You may also like

problem icon

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?

problem icon

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?

problem icon


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:

Edward and Thomas from Dartford Grammar School worked out that:

Starting at zero (with both ropes parallel), the sequence
twist, twist, twist, turn , twist, twist, twist, turn , twist, twist, twist, turn
takes us to:$$0, 1, 2, 3, -\frac{1}{3}, \frac{2}{3}, \frac{5}{3}, \frac{8}{3}, -\frac{3}{8}, \frac{5}{8}, \frac{13}{8}, \frac{21}{8}, -\frac{8}{21}$$

Oliver from Olchfa School also worked this out and noted that:

If the 'moves' "twist (x3), turn" are repeated, the fractions produced only include numbers from the Fibonacci sequence in the numerators and denominators

Terence from The Garden International School in Kuala Lumpur, William from Shebbear College and Akshita from Tiffin Girls' School worked out how to disentangle themselves:

The following sequence takes us back to zero:
twist, turn , twist, twist, turn , twist, twist, twist, turn , twist, twist, twist, turn , twist, twist: $$ \frac{13}{21}, -\frac{21}{13}, -\frac{8}{13}, \frac{5}{13}, -\frac{13}{5}, -\frac{8}{5}, -\frac{3}{5}, \frac{2}{5}, -\frac{5}{2}, -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, -2, -1, 0$$

Both sequences were written up by Sakib from Swanlea Secondary School - you can find his work here .
Oliver from Olchfa School worked this out and added:

To distangle this, we must reduce the fraction to the form 1/n.
We keep twisting the negative fraction until we get the first positive fraction (< 1) which is then turned.
The procedure is repeated until we get to 1/n.
Now we turn again, then twist n times to get the fraction back to 0.

The full proof is given in the problem More Twisting and Turning