DE-MO 2017 Problem 3

General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!). To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns. Both of them must give a command and execute it in their turn.

Only two commands are possible: The command "Divide!" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier. On the other hand, the command "Rule!" removes exactly one tin soldier from each company.

The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command.

a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each?

b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?