7.5. Once in a swamp there were 6 perch, 7 pike, and 8 shuk. Since there was no food in the swamp, the perch started eating the pike, the pike started eating the shuk, and the shuk started eating the perch, with the condition that no fish could eat another fish that had previously eaten an odd number of other fish. After some time, only one fish remained in the swamp. Which one? Justify your answer.
Solution: Suppose the pike remained. Then the pikeperch ate exactly 7 pikes, so one of the pikeperch must have eaten an odd number of pikes. This means that this pikeperch could not have been eaten. Contradiction.

Similarly, we get a contradiction if we assume that the perch remained: in this case, the pikes ate an odd number of perch (exactly 5), so one of the pikes must have eaten an odd number of perch and must survive.

The only remaining option is to show that a pikeperch can remain. Indeed, it is possible that first three perch each ate two pikeperch, then one of the pikes ate all six perch, and finally the last pikeperch ate all 8 pikes.

Answer: a pikeperch remained

Note: The example of fish eating each other, given in the solution, is not the only possible one.

Recommendations for checking:

| included in the work | points |
| :--- | :--- |
| Correct answer and an example of the sequence of fish eating each other with justification that no fish other than the pikeperch could remain the last | 7 points |
| Proven that the last fish could only be a pikeperch, but the example of fish eating each other is missing or incorrect | 6 points |
| Proven that the last fish could not be a perch | 3 points |
| :--- | :--- |
| Proven that the last fish could not be a pike | 3 points |
| Correct answer and an example of the sequence of fish eating each other, but without proof that no fish other than the pikeperch could remain the last | 1 point |
| Correct but unproven answer without a confirming example of fish eating each other | 0 points |

![](https://cdn.mathpix.com/cropped/2024_05_06_3160d2944af22fb50369g-8.jpg?height=525&width=540&top_left_y=771&top_left_x=127)

To the condition of problem 7.6