A trough has holes with diameters 1, 2, 3, . . . , n along its base in that order. It is called an n-trough. The trough is mounted on legs so it is tilted with the smallest hole uppermost.
With each n-trough there are n balls, each with integer diameter at most n. Two or more balls may have the same diameter. They are rolled down the trough one at a time, not necessarily in size order. If a ball drops through a hole, it triggers a trapdoor that closes the hole. A ball with diameter d is called a d-ball.
For example, if a 1-ball and two 2-balls are rolled down a 3-trough in the sequence 1, 2, 2, then each ball will drop through a hole. However, not all balls in the sequence 3, 1, 3 will drop. A sequence of balls is called a sinkable sequence if each ball drops through a hole.
a) Explain why the sequence 3, 1, 4, 3 is not sinkable on a 4-trough.
b) Explain why every sequence of five balls with diameters 1, 2, 3, 3, 4 is
sinkable on a 5-trough.
c) List all sinkable sequences for a 3-trough.
d) How many sinkable sequences for a 5-trough contain exactly one 4-ball and exactly one 5-ball?
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A trough has holes with diameters 1, 2, 3, . . . , n along its base in that order. It is called an n-trough. The trough is mounted on legs so it is tilted with the smallest hole uppermost. With each n-trough there are n balls, each with integer diameter at most n.
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