Kayles

Use the plus/minus buttons to add or subtract dots, three minimum) and then each player takes turn removing one or two adjacent dots. To remove a pair of adjacent dots, press the small circle between the dots.

The last player who can make a move, wins.

Strategy

The strategy is to leave your opponent with a zero nim sum where the nim values for each heap size are taken from the following table. First check the list of exceptional values and if the number of dots does not appear there, take the remainder after dividing by 12 and look up the value n the first table. Kayles has the property that apart from the exception list, the first table describes any possible number of dots. Discussion of how to add nimbers is here and a discussion of how this table was computed is here:

Heap Size Mod 12	0	1	2	3	4	5	6	7	8	9	10	11
Nimber	4	1	2	8	1	4	7	2	1	8	2	7

Exceptional Heap Sizes	0	3	6	9	11	15	18	21	22	28	34	39	57	70
Nimber	0	3	3	4	6	7	3	4	6	5	6	3	4	6