The Mutilated Chessboard: Part II

Shadows and chessboard

Last post, we looked at a tiling of a complete chessboard:

Chess Board

Dominoes tiling 8x8

Then we cut off the corners of the board:

Mutilated checkerboard 3

The puzzle was to figure out whether or not there is a tiling of this board, and to explain either how to do it or why it can’t be done.

Don’t read further unless you’re ready to hear the solution!

Domino Spiral

Okay, here it is.

There is no way to tile the mutilated chessboard. It’s impossible.

One explanation is this.

Each domino covers exactly two squares: a white square and a black square. Try it yourself: imagine placing a domino on the board. You’ll see that no matter where you put your imaginary domino, it has to cover exactly one white and one black.

Mutilated checkerboard 3Since each domino covers exactly one white square and one black square, any board that can be tiled must have the same number of black squares as white squares. Otherwise, the dominos can’t match up with the board.

The mutilated chessboard has 32 black squares and 30 white squares. So it cannot be tiled by dominos. No matter how hard you try, there will always be two black squares left over.

That’s the reason it can’t be done.

I hope this explanation was clear. This is kind of a surprising answer. I for one would never have thought to look at the colors of the squares on the board. That’s a pretty creative way to go about solving the puzzle.

Keep in mind that there is more than one solution to this problem. The board can’t be tiled: that’s a fact. But there are many ways to explain why. I like this reasoning. You might not. Whether or not you find this interesting is a matter of taste.

Hope you enjoyed thinking about this!

The Mutilated Chessboard: Part I

Lafayette Park Chess Tables
Here is a little puzzle, just for fun. I may post hints in the comments if people ask.

Take an 8×8 chessboard.

Chess Board

Imagine we have a set of dominoes and we want to cover this chessboard. Each domino covers up exactly two squares. The goal is to cover the chessboard completely, in a way that none of the dominoes overlap or stick off the board.

First question: is that possible?

Answer: It is. Here is one way to do it. Each of the colorful rectangles below represents a domino.

Dominoes tiling 8x8

A setup like the one in the picture above, where the board is completely covered with dominoes that don’t overlap and don’t go off the board, is called a tiling.

Now take an 8×8 board and cut off the top right corner and the bottom left corner.

Mutilated checkerboard 3

Here comes the puzzle.

Can this board, without the two boxes on the corners, be covered completely with dominoes? Is there a tiling of this board?

There are two possible types of solutions to this puzzle. The format of the solution depends on whether a tiling exists or not.

  1. There might be a tiling that works. In that case, a solution would be the tiling: a picture of how to cover the board.
  2. There might be no tiling that works. In other words, it might be impossible to cover this board with dominoes, no matter how many combinations you try. In that case, a solution would consist of a reason why no tiling can work. If there can’t be a tiling, then why not? Can you prove it?

This is called the “Mutilated Chessboard” problem. It doesn’t involve a lot of mathematical sophistication. It does require creative problem solving. My favorite solution to this puzzle is clever, but simple.

Feel free to post thoughts or questions in the comments! I’ll try to respond, and maybe even post hints.

And no full solutions, please. We’ll get to that in the next post.

Featured Links: Fibonacci Sculptures

This video is not an animation. These are solid sculptures that, when spun under a strobe light, seem to move. The result is a little dizzying, but beautiful. Video here.

Screen Shot 2015-03-17 at 9.18.42 PM

Each piece is designed with the golden ratio in mind, and each has a Fibonacci number of spirals.

The creator of these sculptures, artist John Edmark, says he uses mathematics throughout his work:

I employ precise mathematics in the design and fabrication of my work. I do this neither out of a desire to exhibit precision per se, nor to exalt the latest technology, but because the questions I’m trying to formulate and answer about spatial relationships can only be addressed with geometrically exacting constructions. Mathematical precision is an essential ally in my goal of achieving clarity.

Many of his pieces have a sort of peaceful symmetry.

Articulated Spiral III

More is available on his website.

Math has beautiful order and symmetry, but it can be hard to see it in words on a page. It is interesting to see the ways that mathematical patterns are expressed in this artist’s work.

Habits of a Math Major

I’ve noticed that a lot of math people share the same quirks. While not everyone has the exact same habits, there are a few things I’ve noticed in myself as I’ve become more involved in mathematics.

Reams

1. Paper. There is scrap paper everywhere. Desk, backpack, folder. Stuck in between books. I have entire notebooks devoted to scratch work.

Buried Alive

2. Lots of hand motions when talking. I can’t just say “unit vector” – I have to say “unit vector” while jabbing my pen in the air at ninety-degree angles.

3. Describing mathematical ideas with words like “beautiful,” “fun,” “elegant,” or “cute.” As in, “That’s a beautiful theorem,” or “Here’s a cute problem.” This can surprise people who aren’t familiar with the deeper study of math. But it is actually quite common among people who do math and enjoy it.

Unit-distance:  An artistic drawing of the Petersen Graph

4. I have found that apparently I have given up certain expressions, like “ninety-nine percent sure” to mean “pretty sure.” If I’m really only pretty sure, I end up using a more appropriate number, like eighty percent sure. A bit weird, I know.

5. At unexpected times, I will suddenly, absolutely need to write something down. It could be on an airplane, in the shower, in the middle of some other conversation – all of a sudden, an idea strikes and needs immediate attention. I was recently waiting for a train in Penn Station when I had a great idea about a homework problem I was working on. I only had one piece of paper: a form I had to submit to my college advisor. I was so desperate that I almost wrote on the back of the form! And I probably would have, if I had not remembered that I had a doodle app on my phone.

Blizzard of 2015: Empty Penn Station

6. I’ve seriously considered getting a chalkboard for the wall of my room.

from imgkid.com

It makes me laugh to realize how many math habits I seem to have picked up. Many are actually useful – they help me do better work, or they help me appreciate math better. Some are just silly.

Interestingly, of all the eccentric people I’ve met in my college years, not one has fit the stereotype of the lonely, antisocial mathematician. Some are among the most personable and outgoing people I know.