by Howard Williams
Published Friday February 16 2024 (link)
The Mint’s machine can press circular coins from square plates of precious metal in straight rows with no gaps between coins. Currently, the coins are pressed with the same number of coins in each column and row, with their centres lying on the same vertical and horizontal straight lines.
As newly appointed Warden of the Mint, Newton set about reviewing its operational efficiency. He found that more rows can be squeezed into the same plate by shifting some rows so that each coin in it touches two coins in the row above; each of these rows will have one fewer coin in it. With this method, the best that can be done is to produce exactly 8 per cent more coins per plate.
How many more coins per plate can be produced in this way?
by Mark Valentine
Published Friday February 09 2024 (link)
The racing chief tasked his designer for a new circuit. To create a 100:1 scale plan of the circuit the designer cut paper circles of 10cm radius into 12 equal sectors, then removed the top segments to create isosceles triangles with the two equal sides being 10cm. He arranged them on his desk to make a closed loop with no overlaps. Adjacent pieces always share the entire 10cm edge without overlap, but the short edges remain open.
The chief gave the following requirements. The start-finish line must be located on a straight section of at least 140m in length. Two circular viewing towers of 19m diameter must fit into the internal area, with one at either end of the circuit.
The designer minimised the internal area. How many triangles did he use?
by Victor Bryant
Published Friday February 02 2024 (link)
A group of fewer than twenty of us play a simple gambling game. We have a set of cards labelled 1, 2, 3, … up to a certain number and each player is allocated one of the cards at random. There is a prize for the player with the highest number from those allocated, and a booby prize for the lowest.
In a recent game I was unfortunately allocated a very middling number (in fact exactly half of the highest possible number). I calculated that my chance of winning the booby prize was 1 in N (where N is a whole number) and that my chance of winning the top prize was even lower at 1 in 2N.
How many cards are there in the set, and how many players?
by Colin Vout
Published Friday January 26 2024 (link)
Three companies run scenic coach trips either way between some pairs of towns in my holiday destination. Each route between two towns is served by just one company, but has one or more alternatives via another town. In particular, every Redstart route could be replaced by a unique combination of two Bluetail routes, every Bluetail by a unique combination of two Greenfinches, and every Greenfinch by a unique combination of a Redstart and a Greenfinch. This wouldn’t be possible with fewer towns or fewer routes.
I toured one route each day, starting from the town I’d arrived at the previous day but changing the company, never repeating a route in either direction, and involving as many of the routes as I could.
Which companies did I use, in order (as initials, eg, R,B,G,B)?
by Andrew Skidmore
Published Friday January 19 2024 (link)
Liam was given a set of addition sums as part of his homework. The teacher had cleverly chosen sums where the answer, together with the two numbers to be added, used all the digits from 1 to 9 just once (and there were no zeros). For instance, one of the sums was 193 + 275 = 468. I noticed that one of them was particularly interesting in that the two numbers to be added were perfect powers (squares, cubes etc.).
In the interesting sum, what are the two numbers that were to be added?
by Edmund Marshall
Published Friday January 12 2024 (link)
In our county football competition, fewer than 100 teams compete in two stages. First, the teams are allocated to more than two equally-sized groups, and in each group there is one match between each pair of teams. The top two teams in each group proceed to the first round of the knockout stage, where a single match between two teams eliminates one of them. If the number of teams entering the knockout stage is not a power of 2, sufficiently many teams are given byes (they don’t have to play in the first round), so that the number of teams in the second round is a power of 2. The knockout stage continues until only one team remains. In one year the competition was played with a single match on every day of the year.
How many teams were in the competition that year?
by Stephen Hogg
Published Friday January 05 2024 (link)
Agent J discovered that S.P.A.M.’s IQ booster drug comprises five elements with atomic numbers Z under 92. Some information had been coded as follows: Z1[4;6], Z2[1;4], Z3[4;6], Z4[0;6] and Z5[7;2], where Z5 is lowest and Z3 is highest. Code key: [Remainder after dividing Z by 8; Total number of factors of Z (including 1 and Z)]. The drug’s name is the elements’ chemical symbols concatenated in encoded list order.
J subsequently discovered the Z3 and Z5 values. Now MI6 had just fifteen possible sets of atomic numbers to consider. Finally, J sent a sixth element’s prime Z value (a catalyst in the drug’s production, not part of it). She’d heard it was below Z1 and above Z2, without knowing these. MI6 boffins were now sure of the drug’s name.
Give the catalyst’s atomic number.
by Victor Bryant
Published Friday December 29 2023 (link)
Here are some clues about my PIN:
• It has 3 digits (or it might be 4).
• The first digit is 3 (or it might be 4).
• The last digit is 3 (or it might be 4).
• It is divisible by 3 (or it might be 4).
• It differs from a square by 3 (or it might be 4).
In each of those statements above just one of the guesses is true, and the lower guess is true in 3 cases (or it might be 4).
That should enable you to get down to 3 (or it might be 4) possibilities for my PIN. But, even if you could choose any one of the statements, knowing which guess was correct in that statement would not enable you to determine the PIN.
What is my PIN?
by Mark Valentine
Published Friday December 22 2023 (link)
Kate and Lucy play a pub game on an isosceles triangle table top. They take turns to push a penny from the centre of the table top’s base towards the triangle’s apex, 120cm distant, scoring the sum of their distances from the base, or zero if it ever falls off the table.
Each player aims to maximise their distance, avoiding the chance that the penny might fall off the table. Their penny has an equal chance of landing anywhere within an error radius around the aimed point. This radius is proportional to the distance aimed. As a skilled player Lucy’s error ratio is half of Kate’s.
They both expect to score a whole number of cm per push, but to give each an equal chance of winning Kate gets one more push than Lucy. This number of pushes is the largest possible, given the above information.
How many pushes complete a game between the two?
by Howard Williams
Published Friday December 15 2023 (link)
I have a triangular-shaped garden, the sides of which are an exact number of feet long. To improve its usefulness, I’ve decided to partition it by building a straight fence from one corner to the centre of the opposite side. The length of this fence is exactly 51 feet and the side it attaches to is now 26 feet long each side of the fence.
What, in ascending order, are the lengths of the other two sides of my garden?