Goblin Tribe game
Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p8 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
November 11, 2022
combinatorics
Problem Statement
Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level () comes up. Let's assume Goblin . Goblin explodes itself. Goblin A's Level becomes . After that, Level of Goblin next to Goblin clockwise gets Level . Prove that:
1.) If after that Goblin next to Goblin explodes itself and keep doing this, next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again.
2) 2.) If after that we can choose any Goblin whose level is not to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning.(gools)