Problem 6. Piglet and Winnie-the-Pooh decided to eat a $7 \times 7$ square chocolate bar. They take turns eating pieces from it: Piglet eats $1 \times 1$ pieces, Winnie-the-Pooh eats $2 \times 1$ or $1 \times 2$ pieces (pieces can be eaten not necessarily from the edge). Piglet makes the first move. If before Winnie-the-Pooh's turn, there are no $2 \times 1$ or $1 \times 2$ pieces left in the chocolate bar, the entire remaining chocolate goes to Piglet. Who among the friends can eat more than half of the entire chocolate bar regardless of the actions of the second?
Answer: Piglet.

Solution. Let's mentally color all the cells of the chocolate bar in two colors in a checkerboard pattern. We will get 24 black cells and 25 white cells.

![](https://cdn.mathpix.com/cropped/2024_05_06_90b2af0281b2e8bdc481g-5.jpg?height=408&width=418&top_left_y=91&top_left_x=518)

Let's describe one of the possible strategies for Piglet. On each move, Piglet will eat some black cell. Note that with each move, Winnie-the-Pooh eats exactly one black cell (and exactly one white cell). Thus, after the first 12 pairs of moves, all black cells will be gone, Winnie-the-Pooh will no longer be able to move, and the remaining chocolate will go to Piglet. In total, over the first 12 of his moves, Winnie-the-Pooh will eat no more than 24 cells, which is less than half of the entire chocolate bar, and he will not eat anything more. Therefore, Piglet, by following the given strategy, will eat more than half of the chocolate bar.

## Criteria

0 6. Only the correct answer.

0 6. A strategy for Piglet is given that works only against one or several possible strategies of Winnie-the-Pooh. Usually, such works unjustifiably assume that some moves of Winnie-the-Pooh are "more advantageous" than others and consider only "more advantageous" moves. This also includes works with incomplete consideration of possible actions by Winnie-the-Pooh.

1 6. The idea of a checkerboard coloring of the chocolate bar is present in the work, but there is no further progress.

2 6. The work describes the correct strategy for Piglet, which actually works against any possible actions of Winnie-the-Pooh, but there is no justification at all or the justification considers only one or several possible strategies of Winnie-the-Pooh.

4 points. Any complete and correct solution. A typical solution should include a description of Piglet's strategy and an explanation of why the given strategy works against any possible actions of Winnie-the-Pooh.