Chebotarev A.S.

Ilya Muromets, Dobrynya Nikitich, and Alyosha Popovich were given 6 coins for their faithful service: 3 gold and 3 silver. Each received two coins. Ilya Muromets does not know which coins went to Dobrynya and which to Alyosha, but he knows which coins he received. Come up with a question to which Ilya Muromets will answer "yes," "no," or "don't know," and from the answer, you will be able to determine which coins he received.
Let's check if the question is suitable: "Is it true that you have more gold coins than Alyosha Popovich?"

If Ilya Muromets has two gold coins, he will say "yes," since Alyosha Popovich cannot have more than one gold coin.

If both coins are silver, then Alyosha has at least one gold coin, and Ilya Muromets will answer "no." If he got different coins, he will answer "I don't know," because Alyosha could have either two gold coins or two silver coins.

Of course, other questions could have been asked, for example:

-- Is it true that one of the other two heroes got two silver coins?

-- Is it true that the other two heroes each got at least one gold coin?

-- If I take one of your coins and give you a gold one instead, will you have more gold coins?

(Note that the last question only mentions Ilya Muromets' coins and does not mention the coins of the other two heroes!

## Answer

For example: "Is it true that you have more gold coins than Alyosha Popovich?"

A checker is placed at the bottom-left corner of a 5x9 rectangle drawn on graph paper. Kolya and Seryozha take turns moving it any number of cells either to the right or up. Kolya moves first. The player who places the checker in the top-right corner wins. Who wins with correct play?

## Solution

With each move, Kolya places the checker on one of the marked diagonal cells (see the figure). Seryozha, with his move, removes it from there. Since they can only move to the right or up, the game will eventually end.
![](https://cdn.mathpix.com/cropped/2024_05_06_e217990455cb7ab9a875g-09.jpg?height=388&width=714&top_left_y=566&top_left_x=677)

The problem can also be solved by working backwards using the analysis of winning and losing positions.

## Answer

Kolya wins.

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