It seems that everyone with a blog is always apologizing for not posting in a while, as if this has somehow inconvenienced their readers. Of course, with the magic of feed readers, email notifications, and the like, this is not true at all. You, gentle reader, in all likelihood have not even thought about my blog once since the last time I posted. (Yes, I forgive you.) So I won’t apologize for not posting in a while. If I did, I might ask you to please forgive me due to the fact that I am hard at work on graduate school applications. But I won’t, remember? Onwards!

You may recall that my last post concerned the game of Nuclear Pennies. If you don’t remember it–or if you haven’t had a chance to try solving it yourself–now is a good time!

From George Bell (who also has a great site about peg solitaire–if you enjoy these sorts of mathematical games, you should check it out) comes the following excellent solution:

This puzzle seems rather difficult to solve by trying random moves that appear to get one closer to the solution. However, consider the board a sequence of integers, each being the number of pennies at that location, and the moves as adding or subtracting (+1, -1, +1) to three consecutive values.

Then by inspection we can come up with a solution, not worrying about whether the number of pennies is always positive, by adding the columns:

  0  0  0  0  0  0 +1 (start)
  0  0  0  0 -1 +1 -1
  0  0  0 -1 +1 -1  0
  0 +1 -1 +1  0  0  0
 +1 -1 +1  0  0  0  0
---------------------------
 +1  0  0  0  0  0  0 (end)

This gives you the four critical moves that must be in any solution. However, applying them directly is illegal. [But you can] just [keep] splitting the left penny (7 times), then [do the] “critical 4 moves” and you come up with the mirror image position, so you now combine to the final position. Ergo,

0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0 1
0 0 0 0 0 1 0 1 1
0 0 0 0 1 0 1 1 1
0 0 0 1 0 1 1 1 1
0 0 1 0 1 1 1 1 1
0 1 0 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1
1 1 0 2 1 1 1 1 1  <--  +1 -1 +1
1 1 1 1 2 1 1 1 1  <--     +1 -1 +1
1 1 1 1 1 2 0 1 1
1 1 1 1 1 1 1 0 1  <--           -1 +1 -1
1 1 1 1 1 1 0 1 0  <--              -1 +1 -1
1 1 1 1 1 0 1 0 0
1 1 1 1 0 1 0 0 0
1 1 1 0 1 0 0 0 0
1 1 0 1 0 0 0 0 0
1 0 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0

I added labels to show where the four critical moves are happening in the midst of George’s solution above.

And here’s a video of a slightly different yet essentially similar solution (can you see how they are different, and why this difference is unimportant?):

If you tried the second challenge, that of moving a penny only five spaces to the left instead of six, you probably didn’t succeed: it turns out that this is impossible! A single penny can only be moved by multiples of six. Tomorrow I’ll prove it to you — using, of all things, complex numbers! “Complex numbers!?” I hear you exclaim. “What on earth can complex numbers possibly have to do with penny-shuffling games?” Just wait and see, dear reader!