User blog:Wythagoras/Coins in boxes

Cite from Goucher's blog:

There are two different operations you’re allowed to do: Remove a coin from box i and add two coins to box i + 1. Remove a coin from box i and swap the contents of boxes i + 1 and i + 2.

Compute the largest number obtainable from any possible expression with these rules and arbitrary number of boxes.

Trivial values
\(f(1) = 1\), of course

\(f(2) = 3\)

For \(f(3)\) it is somewhat harder.

1,1,1

0,3,1

0,0,7

or

1,1,1

1,0,3

0,3,0

0,0,6

So \(f(3) = 7\)

\(f(4)\)
1,1,1,1

1,0,0,7

0,2,0,7

0,1,0,14

0,0,0,28

\(f(4) = 28\)

\(f(5)\)
1,1,1,1,1

0,3,0,0,7

0,2,2,0,7

0,2,1,0,14

0,2,0,14,0

0,1,14,0,0

0,1,13,0,4

0,1,12,0,8

...

0,1,1,0,\(2^{14}\)

0,1,0,\(2^{14}\),0

0,0,\(2^{14}\),0,0

0,0,0,0,\(2^{2^{14}}\)

\(f(5) \geq 2^{2^{14}}\)

\(f(6)\)
Ikosarakt found \(f(6) \geq 2\uparrow\uparrow 2 \uparrow\uparrow 2^{14}\).

It is very likely that \(2\uparrow\uparrow 2 \uparrow\uparrow 2^{14}\) is the best possible bound