## Cheese and Butter at the Fall Tournament of the Towns

Here’s a fresh challenge from the recent Tournament of the Towns, crafted by Alexander Shapovalov.

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Puzzle.A mother and her son are playing a game involving cheese and butter. The son starts by cutting a 300-gram block of cheese into 4 pieces. Then, the mother divides 280 grams of butter between two plates. Afterward, the son places the cheese pieces onto these same plates. The son wins if, on both plates, there is at least as much cheese as butter. If not, the mother wins. Can either the mother or the son guarantee a win, regardless of the other’s moves?

## Kai:

The son divides the cheese into 4 pieces of 20, 40, 80, and 160 grams, which guarantees his win with two margins of 0 – 20 grams evolving in the both plates.

23 October 2024, 10:16 am## Sanandan Swaminathan:

Assuming that the son has a way to cut the cheese accurately and knows how much the butter weighs on one plate, he can guarantee that he wins the game. Initially, he can split the 300 grams of cheese into 20, 40, 80, and 160 gram pieces. His mom could either split the 280 grams of butter into two equal parts (140 grams each), or into two unequal parts.

No matter how the mom splits the butter, the son can place appropriate cheese pieces on the plates such that there is at least as much cheese as butter on both plates. WLOG, assume that plate1 has less butter than plate2. Here are the various cases based on the amount of butter in plate1:

If plate1 has 0-20 gram butter, the son would place the 20-gram cheese piece on plate1 and the other cheese pieces (totaling 280 grams) on plate2. In this case, plate2 would have 0-280 grams of butter, so the son would have at least as much cheese as butter on both plates.

If 20-40 gram butter on plate1, place the 40-gram cheese piece on plate1 and the remaining (totaling 260 grams) on plate2.

If 40-60 gram butter on plate1, place the 20 and 40 gram cheese pieces on plate1 and the remaining (totaling 240 grams) on plate2.

If 60-80 gram butter on plate1, place the 80-gram cheese pieces on plate1 and the remaining (totaling 220 grams) on plate2.

If 80-100 gram butter on plate1, place the 20 and 80 gram cheese pieces on plate1 and the remaining (totaling 200 grams) on plate2.

If 100-120 gram butter on plate1, place the 40 and 80 gram cheese pieces on plate1 and the remaining (totaling 180 grams) on plate2.

If 120-140 gram butter on plate1, place 20, 40 and 80 gram cheese pieces on plate1 and the remaining (totaling 160 grams) on plate2.

If the mom splits the butter equally between the two plates (140 grams each), then the son would place the 20, 40 and 80 gram cheese pieces in one plate and the 160 gram piece in the other plate.

If the mom puts more than 140 grams of butter on plate1, there would be less than 140 grams on plate2, so the son would respond using the cases given above (but based on the amount of butter on plate2).

This puzzle reminds me of the pan balance puzzle where the weights given are 1,2,4,8… (powers of 2), and different combinations can be used to produce any weight (i.e. represent the weight to be achieved as a binary number).

23 October 2024, 11:30 am## Sanandan Swaminathan:

It is very mom-like of the mom to play the game where she knows that her son will win :).

23 October 2024, 11:40 am## Ivan:

How complicated. If I were the son, I would have:

23 October 2024, 12:04 pm> glued the top of plate 2 to the bottom of plate 1 and

> put the cheese in plate 1.