Problem of the Week 1233

The Generous Automated Teller Machine

Imagine you have five boxes, B1, B2, B3, B4, B5; and each one contains one coin. You may make moves of the following sort:

1. Remove a coin from a nonempty box B(i) and place two coins in B(i + 1) (here, i is 1, 2, 3, or 4).
2. Remove a coin from a nonempty box B(i) and switch the contents of B(i + 1) and B(i + 2) (here, i is 1, 2, or 3).

What is the largest number of coins you can place in B(5)?

In this form, it appears to be unsolved. I would be interested to know how large you can get. The original contest problem here had six boxes; and that is, of course, more interesting. I posed the case of 5 because, perhaps for that, one can prove what the maximum is. Feel free to send me your values for either version.

Source: Invented by Hans Zantema, problem 5, 2010 International Mathematical Olympiad, Astana, Kazakhstan (July 8, 2010)

I saw it in this very very nice new problem book, Half a Century of Pythagoras Magazine, eds A. van den Brandhof, J. Guichelaar, and A. Jaspers, MAA, 2015.

[View the solution]