### 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?

### 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?

### 3388

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:

The mathematician John Conway developed an interesting trick you can do with two skipping ropes and a number which we've reproduced in the video below. He only allows two operations: twisting and turning.

Twisting has the effect of adding 1:

Turning transforms any number into the negative of its reciprocal

Take a look at this video:
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This is how the ropes got tangled:

Twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn.

This is the sequence of numbers it produced:

0, 1, 2, -1/2, 1/2, 3/2, 5/2, -2/5, 3/5, 8/5, 13/5, -5/13...

and this is how they got disentangled:

Twist, turn. twist, twist, turn. twist, twist, twist, turn, twist, twist, twist,

generating these numbers:

...8/13, -13/8, -5/8, 3/8, -8/3, -5/3, -2/3, 1/3, -3, -2, -1, 0.

Starting at zero (with both ropes parallel), what would you end with after the following sequence of moves:

Twist, twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn.

What sequence of moves will take you back to zero?

You may want to take a look at More Twisting and Turning after this.