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8

Answer to the non-bonus part: Proof: Bonus part: From 12 Hence


7

You should To prove, let us first do the following


7

Here's a little Python program to test it yourself: https://repl.it/repls/ScientificIdenticalPixels And here's C++ code written by user @im_so_meta_even_this_acronym https://ideone.com/SfMHqC


4

Let’s see: I assume that all numbers must be greater than or equal to zero, otherwise the possibilities become too painful. If that was the intention, I’ll look harder. 1 2 3 4 5 6 7 8 9 10 11 Therefore the only starting numbers that win for Alice are Bonus Part: 12


4

Code to find the pattern: https://ideone.com/O5S4Qu (The third number printed out on each line is the winning move if the player to move is in a winning position)


3

User hexomino already figured out the puzzle, and managed to actually find the very complicated path that was exactly how I came up with the game. To recap: The game itself is a lot easier to play than that, though, so I'm posting this self-answer to show how. First, it's very useful to note that in a given board position, every move is uniquely defined by ...


3

We have a winning strategy for: The first move is:


3

Any permutation of the rows and columns is treated as an "equivalent" board. Part 1: Matt chooses to go second. (Case A) If Ben takes 2 or 3 tokens, then Matt can turn it into a 2x2 board and win. (Case B) If Ben takes 1 token: $$\begin{array}{ccc} O & O & O \\ O & O & O \\ \_ & O & O \end{array}$$ then Matt can take 2 tokens to ...


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