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Plutarch's Boxes

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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:

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:

x becomes x+1
Turning transforms any number into the negative of its reciprocal

x becomes negative reciprocal of x
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.