We are going to play a game with checkerboards and dominos. So that we all have the same picture in our heads, a *checkerboard* is an 8×8 grid of squares (64 squares total), colored alternatingly black and white:

A *domino* is a 1×2 block, which is exactly the size of two squares on the checkerboard (the dots on the domino do not mean anything in this game):

Your goal is to cover the checkerboard with dominos, according to the following rules:

- You can use as many dominos as you wish.
- Dominos can be arranged horizontally or vertically, as long as each domino covers exactly two squares on the checkerboard.
- Dominos cannot overlap each other, and cannot extend off the edge of the board.

**
**EXAMPLE 1: Can you cover the checkerboard with dominos?

EXAMPLE 2: What if we remove one of the corner squares from the checkerboard (a “mutilated checkerboard”) – now can you cover it with dominos?

### Assignment, Due TUESDAY 10/28 (end of day)

**Warm up ***(This Warm Up is** just for practice – you must complete it but you do NOT need to submit your work – see below for the three-part Assignment to be submitted).** *Draw or print out a checkerboard, and complete the two examples above (you can draw dominos on your checkerboard in pencil). For your convenience, here is a picture of the checkerboard in EXAMPLE 2:

If you are NOT able to cover the checkerboard with dominos, try to think of an explanation *why *you cannot. Now complete Examples 3 and 4 below.

EXAMPLE 3. Now, consider removing two corners – I’ll remove the top right and bottom right corners. Can you cover the 62-square checkerboard below with dominos? Do it, or try to find an explanation of why you can’t.

EXAMPLE 4. Now lets remove opposite corners (upper right and lower left). Can you cover the resulting 62-square checkerboard with dominos? Do it, or try to find an explanation of why you can’t.

**Assignment.** Your assignment has 3 parts:

PART 1. For each of the four Examples, post your conclusion – is it possible to cover the given checkerboard with dominos, or not? *Do NOT include any explanation at this time – just tell us whether it’s possible or not, for each Example*. Leave your response in the comments.

PART 2. Challenge your friends: Create a puzzle for your classmates by removing TWO squares from a checkerboard – they can be ANY TWO SQUARES (not just corners). Decide for yourself if the resulting mutilated checkerboard can be covered by dominos (but don’t tell us!). Remember, the most challenging puzzles are the ones where the answer is difficult to determine. Post two puzzles in the comments. See the note “POSTING YOUR PUZZLE ONLINE” below for instructions (you do NOT need to draw your puzzle or post an image).

PART 3. The third part of your assignment is to write a short paragraph (at least 3 sentences) responding to the following prompt. Be sure to respond to each part:

*Writing Prompt**: **Did you enjoy this assignment? Why or why not? Describe a connection between this assignment and our work in the class. (If you don’t believe there is a connection, try to *imagine* why we are doing this). Leave your response in the comments.*

POSTING YOUR PUZZLE ONLINE: To describe your puzzle, you just need to tell us which squares you are removing. We can do this in symbols by giving the coordinates of the squares on the checkerboard as ordered pairs — the bottom left square is (1,1), and we count x horizontally and y vertically as usual. The top right square is (8,8). For example, to describe the checkerboard in EXAMPLE 3, I would say ** “EXAMPLE 3: remove squares (8,8) and (8,1)”** (the top right and bottom right squares, respectively). Post your puzzle online by describing which squares to remove.

1) Example 1: yes you can cover the checker board with dominoes.

Example 2: no you cannot cover the check board with dominoes.

Example 3: yea you can cover the checker board with dominoes.

Example 4: no you cannot cover the checkerboard with dominoes

2) My Puzzle 1 : remove square (3,4) and (7,7), is it possible to cover the check board yes or no?

My Puzzle 2: for my second puzzle remove square (8,1) and (5,4). Is it possible to cover the checkerboard with dominoes?

3) I enjoyed this assignment because it was fun to do and it was challenging. I believe we are doing this to challenge ourselves that their is a proof to this reasoning or trying to come up with a proof on why this works. I also believe we are trying to think outside the box looking into things deeply. Overall this was a fun assignment. All assignments should be like this

Great! I like your puzzles…

PART 1) Example 1: Yes, it is possible to cover the checker board with dominoes.

Example 2: No, it’s not possible to cover the checkerboard with dominoes.

Example 3: Yes, it is possible to cover the checker board with dominoes.

Example 4: No, it’s not possible to cover the checkerboard with dominoes.

PART 2) Puzzle 1 : Remove squares (5,8) and (1,7), Can you cover the resulting 62

squares with the dominoes?

Puzzle 2: Remove squares (7,6) and (3,2). Now, is it possible to cover the

checkerboard with dominoes?

