$\underline{\text { Game } A .}$

We will call a rectangle $1 \times 3$ a trimino. Petya and Vasya independently divide a $20 \times 21$ board into triminos. Then they compare the resulting divisions, and Petya pays Vasya as many rubles as there are triminos that coincide in both divisions (occupy the same positions). What is the largest sum of winnings that Vasya can guarantee for himself regardless of Petya's actions?
Example. Vasya tiles a board with horizontal triminoes. Let the number of Petya's horizontal triminoes, the center of which lies in the -th column, be . Since there are 20 cells in the column and a vertical trimino covers three cells, then $a_{i-1}$ \# $2\left(\right.$ mod $\left.d_{i} 3_{y}\right)$. From this, $\equiv 2-a_{i} \equiv(\bmod 3)$. Since by definition $a_{i} \equiv$ 0, then $\equiv 2(\bmod 3)$, therefore in each of the columns $2,5,8,11,14,17,20$ there are centers of at least two of Petya's horizontal triminoes. They will coincide with Vasya's, giving him at least 14 rubles.

Estimate. We can assume that Petya knows Vasya's tiling. Petya tiles the top two rows with horizontal triminoes, which will give Vasya no more than 14 matches. The remaining board Petya divides into $3 \times 3$ squares. If a Vasya's horizontal trimino is completely inside one of these squares, Petya tiles it with vertical triminoes, otherwise - with horizontal ones, and there will be no matches in this square.

## Answer

14 rubles.