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 Howmanysinkablesequencesfora5-troughcontainexactlyone4-ball and exactly one 5-ball?
a) To determine if a sequence is sinkable on a 4-trough, we need to check if each ball drops through a hole. In this case, we have the sequence 3, 1, 4, 3 on a 4-trough.
Starting with the first ball of diameter 3, we check if it drops through a hole. Since there are no existing holes, it opens a hole in the trough.
Moving on to the second ball of diameter 1, we check if it drops. However, since there is no hole with a diameter equal to or larger than 1, it does not drop through any hole.
Next, we have a ball of diameter 4. We check if it drops, but again, there is no hole with a diameter equal to or larger than 4, so it does not drop through any hole.
Finally, we have a ball of diameter 3 again. We check if it drops, but there are no holes with a diameter equal to or larger than 3, so it also does not drop through any hole.
Since not all the balls in the sequence drop through a hole, the sequence 3, 1, 4, 3 is not sinkable on a 4-trough.
b) To show that every sequence of five balls with diameters 1, 2, 3, 3, and 4 is sinkable on a 5-trough, we need to demonstrate that each ball drops through a hole.
Starting with the first ball of diameter 1, we check if it drops through a hole. Since there are no existing holes, it opens a hole in the trough.
Moving on to the second ball of diameter 2, we check if it drops through a hole. Since there is a hole with diameter 2 (opened by the previous ball), it drops through that hole.
Next, we have a ball of diameter 3. We check if it drops through a hole. Again, there is a hole with diameter 3 (opened by the previous ball), so it drops through that hole.
We then have another ball of diameter 3. Since there is still a hole with diameter 3 (opened by the previous ball), it drops through that hole.
Finally, we have a ball of diameter 4. We check if it drops through a hole. There is a hole with diameter 4 (opened by the previous ball), so it drops through that hole.
Since all five balls in the sequence drop through a hole, the sequence of balls with diameters 1, 2, 3, 3, and 4 is sinkable on a 5-trough.
c) To list all sinkable sequences for a 3-trough, we need to consider each possible arrangement of balls.
Since the trough has a capacity of 3, there can be at most 3 balls in a sinkable sequence. The possible sinkable sequences for a 3-trough are:
- 1-ball, 2-ball, 3-ball
- 1-ball, 3-ball, 2-ball
- 2-ball, 1-ball, 3-ball
- 2-ball, 3-ball, 1-ball
- 3-ball, 1-ball, 2-ball
- 3-ball, 2-ball, 1-ball
These are all the sinkable sequences for a 3-trough.
d) To determine how many sinkable sequences for a 5-trough contain exactly one 4-ball and one 5-ball, we need to consider the possible arrangements.
In this case, we have two fixed balls with diameters 4 and 5, and three remaining spots in the sequence. The possible arrangements are:
- 4-ball, 5-ball, x, x, x
- 4-ball, x, 5-ball, x, x
- 4-ball, x, x, 5-ball, x
- 4-ball, x, x, x, 5-ball
- x, 4-ball, 5-ball, x, x
- x, 4-ball, x, 5-ball, x
- x, 4-ball, x, x, 5-ball
- x, x, 4-ball, 5-ball, x
- x, x, 4-ball, x, 5-ball
- x, x, x, 4-ball, 5-ball
There are a total of 10 possible arrangements where there is exactly one 4-ball and one 5-ball in a sinkable sequence for a 5-trough.