PART 3) Did you enjoy this assignment? Why or why not? Describe a connection between this assignment and our work in the class. (If you don’t believe there is a connection, try to imagine why we are doing this). Leave your response in the comments.

Yes. I really did enjoy the assignment. At first I thought I was able to complete the fourth example in the exact way that I had visualized example 3 until I noticed something. I believe there must be a connection to our work in class. All I can think of is proof by contradiction. For instance in the fourth example I was assuming that I was able to do it just as I used my logic of even number of squares the dominoes can cover in the 3rd example. But if I contradicted it then I will be able to say if it’s possible to cover with the dominoes or not by looking at the significant coordinates on the checkerboard.

Thanks! I appreciate your observation that examples 3 and 4 are similar, but have some key difference – you’ll have a chance to explore this more as the project progresses…

-Prof. Reitz

Examples 1. Yes you can cover the board.

Example 2. No you can’t cover the board

Example 3. No you can’t cover the board

2) Now try covering the board without tile (1,4) and tile (6,2). Is it possible.

Now remove tiles (1,1) and (8,4). Can it be done without these tiles.

3) This was a very interesting challenge. I made me think outside the box and not at the obvious choice which wrong. As far as how it relates to our class work, I would say that by looking at the puzzle and reading the direction I started to think if it was possible before actually doing the puzzle. Logic didn’t always work.

I made a mistake . For example 3 it can be done and example 4 is no it can’t be done.

Great work – I’m glad to see you’ve kept thinking about it!

3) Yes, I do enjoy playing the game a lot. At the beginning, I think that it is impossible to cover the board because if the board is 8×8, the entire board consists of 64 small squares. A domino is 2×1, so it can be viewed as an even number. Therefore, when one of the corners is missing, the entire area of the board becomes 63 squares. It is an odd number. Thus there should be no way to cover an odd number area using even number area dominos. However, when I simply tried to place the dominos, it worked perfectly. I think that this game has some connections with being symmetric or not, and it reminds me of figurative numbers which were used in ancient mathematics.

the entire procedure of playing the game is exactly the same as proving a statement

because you know what you want, but you need to figure out how to get your desired result.

1) For example 1, yes, it can be covered completely using dominos.

For example 2, yes, it can be covered with dominos

For example 3, yes, it is possible.

For example 4, no, it is impossible to cover the board using dominos

2) Puzzle 1: remove the squares (4, 4) and (8, 8), can you cover the entire board using dominos?

Puzzle 2: if you remove the squares (1, 1) and (6, 2), is that possible to cover the entire board using dominos?

Great puzzles – can’t wait to see if someone can solve them!

Part One

Example 1- Yes, it’s possible

Example 2- No, it’s impossible

Example 3- yes, it’s possible

Example 4- no, it’s impossible

Part two

First puzzle: Can u cover the checker board if I remove tiles (2,2) and (4,5)?

Second puzzle: Can u cover the checker board if I remove tiles (6,3) and (4,7)?

Part three

Yes, I really did enjoy this assignment because at first it looks really simple because if u have a 2×1 domino and a 8×8 checker board one can conclude that if the amounts of space is even its possible and if its not than its impossible. But that not the case in this puzzle the amount of space does matter but the order matters as well so its a bit tougher when u remove tiles that aren’t next to each other. I believe that this assignment relates to proofs when we do cases (for ex 3 and 4 you remove 2 tiles but in each example you remove tiles in different locations therefore creating different cases where it may be true in one case and false in the other).

I like your observation that “space does matter, but the order matters as well” – Great connections to our class work!

Part 1: Ex 1- Yes you can cover the checkerboard

Ex 2- No you cannot cover the board

Ex 3- Yes you can cover the board

Ex 4- No you cannot cover the checkerboard with dominoes

Part 2: For my second puzzle challenge I would want to remove (8,2) and (8,7).

Part 3: I think the assignment is interesting to look at a checkerboard in a different way by seeing what would happen if the corners were removed. At first, I didn’t see a connection with what we are doing in class however, after observing the rows I can see the answers change when there is an even and odd amount of squares left on the board in the rows vertically. Which does tie in with proofs that we have being learning because we can show with visually and with words why one way might work while another may not.

I like your puzzle! (don’t forget to post one more for full credit).

example1-yes you can cover

example2-no you can’t

example3-yes you can cover

example4-no you can’t

2) can you cover the checkerboard without tiles (1,8) and (7,8)

3) In this game, we are moving case after case to figure out where it is possible or not to cover the checkerboard and this is the same method (case-per-case ) mathematicians and other scientists use in proving theorems or to establish the veracity of their experiments