The other day I ran back home in order to visit my family, and, while I was there, I made arrangements to visit my old friend Bubba. Bubba has acquired a lot of land due to the fact he comes from a large family and has managed to outlive a lot of them. He told me he wanted me to meet him at a particular spot and gave me directions to find him. One catch was that last leg of the journey required that I walk a mile or two.
I drove back into the boonies quite a ways following his directions as best as I could. There was one point where I could have chosen to go this way or that and I went that. Then I got to a place where I thought I was to begin walking and walked way more than two miles. I was remote enough that my cell phone wouldn’t work, of course, and I was about to retrace my steps when I saw what appeared to be an abandoned old church.
Drawing closer to it out of curiosity, I saw that I was mistaken, not about it being an old church, but about it being abandoned. On the front of it there was a sign that read, “The First Church of the Prophet Roy,” and off to the side there was a man who was staring inexplicably at a stack of old tires while scratching his head as if involved in some sort of a huge mental calculation.
“Hello,” I said from a distance so as not to startle him.
“Howdy,” he said in reply, continuing to stare at the stack of tire and scratch his head.
“That’s quite a stack of tires there,” I said as a way of trying to engage him in conversation.
It was indeed an interesting stack of tires. There were at least a dozen tires in the stack. They were stacked in a conical pile with each tire smaller than the one beneath it. The bottom one was a tractor tire, the top one was a tire from a wheel barrow, and each was mounted on a wheel. They were stacked with a fence post as their central axis that kept them from sliding to one side or the other.
“Yup,” he answered. “Quite a stack o’ tars.”
That not being terribly informative, I thought I’d try a direct question.
“What are you staring at them for?”
“I’m trying to figure out how to move them,” he said.
There were only about a dozen, so I thought I’d do my good deed for the day.
“I’ll help you,” I said and begin removing the smallest tire from the top of the pile, but, much to my surprise, he stopped me.
“No, you cain’t just do that,” he said with his hand gently on my elbow. “They’s rules.”
It was then I noticed in his had a book with a black leather cover. I initially took it to be a Bible but closer inspection revealed a title emblazoned in gold on its cover: The Prophecies of Roy.
He pointed to a passage with contained directions. It was written in archaic English much like the language of the King James Bible. After a bit of deciphering, I discerned the rules. There were two other posts besides the one the tires were stacked around. I hadn’t even noticed them until then. The passage directed the reader to move the tires from their current post to one of the other posts—it didn’t matter which one. What did matter was they had to be moved one tire at a time and at no time could a larger tire be stacked upon a smaller one.
The rules were simple, but the problem was how to carry the process out systematically. After little figuring myself, I explained it to my new friend, Daryl, who I learned was a new follower of the Prophet Roy.
In order to explain it to Daryl, I named the three posts John, Jack, and Jick with John as the name of the post holding all the tires originally.
“Daryl,” I began, “the idea is to first move a stack of one, then move a stack of two, then move a stack of three, and so on until you are done.”
He nodded his head as if he understood.
“First move the top tire to the post I named Jack,” I said. “If there was only one tire you’d be done. Now move the second tire to Jick and then move the tire from Jack over to Jick. If there were only two tires, you’d be done. Now, it gets harder.”
He looked at me with kind of a sad look.
“...but not that hard,” I said, trying to soften it. “Take the third tire and put it on Jack. Now, take the smallest tire and put it over on John. Then move the second tire to Jick and cap it off with the smallest tire you that’s been back on John.
“Notice,” I said, “that all of the piles with an odd number of tires are over on Jick but when it’s an even number, it’s over on Jack.”
He nodded again. Encouraged by this, I decided to expand a little more.
“You see,” I said, “this is an example of what is called a recursive process. When I build a pile of size n on Jick, for example, I then pull the new big tire off of John on to Jack. Then I proceed to build a new pile of size n on top of the new tire so as to make a tower of height n+1.”
Daryl was beginning to look uncertain, but I was undeterred.
“You see each time I move all of the tires I did before, then introduce the bigger tire, and then move all of the tires again. The number of moves adds up quickly: 1 move for 1 tire, 3 moves for 2, 7 for three, and so on. This is because each time we double what came before and add one.”
I looked up at him and he had the look my cat gets when she watches me program the DVD.
Whether he understood the mathematics or not, Daryl understood the procedure, and he was following it when I left. It turns out he knows Bubba and he explained to me how to get to Bubba’s place in a much clearer fashion than Bubba had.
As I walked back to my car, I recalled the story about a similar problem call the Towers of Hanoi. According to this story, there are monks in a temple in Hanoi who are moving disks from peg to peg using the same rules as the Prophet Roy insisted upon, i.e. never put a larger disk on a smaller one. Legend says that the world will end when they are done. They began with 64 disks, however, and it will take them more than the life of the universe to finish.
When I finally found Bubba, I told him the story and gave him The Prophecies of Roy that had described the puzzle. I’d put it in my pocket and had forgotten about it.
“Yes,” he said, “Daryl is a member of the Church of the Prophet Roy. They believe they are going to make the world come to an end, but none of them ever understood how to work the puzzle before. How many moves will it take Daryl to be done?”
“With a dozen tires,” I said, “that would be 4095 moves. Assuming that he moves one tire a minute, it would take 68 hours or so.”
Bubba was now reading The Prophecies of Roy.
“Then, according to this,” he said indicating the book, “the world has about 68 more hours to exist.” He closed the book and said, “I think I’ll have a beer. Better yet, I think I’ll take a few over to Daryl it might slow down the work.”
And so we did.