6
Part of 2020 Tournament Of Towns
Problems(3)
Alice has a deck of 36 cards, 4 suits of 9 cards each, game with Bob
Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p6
6/10/2020
Alice has a deck of cards, suits of cards each. She picks any cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bob gains a point. What is the maximum number of points he can guarantee regardless of Alice’s actions?Mikhail Evdokimov
combinatoricsgame
2n consecutive integers on a board, replace pairs by their difference and sum
Source: Tournament of Towns, Senior A-Level Paper, Spring 2020 , p6
6/3/2020
There are consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any consecutive integers again.Alexandr Gribalko
consecutivenumber theorycombinatorics
N white, blue and red cubes around a circle, a robot destroys 2, puts 1 back
Source: Tournament of Towns 2020 oral p6 (15 March 2020)
5/18/2020
Given an endless supply of white, blue and red cubes. In a circle arrange any of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors.
We will call the arrangement of the cubes good if the color of the cube remaining at the very end does not depends on where the robot started. We call successful if for any choice of cubes all their arrangements are good. Find all successful .I. Bogdanov
combinatoricswinning strategygame strategygame