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Puzzles
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07-31-17
Can all 7 Tetris pieces fit into a 4x7 rectangle? The pieces can be rotated, but only one of each piece may be used. |
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SOLUTION
Nope! You can color the 4x7 rectangle like a checkerboard. Notice that there are 14 white squares and 14 black ones. Since all the Tetris pieces have 4 squares, they should each occupy 2 black and 2 white squares. However, all the pieces satisfy the aforementioned condition except for one. The piece on the right occupy either 3 black and 1 white or 3 white and 1 black, making it impossible to fit all the Tetris pieces in a 4x7 rectangle. |
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08-07-17
You and your friend are playing a game on a round table. There is an infinite supply of identical quarters which you use for the game. You and your friend will take turns placing quarters onto the table (assuming the table can hold more than 50 quarters), and the winner will be the one who places the final quarter (that is, the last quarter that makes it impossible for your friend to put down another quarter), thus completely filling the table. You cannot place quarters on top of other quarters, but you may and must place the quarter anywhere on the table. The quarter must be flat on the table. To guarantee a win, who should go first, and what is the optimal strategy?
You and your friend are playing a game on a round table. There is an infinite supply of identical quarters which you use for the game. You and your friend will take turns placing quarters onto the table (assuming the table can hold more than 50 quarters), and the winner will be the one who places the final quarter (that is, the last quarter that makes it impossible for your friend to put down another quarter), thus completely filling the table. You cannot place quarters on top of other quarters, but you may and must place the quarter anywhere on the table. The quarter must be flat on the table. To guarantee a win, who should go first, and what is the optimal strategy?
Solution:
You want to go first. Let's consider the table. Since it is a circular table, we can essentially "mirror" your friend's move--except for one spot: the middle of the board. By placing a quarter in the middle of the board, your friend must move somewhere else, and the optimal strategy is to mirror the move.
You want to go first. Let's consider the table. Since it is a circular table, we can essentially "mirror" your friend's move--except for one spot: the middle of the board. By placing a quarter in the middle of the board, your friend must move somewhere else, and the optimal strategy is to mirror the move.
Placement Test Solutions
| rmc_novice_solutions.pdf |
| rmc_intermediate_solutions.pdf |
| rmc_advanced_solutions.pdf |
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Math Contest Information
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