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Jun 11 21

Sunday Times Teaser 3064 – Turnip Prize

by Brian Gladman

by Victor Bryant

Published Sunday June 13 2021 (link)

The Turnip Prize is awarded to the best piece of work by an artist under fifty. This year’s winning entry consisted of a mobile made up of many different plain white rectangular or square tiles hanging from the ceiling. The sides of the tiles were all whole numbers of centimetres up to and including the artist’s age, and there was precisely one tile of each such possible size (where, for example, a 3-by-2 rectangle would be the same as a 2-by-3 rectangle). Last week one of the tiles fell and smashed and then yesterday another tile fell and smashed. However, the average area of the hanging tiles remained the same throughout.

How old is the artist?

Jun 5 21

Sunday Times Teaser 3063 – Pie by Five

by Brian Gladman

by Colin Vout

Published Sunday June 06 2021 (link)

We had a delicious pie, rectangular and measuring 20 centimetres along the top and 13 centimetres in the other direction. We divided it into five pieces of equal area, using five straight cuts radiating from one internal point. This internal point was rather off centre, in the top left-hand quarter, although the distances from the left and top sides were both a whole number of centimetres. The points where the cuts met the edges were also whole numbers of centimetres along the edges; one edge had two cuts meeting it, and the other three edges had one each.

How far was the internal point from the left and top sides, and how far along the four sides (starting at the top) did the cuts reach the edges (measured clockwise along the edges)?

May 28 21

Sunday Times Teaser 3062 – Family Stock

by Brian Gladman

by Howard Williams

Published Sunday May 30 2021 (link)

I have been transferring shares in the family business to my grandchildren, which I’ve done this as part of their birthday presents. On their first birthday I transferred one share, on their second birthday three shares, on their third birthday five shares etc. I have now four grandchildren and at the most recent birthday they were all of different ages. From my spreadsheet I noticed that the number of shares most recently transferred to each grandchild were all exact percentages of the total number of shares transferred to all of them over their lifetimes.

In increasing order, what are the ages of my grandchildren?

May 21 21

Sunday Times Teaser 3061 – Ratio

by Brian Gladman

by Andrew Skidmore

Published Sunday May 23 2021 (link)

Liam has split a standard pack of 52 cards into three piles; black cards predominate only in the second pile. In the first pile the ratio of red to black cards is 3 to 1. He transfers a black card from this pile to the second pile; the ratio of black to red cards in the second pile is now 2 to 1. He transfers a red card from the first pile to the third pile; the ratio of red to black cards in this pile is now a whole number to one.

Liam told me how many cards (a prime number) were initially in one of the piles; if I told you which pile you should be able to solve this teaser.

How many cards were initially in the third pile?

May 14 21

Sunday Times Teaser 3060 – Even and Odd

by Brian Gladman

by Edmund Marshall

Published Sunday May 16 2021 (link)

My daughter and I once played a game based on the number 13 and the following rule:

Think of a positive whole number greater than 1. If it is even, halve it. If it is odd multiply it by 13 and add 1. Either of these operations is to be regarded as one step. Apply another step to the outcome of the first step, and then further steps successively.

For our game, we chose different starting numbers that were odd, and the winner was the person who by the fewer number of steps reached the number 1. My daughter won because she started with the number that leads to 1 in the fewest number of steps.

What was my daughter’s starting number?

May 8 21

Sunday Times Teaser 3059 – Nine Blocks to the Diner

by Brian Gladman

by Colin Vout

Published Sunday May 09 2021 (link)

“Sure, I can help. From this hotel it’s nine blocks to the diner; from there it’s five blocks to the library, and then six blocks back here. I guess instead you could come back from the diner to the museum — that’s four blocks — and then seven blocks back here. Or three blocks to the gallery and then eight blocks back here.”

“So the diner’s straight along here?”

“No sir, it’s not straight along one road for any of these trips; I just meant so many edges of blocks. The city’s on a square grid, and all these places are on corners, but, fact is, none of them is on the same street or avenue as any other one.”

How many blocks between the library and the museum, museum and gallery, and gallery and library?

Apr 30 21

Sunday Times Teaser 3058 – Total Resistance

by Brian Gladman

by Peter Good

Published Sunday May 02 2021 (link)

A physics teacher taught the class that resistors connected in serial have a total resistance that is the sum of their resistances while resistors connected in parallel have a total resistance that is the reciprocal of the sum of their reciprocal resistances, as shown in the diagrams. Each pupil was told to take five 35-ohm resistors and combine all five into a network. Each pupil then had to calculate theoretically and check experimentally the resistance of his or her network. Every network had a different resistance and the number of different resistances was the maximum possible. The total sum of these resistances was a whole number.

How many pupils were there in the class and what was the sum of the resistances?

Apr 24 21

Sunday Times Teaser 3057 – Cut for Partners

by Brian Gladman

by Danny Roth

Published Sunday April 25 2021 (link)

George and Martha are playing bridge with an invited married couple. Before play starts, the players have to cut for partners. Each player draws a card from a standard pack and those drawing the two highest-ranking cards play together against the other two. For this purpose, the rank order is ♠ A, ♥ A, ♦ A, ♣ A, ♠ K, ♥ K etc. down to ♦ 3, ♣ 3, ♠ 2, ♥ 2, ♦ 2, ♣ 2 (the lowest).

George drew the first card, then Martha drew a lower-ranking card. “That is interesting!” commented the male visitor to his wife. “The probability that we shall play together is now P. Had Martha drawn the ♥ 7 instead of her actual card, that chance would have been reduced to P/2, and had she drawn the ♥ 6, the chance would have been reduced to P/3.

Which cards did George and Martha draw?

Apr 16 21

Sunday Times Teaser 3056 – Rose Garden

by Brian Gladman

by Andrew Skidmore

Published Sunday April 18 2021 (link)

The six rose bushes in my garden lie on a circle. When they were very small, I measured the six internal angles of the hexagon that the bushes form. These were three-digit whole numbers of degrees. In a list of them, of the ten digits from 0 to 9, only one digit is used more than once and only one digit is not used at all. Further examination of the list reveals that it contains a perfect power and two prime numbers.

In degrees, what were the smallest and largest angles?

Apr 10 21

Sunday Times Teaser 3055 – Proceed to Checkout

by Brian Gladman

by Colin Vout

Published Sunday April 11 2021 (link)

The dartboard at the Trial and Arrow pub is rather different from the standard one: there are only 3 sectors, each with a positive whole number label, and no central bullseye scoring region. There are still double and treble rings: for instance, if the sector label is 3, a dart in that sector can score 3, 6 or 9.

As usual, scores are counted down from a starting value, the idea being to finish (“check out”) by reaching exactly zero. Players take turns throwing three darts, or fewer if they check out before that. Unusually, the checkout doesn’t have to finish on a double.

The lowest impossible checkout is the smallest value that can’t be achieved in one turn; on this board that value is 36.

What are the sector labels?