Card game
Solution
Answer 1: The probability for Father Christmas to win the game
is zero.
First of all, a card lies either face up on the table, in one’s hand or in
the opponent’s hand. Therefore, both players know their opponent’s hand,
so basically, we can assume that the game is played with open hands.
Second of all, if we consider the situation where it’s the Grinch’s turn and
he has k cards in his hand, Santa Claus must have k − 1 cards left. Let S
denote the sum of the numbers visible on the cards already on the table. A
card x is bad for the Grinch if Santa Claus has the card y in his hand such
that S + x + y is divisible by 17. In this case, the Grinch cannot play the
card x, otherwise Santa would win by playing the card y. Since the Grinch
has more cards than Santa in his hand, he has at least one card in his hand
which is not bad. Therefore, the Grinch can play an arbitrary card in his
hand which is not bad and continue to his next turn.
Third of all, Santa Claus cannot win with his first card. The Grinch then
keeps playing his cards which are not bad and prevents Santa from winning
on his next turn. The game keeps going on and on until Santa has no more
cards and the Grinch still has one left. The Grinch plays his last card. The
sum on the table now equals to
1 + 2 + 3 + · · · + 16 = 8 · 17,
1
thus the Grinch wins.
We deduce that answer #1 is correct. Father Christmas has no chance of
winning at this game.
2