PROBLEM OF THE WEEK-1

Here I start a new section called problem of the week. Solution of this question will be posted in the coming week. Till then have fun.

Alphonse and Beryl are back! They are playing a two person game with the following
rules:
• Initially there is a pile of N stones, with N >= 2.
• The players alternate turns, with Alphonse going first. On his first
turn, Alphonse must remove at least 1 and at most N −1 stones from
the pile.
• If a player removes k stones on their turn, then the other player must
remove at least 1 and at most 2k − 1 stones on their next turn.
• The player who removes the last stone wins the game.
(a) Determine who should win the game when N = 7, and explain the winning
strategy.[Easy]
(b) Determine who should win the game when N = 8, and explain the winning
strategy.[Medium]
(c) Determine all values of N for which Beryl has a winning strategy. Explain this
strategy.Where N<=100[Difficult]