1. Try starting with fewer objects to investigate.
Nim
The rules:
Make a pile of objects (eg 10 pencils).
Take it in turns to remove 1 or 2 items from the pile.
Whoever takes the last object loses.
Is there a winning strategy? Does it matter who goes first?
Try starting with fewer objects to investigate.

2. 1
LOSE! 2 3 4 5 6
Click to successively hide counters. When the last counter is gone, whoever removed it has lost. 7 8 9 10

3. 1
LOSE! 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Click to successively hide counters. When the last counter is gone, whoever removed it has lost. 16 17 18 19 20

4. 1 1 2 2 3 3
The ‘bad’ numbers (the ones to avoid) form a sequence:
1, 4, 7, 10, 13, 16, 19, 22, …
What’s the best way to describe it?
LOSE!
LOSE! 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12
You are given the chance to take first move in a game with 50 counters.
Should you take it? If so, how many counters should you take at the start?
What about for 100 counters? 15 15 13 13 14 14
50 is not one more than a multiple of 3, so it is not a ‘bad’ number. Since 48 does divide by 3, 49 is the next bad number, so take one counter for your first move (from then on, whatever your opponent does, you do the opposite to keep them on bad numbers). For 100 counters, you’ll lose if you play against a perfect player, as 100 is a bad number. You’ll just have to hope they mess up and that you notice when you are not left on a bad number. 16 16 17 17 18 18 19 19 20 20