2010 IMO Problems/Problem 5
Each of the six boxes , , , , , initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box , , remove one coin from and add two coins to ;
Type 2) Choose a non-empty box , , remove one coin from and swap the contents (maybe empty) of the boxes and .
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes , , , , become empty, while box contains exactly coins.
Author: Unknown currently
Let the notation be the configuration in which the -th box has coin,
Our starting configuration is
We need these compound moves:
Compound move 1: , this is just repeated type 1 move on all coins.
Compound move 2: , apply type 1 move on 1 of the coin to get , then apply compound move 1 to the 2 coins to get , apply type 2 move and get , and repeat compound move 1 and type 2 move until is achieve.
Compound move 3: , apply compound move 2 to obtain and use type 2 move to get
Compound move 4: with 's. Apply compound move 3 times.
Compound move 5: with , use type 2 move times.
Let's follow this move:
Using Compound move 1, 4 and 5, We can obtain:
, where where X has 2's and Y has 2's, and is clearly bigger then