MathDB

5

Part of 1996 Estonia National Olympiad

Problems(4)

2 player game with n sweets on a table, winning strategy

Source: 1996 Estonia National Olympiad Final Round grade 10

3/11/2020
John and Mary play the following game. First they choose integers n>m>0n > m > 0 and put nn sweets on an empty table. Then they start to make moves alternately. A move consists of choosing a nonnegative integer k≤mk\le m and taking kk sweets away from the table (if k=0k = 0 , nothing happens in fact). In doing so no value for kk can be chosen more than once (by none of the players) or can be greater than the number of sweets at the table at the moment of choice. The game is over when one of the players can make no more moves. John and Mary decided that at the beginning Mary chooses the numbers mm and nn and then John determines whether the performer of the last move wins or looses. Can Mary choose mm and nn in such way that independently of John’s decision she will be able to win?
gamegame strategywinning strategycombinatorics
game with numbers 1, ... ,mn in a mxn board

Source: 1996 Estonia National Olympiad Final Round grade 9

3/11/2020
Three children wanted to make a table-game. For that purpose they wished to enumerate the mnmn squares of an m×nm \times n game-board by the numbers 1,...,mn1, ... ,mn in such way that the numbers 11 and mnmn lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number 11) in one of the corners but each child wanted to have the final square (with number mnmn ) in different corner. For which numbers mm and nn is it possible to satisfy the wish of any of the children?
tablecombinatoricsboard
n triangles, any 3 of them have common vertex, not 4 of them

Source: 1996 Estonia National Olympiad Final Round grade 11 p5

3/11/2020
Suppose that nn triangles are given in the plane such that any three of them have a common vertex, but no four of them do. Find the greatest possible nn.
combinatorial geometrycombinatoricsTriangles
n tetrahedra, any 2 have at least 2 common vertices, any 3 at most 1

Source: 1996 Estonia National Olympiad Final Round grade 12 p5

3/11/2020
Suppose that nn teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible nn.
combinatorial geometrycombinatoricstetrahedron