In a game of tic tac toe, noughts and crosses are drawn inside an unoccupied cell of a 3 x 3 matrix by two players I, II in alternating moves. Player I draws crosses and Player II draws noughts. The game is won by the first player who draws a row or a column with the same symbol. If a game is not won by any player, which of the following statements are true and which are false.
(a) There is a column with at least 2 noughts
(b) there are 2 rows each with 2 noughts
(c) there are 2 rows and 2 columns with 2 crosses in each of these rows and columns
(d) there is on diagonal with at least 2 crosses.
How I am suppose to apply the pigeon-hole principle to this question. I understand that (a) is True but i do not understand why (b) to (d) are False.
Appreciate any help.
Thanks in advance.
There are 9 total turns in a game of Tic Tac Toe, with Player I getting 5 turns and player II getting 4 turns. Since the game was a draw, there is no column or row with three crosses or three noughts.
a) TRUE because Player II (noughts) has 4 turns and only 3 columns to choose from, hence one of his turns must have placed a second naught in a column with an existing naught.
b) FALSE because Player II had 4 turns. If he used all 4 to place 2 naughts in 2 rows then row 3 would have be all crosses and Player I would have won. Since no one won, this is false.
c) TRUE because Player I had 5 turns, there are 3 rows, 3 columns and each turn must be placed in a row and a column. Pigeonhole principle says 5/3 there must be two rows and two columns with at least 2 crosses.
Not sure about d.