Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Twisting and Turning

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

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.

## You may also like

### Tweedle Dum and Tweedle Dee

### Sum Equals Product

### Special Sums and Products

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.

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?

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...

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 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.