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Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

Twisting and Turning

Age 11 to 14 Challenge Level:

Twisting has the effect of adding 1: $$x\mapsto x + 1$$ Turning transforms any number into the negative of its reciprocal: $$x\mapsto -\frac{1}{x}$$ Starting at zero, these five moves: Twist, twist, twist, turn, twist
produce:$$0, 1, 2, 3, -\frac{1}{3}, \frac{2}{3}$$

Can you continue from there and then return to zero? You might find it helpful to record each step in a table.

Take another look at the video.
The team use a strategy to help them get back to zero.
Can you figure out how they decide when to stop twisting and start turning?

If you want to have a go at the trick for yourself, but don't have enough people or skipping ropes, you can also perform the tangling and untangling process using a small piece of card and two pieces of string.

Photo of a square piece of card with a small slit diagonally at each corner. Each slit holds in place one end of a piece of string. There is a tangle of string in the middle of the square.