OF DOMINOES, TROMINOES, TETROMINOES AND OTHER GAMES

Transcription

1 OF DOMINOES, TROMINOES, TETROMINOES AND OTHER GAMES G. MARÍ BEFFA This project is about something called combinatorial mathematics. And it is also about a game of dominoes, a complicated one indeed. What does one need to know to play this game? You need to know how to count (no fingers!), you will be using your mind, colors and perhaps some drawing but nothing else. Above all you need to use your imagination! 1. Let s play dominoes! In this play of dominoes we start with a rectangle, choose your favorite size of rectangle. Mine is this one A domino will cover two squares, so I divided the rectangle in squares of size 1 by 1, the same size as the dominoes squares. Here is a very simple question: can we cover the rectangle with dominoes? when can we do that? Think a little bit about your rectangle and my rectangle and see if you can answer the more general question For which values of n and m can we cover a n by m rectangle with dominoes? That part was easy, so let s complicate it a little more. Assume we have dominoes and two single squares (single squares are sometimes called monominoes). Can we place the squares anywhere and still be able to cover the rest of the rectangle with dominoes? For the business-minded people in the group, think of a rectangular area with a water tower and a police station and divide the rest of the terrain with house parcels the shape of dominoes. Now this gets tricky, because what we want to do here is to give an absolute answer that will be good for any rectangle you have. We want to cover all possible situations, we like to write general statements and we call then Theorems. Mathematicians are that way. 1

2 2 G. MARÍ BEFFA It helps to introduce colors, only two of them, black and white. Let s alternate the colors so my rectangle looks now like How would that help anyone? Well, since any domino covers a black and a white square, it is at least clear that if I place the tower and the police both in white squares there is no way I can cover the rest with dominoes! The question that you need to resolve is If we place a square in a white portion and the other one in a black portion, can we cover the rest with dominoes? The answer is yes, but why? You need to explain why, and here is a small hint: try to cover the rectangle with a path one square wide. Cover it completely and choose a proper one, not any path will work for you. Use it to answer the question and have fun! After you had fun, try to write your personal Theorem. Writing theorems is a tricky business so you really have to be sure what you write is correct! 2. What are trominoes, tetrominoes and pentominoes? Can t you guess? Indeed, they are figures formed by 3, 4 or 5 squares. Any shape is good as far as you can go from one square to the next by moving horizontally or vertically. (A mathematician called Solomon Golomb - have you ever seen so many o s put together in one name?! - introduced the n-omino in You can pick your favorite number in the place of n). Can you find all possible shapes of trominoes, tetrominoes and pentominoes? Covering a rectangle with a certain shape is called tiling the rectangle (and if you have tiles in your kitchen or bathroom now you know why!). We are going to look into tilings by trominoes. You already saw above that there are two different types of trominoes. One of them could be called the L-tromino. Let s dive into the world of L-trominoes tilings! Here is my first tiling

3 3 It is the easiest one, I get to pick since I am writing this project! The one for you will not be this easy! So I am going to give you a bunch of practice questions so you can get to be an expert L-tromino tiler. Can you tile the 5 6 rectangle? What about the 4 5 rectangle? And the 5 9 rectangle? What is the smallest square that can be L-tiled? The 5 9 will be a little tougher, but I hope all this tiling gives you a feeling for what is possible and what is not. Let us look first at those rectangles with one side having length 2. And then think about rectangles with one side having length 3. How should the other side be in either case so that we can L-tile the rectangle? Don t you feel like you are and expert by now? Perhaps you are ready to write your second Theorem! Theorem: if 2 n 3 and m n, then a n m rectangle can be tiled by L-trominoes if, and only if the product nm can be divided by the number...which number?! Now we need to build our tiling muscles to attack the cases we have not covered here. Let s start by tiling a 9 9 square. It will be simpler if you divide the square into rectangles. And talking about dividing the square into rectangles, can you use that technique (and this is a hint!!) to show why the following is true If n 4 and m 5 and m can be divided by the number... (which number!?), then the n m rectangle can be tiled by L-trominoes. Did we look into all possible situations? Are we missing something? Be sure that you have the solution for all possible cases!! 3. Tetrominoes and pentominoes?! Shouldn t we do something else?! As you see one can go on and on with different sizes and shapes trying to solve the problem of rectangle tiling. In fact, many research papers have been written (and are written) on the subject of n-tiling, but we are going to do something a little different. What we will do is to fold our rectangle and try to tile other figures. Take any of your favorite rectangles and glue it along one side. What you get is a cylinder.

4 4 G. MARÍ BEFFA The size of the cylinder depends on which side of the rectangle you glued. Now, take the cylinder and glue the other side. What you then get is a donut or a bagel, depending on your culinary taste. Before we start learning how to tile donuts we will play a little game to get a familiar feeling about living in a donut world. There used to be a popular video game pitting two players in combat planes on the screen. When one of the planes flew off one edge of the screen, it didn t crash but rather it came back from the opposite edge of the screen. This is exactly what would happen if the planes were fighting in a donut world, rather than in a flat screen. Play a few games of donut tic-tac-toe with someone else in the group. The rules are the same as the traditional tic-tac-toe, except here the opposite sides of the board are glued to form a donut. Be sure you know what a winning three-in-a-row is! Back to tiling, you might think: what is the problem here? Tiling a rectangle and one of these two figures is the same, just take a tiling from one to the other.. big deal! Well, it is true that when we tile a rectangle we have also tiled a cylinder and a donut. But sometimes we can tile a donut of a given size but we can t tile any rectangle producing that one donut!! Likewise for the cylinder!! Can you think of a tiling for which that might be true? If you can t, do not worry about it, we will do a little practice and things will be crystal clear. Because mathematicians do not have a very distinctive culinary taste, the figure we get when we glue the rectangle twice is usually referred to as a torus (in spanish the word for bull is toro and the word for torus is also toro. I was always highly confused about what a donut had to do with a bull). First problem you want to solve: Find all n such that the torus T (n n) can be tiled with L-trominoes. Now we make life a little twisted. We are going to glue the rectangle along one of the sides, but we will give a turn to that side first so that you glue sides that are in opposite parts of the rectangle. The resulting surface is called the Möbius band. This band is very popular among students and teachers alike and it is usually said that the Möbius band has only one side. For example, if you cut a cylinder along the center you will get two cylinders (the top and the bottom). Try to cut a Möbius band along the center. If you think it will become two Möbius bands, think twice!. Cut again whatever you obtained again along the center and look at what you get. These are geometric properties have been studied and classified by geometers. You can fool your parents and friends with this trick.. Back to our tiling business, why don t you try to do the following A rectangle 4 9 cannot be tiled with L-tetrominoes you have to show why but a 4 9 Möbius band can! Careful with this question, I am asking to find tiling by tetrominoes not trominoes. An easier question, which you might want to do first, is Show that the 4 2 cylinder can be tiled by T -tetrominoes, but the 2 4 cylinder can t.

5 There many, many more questions one can ask. For example, we could cut the tiling of a torus not along straight lines but along the tetrominoes, winding along. What you get when you open it up will not tile a rectangle. But can I cut in a way such that by putting many copies of the figure we get we can obtain the tiling of the entire plane? Of course you need many many copies (infinite) and they need to fit in properly. These tilings are called periodic tiling of the plane, because you repeat on and on the same pattern. Can you think of a non-periodic tiling? Keep on going, there is never an end to tiling... 5