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Problems Discrete Mathematics Algorithm Theory Game theory Symmetric strategies

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?

Problem #PRU-30453

Problems Discrete Mathematics Algorithm Theory Game theory Symmetric strategies

12–13 3.0

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?

Problem #PRU-30454

Problems Discrete Mathematics Algorithm Theory Game theory Winning and loosing positions

12–13 3.0

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?

Problem #PRU-30456

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

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?

Problem #PRU-30467

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

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.

Problem #PRU-30611

Problems Number Theory Numeral systems Decimal number system Divisibility Divisibility rules Divisibility tests for 2 and 4

11–13 2.0

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

Problem #PRU-30648

Problems Algebra Algebraic equations and systems of equations Diophantine equations

12–15 3.0

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

Problem #PRU-30649

Problems Algebra Algebraic equations and systems of equations Diophantine equations

11–13 1.0

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

Problem #PRU-30651

Problems Algebra Algebraic equations and systems of equations Diophantine equations

13–15 3.0

Determine all the integer solutions for the equation $21 x + 48 y = 6$ .

Problem #PRU-30653

Problems Number Theory Divisibility Equations in integer numbers

11–12 2.0

A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by $m$ squares to the right or by $n$ squares to the left. For which $m$ and $n$ can it move to the next cell to the right?