Equal averages Comment How to derive the formula? Enumerate the first 15 possibilities: N: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 NS: 0 1 0 1 2 1 2 3 2 3 4 3 4 5 4 NS is number of sets See the pattern: 1: N: 1 4 7 10 13 NS: 0 1 2 3 4 2: N: 2 5 8 11 14 NS: 1 2 3 4 5 3: N: 3 6 9 12 15 NS: 0 1 2 3 4 or 1: N mod(3) = 1 -> NS = (N - 1) / 3 2: N mod(3) = 2 -> NS = (N + 1) / 3 3: N mod(3) = 0 -> NS = (N - 3) / 3 and 1: (N – 1) / 3 = (N - 3 + 2 * 1) / 3 2: (N + 1) / 3 = (N - 3 + 2 * 2) / 3 3: (N – 3) / 3 = (N - 3 + 2 * 0) / 3 All together: NS = (N - 3 + 2 * N(mod(3)) / 3 I verified the result for N = 1 up to 10000. Everything OK. Hub. Reply

How to derive the formula?

Enumerate the first 15 possibilities:

N: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

NS: 0 1 0 1 2 1 2 3 2 3 4 3 4 5 4

NS is number of sets

See the pattern:

1: N: 1 4 7 10 13

NS: 0 1 2 3 4

2: N: 2 5 8 11 14

NS: 1 2 3 4 5

3: N: 3 6 9 12 15

NS: 0 1 2 3 4

or

1: N mod(3) = 1 -> NS = (N - 1) / 3

2: N mod(3) = 2 -> NS = (N + 1) / 3

3: N mod(3) = 0 -> NS = (N - 3) / 3

and

1: (N – 1) / 3 = (N - 3 + 2 * 1) / 3

2: (N + 1) / 3 = (N - 3 + 2 * 2) / 3

3: (N – 3) / 3 = (N - 3 + 2 * 0) / 3

All together: NS = (N - 3 + 2 * N(mod(3)) / 3

I verified the result for N = 1 up to 10000. Everything OK.

Hub.