### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

### Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

### Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

# Twisting and Turning

##### Stage: 3 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.