# What kind of puzzle is solved by these sequences of numbers?

Samantha had been fiddling with something at her desk for ages, and constantly scribbling notes on a piece of paper. When my curiosity eventually got the better of me, I wandered over to ask what she was up to.

"No, no!" she exclaimed. "Don't look yet! I'm still on the simple version. Once I work out how to solve the more difficult version, then I'll show you the puzzle. Just give me another few minutes."

"Oh all right, you can look at this." She handed me a sheet of paper with two sequences of numbers written on it. "That's the solution to one case of the simple version. You won't be able to deduce anything from it though, not until I tell you what it's all about."

I looked at the paper in my hand. It read as follows:

(6 4 8 7 5 3 1 2)

3 5 2 1 7 6 4 2 1 3 5 8 2 1 3 5 8 3 6 4 1 2 3 6 5 8

"Challenge accepted!" I said.

But damn, I can't actually figure out what kind of puzzle Samantha is studying. Can you help?

• Probably unnecessary disclaimer: this isn't a true story, and in fact I do know what the solution is. Commented Sep 21, 2016 at 14:15
• "That's the solution to one case". Does this mean that there are/may be more solutions and all of them consist of 26 numbers? Commented Sep 21, 2016 at 14:19

It's a...

sliding block puzzle!

The board is originally in the state

given by the numbers in the parentheses: top row 648, middle row 753, bottom row 12(blank).

The sequence of numbers

gives the order in which to slide the tiles into the empty space.

• You my sir, are a genius. Commented Sep 22, 2016 at 4:20
• Now here is a second challenge: find a better solution to the original puzzle without using the original board. Commented Sep 22, 2016 at 4:20
• @TheGreatDuck: I don't understand what you mean. A "better solution"? But it's a different puzzle if the original board is different.
– Deusovi
Commented Sep 22, 2016 at 4:23
• I mean find a way to somehow simplify the motions in the solution without looking at WHAT the original board looked like. I.E. simplify the operations 'blindly'. Commented Sep 22, 2016 at 4:29
• @TheGreatDuck The process for doing so would generate the original board midway through; I'm not sure it would be possible to find a simpler path without resetting to the origin state unless there are loops in the given path. Commented Oct 27, 2016 at 15:05