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.

Solution:

I claim we can accomplish this final figuration. In the following lemmas, an ordered -tuple will represent the number of coins in each box (in numerical order). Arrows between ordered -tuples will denote a sequence of operations (that should be easy to figure out from context) that brings the first -tuple to the second

Lemma 1: It suffices to prove that is obtainable where

Proof: If we have at some point then we can perform a number of Type 2 switches on the fourth box and get Then performing as many Type 1 switches as possible on box 4 and then as many as possible on box 5 we obtain the desired configuration.

Lemma 2: From we can get for any three consecutive boxes

Proof: I will show by induction on that from we can get for any positive integer The base case is trivial, and the fact that we can get proves the induction hypothesis. Letting now yields the desired result.

Lemma 3: From we can get where there are 's in the tower of exponents for any four consecutive boxes

Proof: I will show by induction on that from we can get where there are 's in the tower of exponents for any positive integer Let where there are 's in the tower of exponents. Now the base case is trivial, and the fact that by Lemma 2 we can get proves the induction hypothesis. Letting now yields the desired result.

Now, letting where there are 's in the tower of exponents and letting where there are 's in the tower of exponents, note that we can get and since it is clear that (by a lot!) by Lemma 1 we have solved the problem.