Times tables on a circle

With 10 dots and times 2, list where every dot 0 to 9 goes.

0→0, 1→2, 2→4, 3→6, 4→8, 5→0, 6→2, 7→4, 8→6, 9→8.

Draw the ten lines. Which dots get two lines arriving?

2, 4, 6 and 8 get two lines each: for example 1 and 6 both go to 2. Dot 0 gets one, from 5, as 0 goes to itself.

With 200 dots and times 2, where do dots 150 and 199 go?

300 − 200 = 100, and 398 − 200 = 198.

With 200 dots, which dots go to themselves under times 2? Under times 3?

Times 2: only dot 0, as 2n = n needs n = 0. Times 3: dots 0 and 100, as 2n must be 0 or 200.

How many cusps do times 2, times 3 and times 4 give?

1, 2 and 3: the cardioid, the nephroid and a curve with three cusps, always k − 1.

With 200 dots, show that times 101 sends every even dot to itself and every odd dot to the opposite side.

101n = 100n + n. For even n, 100n is a multiple of 200, so the dot stays n. For odd n, 100n leaves 100, so the dot is n + 100: half way round.

With 200 dots, what does times 201 draw, and why?

Nothing: 201n = 200n + n, so every dot goes to itself and every line has length zero.