Combinatorics · Combinatorial games · Invariants · Powers of two · Strategy
All powers of two are written on a board in increasing order: \(1, 2, 4, \ldots\). Aca and Braca now take turns, Aca first. A move consists of choosing two numbers that stand next to each other on the board and replacing the two of them by their sum. Braca's aim is that after some move the board carries two numbers, both greater than \(1\), whose difference is \(1\).
(a) Prove that Aca can play in such a way that Braca's aim is never reached.
(b) Suppose Braca knows in advance that at some moment he will be allowed to make two moves in a row. Prove that he can then reach his aim.
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Serbian National Competition (Drzavno takmicenje) 2022, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source