-256x256.png)
In the realm of game theory, a player's strategy refers to the choices they make within a context where the best possible result is influenced not solely by their individual decisions but also by the decisions made by others. The field primarily focuses on how a player's moves within a game can impact and influence the behaviour and choices of other participants.
1. Two people take turns placing rooks on the chessboard (one rook at a time) so that they do not beat each other. (Who placed which rook is not taken into account. You cannot place a rook even under the battle of your own rook.) Whoever cannot place a rook loses. Who will win if played correctly - first or second?
2. George and Frederik write out an 8-digit number, putting the numbers in order, starting with the highest digit. George starts. Can Frederik get the number to be divisible by 9?
3. The rook is on square a1. During a turn, you are allowed to move it any number of cells to the right or any number of cells up. The winner is the one who places the rook on the h8 square. Who has a winning strategy?
4. There are a) 2; b) 3 identical piles of stones. Two players take turns taking any number of stones from any pile, but only from one. The one who takes the last stones wins. Who will win if played correctly?
5.
a) Two people play by moving the king around the chessboard. Moves one field to the left, down, or diagonally left and down are allowed. The one who places the king on square a1 wins. At what initial positions of the king does the beginner win, and at which does his partner win?
b) There are two piles of stones. Two players take turns taking stones. It is allowed to take one stone from any pile or one stone from both piles. The one who takes the last stones wins. Explore this game.
6. There are 12 cells in a row. There is a white chip on the rightmost square, and a black one on the leftmost square. Two players take turns moving their piece one space - forward or backward. (You cannot skip a move.) The loser is the one who does not have a move. Who wins: the beginner or his partner?
7. Two players take turns moving the clock hand 2 or 3 hours forward. The hour hand first points to 12; The winner is declared the one after whose move she pointed to 6. Who will win if the game is played correctly? (The needle may rotate several times before stopping at 6.)
8. There are several disadvantages written in the line. Two players take turns transferring one or two adjacent minuses to a plus. The one who sent the last minus wins. Who will win if played correctly, the beginner or his partner?
9. Two people take turns picking off the petals of a chamomile, and at one time you can tear off 1 or 2 neighboring (growing nearby) petals. The one who makes the last move wins. Who will win if played correctly?
10. On a 7x7 board, two people take turns coloring the squares so that they do not have a single common side; b) points. The one who cannot make a move loses. Who will win if played correctly?
11. There are 20 points on a circle. Two people are playing. Each move the player draws a chord with ends at these points so that the chords do not intersect inside the circle. (Chords can have common ends.) The one who cannot draw the chord loses. Who will win if played correctly?
A chessboard is given. In one move, it is allowed to cover any two previously uncovered squares with a 1×2 domino. The player who cannot make a move loses.
Alice and Charlie play a game, taking it in turns with Alice going first.
On a blackboard is written a three-digit number. If the current number on the blackboard is n, a move consists of choosing a non-zero digit, k, of n and replacing n with n − k on the blackboard. This is repeated until the number 100 is written on the blackboard.
The player who writes the number 100 wins.
(a) If the starting number is 125, Alice can always win. State Alice’s first move and how Alice responds to whatever move Charlie makes at each stage.
(b) Find, with proof, for which starting values Charlie has a winning strategy.
From UKMT - Maclaurin Mathematical Olympiad - 2025 - 2
There are 20 points on a circle. Two people are playing. Each move, the player draws a chord with ends at these points so that the chords do not intersect inside the circle. (Chords can have common ends.) The one who cannot draw a chord loses. Who will win if played correctly?
On a 7×7 board, two people take turns coloring the squares so that they do not have a single common side. The one who cannot make a move loses. Who will win if played correctly?
Two people take turns picking off the petals of a chamomile, and at one time you can tear off 1 or 2 neighboring petals.
The one who makes the last move wins. Who will win if played correctly?
There are several minus signs written in a row.
Two players take turns converting one or two adjacent minuses to pluses.
The one who converts the last minus wins.
Who will win if played correctly - the beginner or his partner?
Two players take turns moving the clock hand 2 or 3 hours forward.
The hour hand first points to 12.
The winner is the one after whose move the hand points to 6.
Who will win if the game is played correctly? (The needle may rotate several times before stopping at 6.)
There are 12 cells in a row.
There is a white chip on the rightmost square, and a black one on the leftmost square.
Two players take turns moving their piece one space – forward or backward. (You cannot skip a move.)
The loser is the one who does not have a move. Who wins: the beginner or his partner?
There are two piles of stones.
Two players take turns taking stones.
It is allowed to take one stone from any pile or one stone from both piles.
The one who takes the last stones wins. Explore this game.
Two people play by moving the king around the chessboard. Moves one field to the left, down, or diagonally left and down are allowed.
The one who places the king on square a1 wins. At what initial positions of the king does the beginner win, and at which does his partner win?
There are a) 2; b) 3 identical piles of stones. Two players take turns taking any number of stones from any pile, but only from one. The one who takes the last stones wins. Who will win if played correctly?
The rook is on square a1. During a turn, you are allowed to move it any number of cells to the right or any number of cells up. The winner is the one who places the rook on the h8 square. Who has a winning strategy?
George and Frederik write out an 8-digit number, putting the numbers in order, starting with the highest digit. George starts. Can Frederik get the number to be divisible by 9?
Two people take turns placing rooks on the chessboard (one rook at a time) so that they do not beat each other. (Who placed which rook is not taken into account. You cannot place a rook even under the battle of your own rook.) Whoever cannot place a rook loses. Who will win if played correctly – first or second?
