6. A stork, a cormorant, a sparrow, and a pigeon decided to weigh themselves. The weight of each of them turned out to be an integer number of parrots, and the total weight of all four was 32 parrots. Moreover,

- the sparrow is lighter than the pigeon;
- the sparrow and the pigeon together are lighter than the cormorant;
- the stork is lighter than the sparrow and the cormorant together;
- the pigeon and the cormorant together are lighter than the stork and the sparrow.

How much does each of them weigh? (Find all possible solutions and explain why there can be no other solutions.)
Solution:

Let's denote the weights of the birds by A, B, V, and G respectively.

Since $\Gamma +$ B $< \mathrm{A} + \mathrm{B}$, then $\Gamma +$ B is no more than 15 parrots, while $\mathrm{A} + \mathrm{B}$ is no less than 17 parrots.

If a sparrow could weigh at least 5 parrots, then a pigeon must weigh no less than 6, and thus a cormorant must weigh no less than 12 (since it is heavier than a sparrow and a pigeon together), but then together with a pigeon, it would already weigh 18, which is impossible. Therefore, a sparrow weighs no more than 4 parrots.

If the stork weighed 14 parrots, then a sparrow and a cormorant together must weigh no less than 15, but then the remaining pigeon would weigh no more than $32 - 14 - 15 = 3$, and the sparrow would weigh even less, i.e., no more than 2, and the sum $\mathrm{A} + \mathrm{B}$ would be no more than 16. If the stork weighed 15 parrots, it would be even worse: a sparrow and a cormorant would weigh no less than 16, a pigeon no more than 1, and then the sparrow could not be lighter than the pigeon. Therefore, the stork cannot weigh more than 13 parrots.

We have determined that A is no more than 13, V is no more than 4, and A + V is no less than 17. This is only possible if $\mathrm{A} = 13$ and $\mathrm{B} = 4$. Since $\mathrm{B} < \Gamma$, then $\Gamma$ is no less than 5, and since $\mathrm{A} < \mathrm{B} +$ B, then B is no less than 10. But since $\Gamma +$ B is no more than 15, then $\Gamma = 5$, and B $= 10$. This is the only possible solution.