Open Math Problem: Bloxorz

Wow! I am totally on a blog posting roll. I think the end of the summer has me going out less, which has me putzing around the apartment more, which has me thinking about school and math more. (It doesn’t help that I don’t really get reception on my TV and I don’t have cable.)

In any case, MathTeacherMambo pointed me to this (warning: VERY ADDICTIVE) game:

BLOXORZ

For those who are starting their pre-start-of-classes meetings, I warn you that you might not ever make it to those meetings. Your principal will knock on your door, but you’ll be drooling and staring vacantly at your computer screen, cursing the day you ever reached level 11, 12, or n.

For those who are wisely taking my advice and holding off playing the game, here are a few examples:

As you can see, the goal is to get the block in the hole.

I think this is a great open math problem ready to be attacked and solved by a high school student. Fundamentally, I think with a lot of work, a student should be able to answer the following question:

Given a particular floorplan and starting block position, can you decide whether the floorplan is solvable? Can you tell, without playing the game, whether there is a way to get the block in the hole? 

To illustrate, two simple examples of floorplans — the first one is obviously not solvable, the second one obviously is.

What about this third one?

The first two are easy to solve by inspection. Even the plan above is easy to solve by inspection — but you’ll notice it gets slightly harder. I want to know — even in the most crazy floorplan — is there a solution? If not, I want proof that there is no solution.

The game itself has a bunch of complicating elements — like transporters, and ways to get more floor to appear by having the block roll over a button. (See the second video above.) But I think the base case is hard enough — with none of that nonsense.

Once the initial problem has been solved, I think a great follow up question would be: what is the fewest number of moves you can solve a particular floorplan in? 

I have a student who approached me about doing an independent study of linear algebra or differential equations this year. I know I’m going to be overwhelmed, so I had to decline. However, I suggested that instead of having a formal course, we could work on an investigative problem together. I just emailed him this idea — but I don’t know if he’ll want to.

PS. I’m guessing a good starting place for this problem is looking at the work that has been done on the Lights Out game. (I’ve never played it, but it seems like it’s similar enough in nature that that solution can inform our approach.)

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