MathDB

4

Part of 2002 Putnam

Problems(2)

Putnam 2002 A4

Source:

3/12/2012
In Determinant Tic-Tac-Toe, Player 11 enters a 11 in an empty 3×33 \times 3 matrix. Player 00 counters with a 00 in a vacant position and play continues in turn intil the 3×3 3 \times 3 matrix is completed with five 11’s and four 00’s. Player 00 wins if the determinant is 00 and player 11 wins otherwise. Assuming both players pursue optimal strategies, who will win and how?
Putnamlinear algebramatrixsymmetrycollege contestsPutnam games
Putnam 2002 B4

Source:

3/12/2012
An integer nn, unknown to you, has been randomly chosen in the interval [1,2002][1,2002] with uniform probability. Your objective is to select nn in an ODD number of guess. After each incorrect guess, you are informed whether nn is higher or lower, and you <spanclass=′latex−bold′>must</span><span class='latex-bold'>must</span> guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than 23\tfrac{2}{3}.
Putnammodular arithmeticcollege contests