Spin the Multiples

Twelve dots sit evenly round a circle, and one number does everything: the step. Start at the top dot, count on by your step, draw a line to whichever dot you land on, and keep counting on until you arrive back where you began. The dots you land on are the multiples of your step, and the lines they leave behind are the figure.

Change the step and the figure changes with it. On a ring of twelve, stepping by 1 walks all the way round and touches every dot; stepping by 4 makes a triangle out of three; stepping by 6 goes straight across and comes straight back, so it visits two. Along the foot of the sheet is a bar for every step the ring has, and each bar grows to the number of dots that step landed on, so the whole ring's behaviour is there to look along once you have tried it.

Then a ring of twenty-four, and the question worth arguing about. Twenty-three is the biggest step you can take on that ring, and it is easy to expect a big step to make a big, complicated figure — or perhaps to fly round so fast it barely touches anything. It visits all twenty-four dots. Twelve, half as big, visits two. Eight visits three. The job is to find every step that lands on all twenty-four, and when the bars are in they stand at 1, 5, 7, 11, 13, 17, 19 and 23 — the small end and the big end together. Size turns out not to be the thing that sorts them. Sharing a factor with twenty-four is.