Two people (call them A and B) are playing the following game:

• Two consecutive positive integers (eg 12 and 13) are chosen. One number is stuck to A's forehead and the other number is stuck to B's forehead. This means that A can see B's number and B can see A's number, but neither can see their own number.
• A goes first. If they know what their number is, they say it and win. Otherwise they say "pass".
• If A says "pass", then B has a turn. They either say what their number is or say "pass".
• The game goes back and forth in this way until someone works out what their number is.
• Naturally, A and B know each other to be experts at logic, and they won't guess - they'll only say their number if they're sure.

 

First, a few warm-up problems to check you've understood the game!

a) If A and B have 5 and 4 respectively, explain why A says "pass" on their first turn.

b) If A and B have 2 and 1, what will A say on their first turn?

c) If A and B have 1 and 2, how will the game go?

d) If A and B have 2 and 3, explain why the game will go A: "pass", B: "3".

e) (A bit harder) If A and B have 4 and 3, how will the game go?

 

Now for the big challenge:

• If A and B have 99 and 100 respectively, how many turns will the game last, and who will win?