how many seconds does this happen?

**by Adrian Somerfield**

**by Adrian Somerfield**

#### Published 22nd September 2002

Throughout the following calculations, each digit has consistently been replaced by a letter, with different letters for different digits:

**BL / WH – AC / IT = K / E**

Here the fractions are not necessarily in their simplest form — indeed, each of the three fractions in the above subtraction is in fact a whole number. Furthermore (though you might not need to know all these):

**IT / WB + CA / KL = H / E**

**WB / TI x KL / CA = E / H**

**BT / IK ÷ EL / WA = H / C**

What number is **BLACK**?

**by Stephen Hogg**

**by Stephen Hogg**

#### Published Sunday June 18 2023 (link)

A heckled lecturer used her megaphone. The power reaching its amplifier’s input was multiplied by a two-figure whole number (the “intrinsic gain”). In each of three stages a fraction of the power was lost but, by good design, each of the three fractions was under one-twentieth. Curiously, each happened to have a different two-figure prime denominator.

It turned out that the effective gain in the process (the ratio of the output power to the input power) was a whole number. In addition, the intrinsic gain and the effective gain both started with the same digit.

In ascending order, what were the three fractions?

*by Colin Vout*

*by Colin Vout*

#### Published Sunday June 11 2023 (link)

The gaslight glittered on the polished brass of the machine. “My Number Cruncher can calculate mechanically the result of a calculation on three positive whole numbers used once each, provided it is restricted to plus, times and brackets operations,” declared the Engineer. He opened the delicate cover and made meticulous adjustments. “There, I have disposed it for one particular expression. Please put it to the test.”

I chose three numbers, all greater than 1, and rotated the three dials to those positions. Then I cranked the handle until a delicate bell rang, and the result was indicated as 451. I made one more try, using the same three numbers although in a different order; this time the machine yielded 331.

In ascending order, what were the three numbers I selected?

**by Mark Valentine**

**by Mark Valentine**

#### Published Sunday June 04 2023 (link)

Grace and Helen play a turn-based card game. Each player is given two cards with a zero printed on them, and unlimited cards with a one printed on them. Before the game starts a 1 is placed on the table. Sitting on the same side of the table, a turn then involves placing a card to the left of the table cards, such that the value shown in binary remains prime or one (allowing leading zeros). The winner is the player who is last able to play a card, earning points equal to the value on the table at that time.

Being expert logicians, they play the best possible cards to maximise their points or minimise their opponent’s points. Grace goes first.

Who wins and with what cards on the table from left to right?

*by Peter Good*

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

The original wording can be obtained by clicking the above link. What follows is a revision to remove ambiguities and clarify the intent of the original wording:

In their art class, Jack and Jill each had a box of spherical glass marbles of at least three different radii, each a whole number of centimetres.

Jack placed three identical marbles from his box onto a desk and then placed each of the others in turn on top with all four marbles touching. He worked out the height of the top of the upper marble above the desk, obtaining a different whole number of centimetres 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 heights obtained were above 30 centimetres.

List all 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?