BMO1 B 2020 7

(a)
We begin with EOE, then Od moves next to one of the pebbles making it EEO or OEE.

If the endpoints are EE, then the next move makes it EOE or EEE. Od needs to make it EEE, unless we have 2 adjacent even pebbles where Od just makes it EOE and ends the game.

If Ev plays EOE then Od moves an outer pebble next to O, then we have OEE or EEO, if instead we have EEE then Od goes to make it EEE again. And we repeat.

This way we have Od win.

(b)
If n is odd, then we use part (a) backwards, and force Od into an OO gap, where we keep doing this (creating OO gaps), we can center these around any odd n.

However if n is even, then Ev must play EOE or EEE after the first move, in which Od just uses the EEE strategy in part (a) and win.

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