Problem 5

Each of the six boxes

B_1,B_2,B_3,B_4,B_5,B_6

initially contains one coin. A type 1 operation chooses a nonempty box

B_j

with

1\le j\le5

, removes one coin from it, and adds two coins to

B_{j+1}

. A type 2 operation chooses a nonempty box

B_k

with

1\le k\le4

, removes one coin from it, and exchanges the contents of (possibly empty) boxes

B_{k+1}

and

B_{k+2}

. Determine whether a finite sequence of operations can leave

B_1,B_2,B_3,B_4,B_5

empty and

B_6

containing exactly

2010^{2010^{2010}}

coins. Here

a^{b^c}

means

a^{(b^c)}

Step 1 of 5: Reach a useful seed configuration

Previous step Next step

(1,1,1,1,1,1)\to(0,0,5,11,0,0)

Detailed analysis

Write a configuration as $(B_1,B_2,B_3,B_4,B_5,B_6)$ . Applying type 1 at $B_5$ and then at $B_1$ gives $(0,3,1,1,0,3)$ . The following legal moves, with unchanged coordinates suppressed, are $(0,3,1,1,0,3)\to(0,0,7,0,0,7)\to(0,0,6,2,0,7)\to(0,0,6,1,2,7)\to(0,0,6,1,0,11)\to(0,0,6,0,11,0)\to(0,0,5,11,0,0)$ . The first arrow uses type 1 at $B_2$ three times, then at $B_4$ once and $B_5$ twice; the other arrows use, respectively, type 1 at $B_3$ , type 1 at $B_4$ , type 1 at $B_5$ twice, type 2 at $B_4$ , and type 2 at $B_3$ . Hence the displayed seed configuration is reachable.