Solution of the Week #356 - Counter Game 2

This can be solved using exactly the same method as the first Counter Game puzzle. For all negative numbers, the number of ways is 0, and for 0 it is 1, and each subsequent number n, the number of ways is the sum of the number of ways of n-2 and n-7.

0:1

1:0

2:1

3:0

4:1

5:0

6:1

7:1

8:1

9:2

10:1

11:3

12:1

13:4

14:2

15:5

16:4

17:6

18:7

19:7

20:11

21:9

22:16

23:13

24:22

25:20

26:29

27:31

28:38

29:47

30:51

31:69

32:71

For 25 counters the number of ways is less than for 24. By coincidence n=25 is also the last time that the number of ways is less than n. To know for sure the sequence doesn’t reverse again after this point, it is only necessary to find 7 in a row that are increasing, hence me continuing the sequence to n=32.