Problems

There are two piles of rocks: one with 30 rocks and the other with 20 rocks. In one turn a player is allowed to take any number of rocks but only from one of the piles. The loser is the player who has no rocks left to take. Who would win in a two player game, if the right strategy is used?

There are twenty dots distributed along the circumference of circle. Consider the game with two players where: in one move a player is allowed to connect any two of the dots with a chord (aline going through the inside of the circle), as long as the chord does not intersect those previously drawn. The loser is the one who cannot make a move. Which player wins?

Consider a rectangular parallelepiped with size a) $4\times 4\times 4$; b) $4\times 4\times 3$; c) $4\times 3\times 3$, made up of unit cubes. Consider the game with two players where: in one turn a player is allowed to pierce through any row with a long wire, as long as there is at least one cube in the row with no wire. The loser is the player who cannot make a move. Who would win, if the right strategy is used?

Two people take turns drawing noughts and crosses on a $9\times 9$ grid. The first player uses crosses and the second player uses noughts. After they finish, the number of rows and columns where there are more crosses than noughts are counted, and these are the points which the first player receives. The number of rows and columns where there are more noughts than crosses are the second player’s points. The player who has the most points is the winner. Who wins, if the right strategy is used?

A rook is on the a1 square of a chessboard. Consider the game with two players where: in one move a player can move the rook by any number of squares to the left, right or up. The winner is the player who places the rook on the square h8. Who would win, if the right strategy is used?

There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?

The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.

Prove that: $$a_1{a}_2{a}_3\cdots a_{n-1}{a}_n\times {10}^3\equiv a_{n-1}{a}_n\times {10}^3(\mathrm{mod}4),$$ where $n$ is a natural number and $a_i$ for $i=1,2,\dots ,n$ are the digits of some number.

Solve the equation $3x+5y=7$ in integers.

Determine all integer solutions of the equation $3x-12y=7$.