Dr Nielsen: How many different ways are there to arrange dominoes?
The students came up with four ways, shown below.
Rules: No leaning a domino on its side, turning it upside down, cutting it in half.
Dr Nielsen: Today's problems have levels!
Level 1.
Question: Can you fill up a 1x1 chessboard with dominoes?
Children: No! Because a domino doesn't fit!
Level 2.
Question: Can you fill up a 2x2 chessboard with dominoes?
How many think you can? (Many hands).
OK you have a piece of paper and a pencil. Try it.
After a minute or two, a girl volunteered to show how to cover a 2x2 chessboard with dominoes.
Notation: draw a line through two squares to represent a domino.
Level 3. Can you fill a 3x3 chessboard with dominoes?
The children worked for a minute. After a while, several raised their hands to say no.
Why not?
Because dominoes have two squares each, so they cover an even number of squares. But there are nine squares in a 3x3 chessboard, which is odd.
Level 4. 4x4. Can you do it?
Again after a couple of minutes, a student came up and showed one solution on the board.
Dr Nielsen: Have you figured out a pattern?
The students decided that you could always fill a chessboard with an even number of squares, but you could never fill a chessboard with an odd number of squares.
Level II.
Dr Nielsen: Can you cover a 2x2 chessboard with dominoes if one of the corners of the board is cut out?
The children thought for a minute, and then answered no.
One child had a suggestion: You can do it if you let the dominoes overlap.
Dr Nielsen: Very good! New rule. No overlapping dominoes.
Then the children could see that there were an odd number of squares, so they couldn't cover this board with dominoes.
Next level: What about a 3x3 board with one corner missing?
One child figured out a solution and showed it on the board.
Next level: What about 4x4 board with a corner missing?
Immediate answer. NO!
Why not?
Because there are an odd number of squares.
One of the children who knew how to multiply explained that an even number times an even number is even, and if you take away one, you'll get odd.
Those children who didn't know how to multiply could just count: 15 squares. Odd, so you can't cover it with dominoes.
Next level: What about a 5x5 board with a corner missing?
After a minute, one child finished and raised his hand. Dr Nielsen asked that child to try the 7x7 board while the others finished.
After a few more minutes, when many hands were raised, Dr Nielsen had the children give him the answer.
Can you cover a 5x5 board with a corner missing? YES!
Can you do 6x6? (Immediate answer) NO!
What’s the pattern?
The children decided you could always do odd sided squares with a corner missing (3x3, 5x5, 7x7), but you could never do even sided squares with a corner missing (2x2, 4x4, 6x6).
LEVEL III
Level 1. Can you fill a 3x3 chessboard with dominoes when there are two opposite corners missing?
Children: No. Because there are seven squares left over. (They seemed to be figuring out a way to solve these problems!)
Level 2. What about a 4x4 board with opposite corners missing?
One child: Yes! I just need to figure it out.
Dr Nielsen: Ok. Figure it out.
After a few minutes, the children changed their minds. No! You can't do it!
Dr Nielsen: Are you sure?
After a couple more minutes, with still no children who had found a solution, Dr Nielsen asked, How many squares are left?
Children: 14.
Dr Nielsen: 14 is even. It’s even! So you can do it right?
Children: No.
Dr Nielsen: Why not?
One child suggested that the board had been "cut up too much."
So Dr Nielsen tried:
(The children were skeptical, but without a good answer as to why not, he moved on.)
Dr Nielsen: We know we can’t
do a 5x5 chessboard, because when we remove 2 corners there will be 23 squares, which is odd.
Let’s try 6x6.
The children worked on it for several minutes, with Dr Nielsen asking every now and then how many people needed more time? Since no one had quite finished, he let them keep working.
After a while, he brought the group together to talk about the chessboard.
Dr Nielsen: Let's color the 6x6 chessboard black and white.
How many squares are there total? 36
How many black squares? 18
How many white squares? 18
If we remove the top left corner and the bottom right corner, how many black squares and how many white?
18 black, 16 white!
What does a domino cover? 1 black square, and one white square.
So can you cover a 6x6 chessboard with opposite corners removed?
No! There aren't enough white squares!
A student noticed that for an odd number, like a 5x5 chessboard, opposite corners had different colors. They asked, could you cover a 5x5 chessboard with opposite squares removed?
Dr Nielsen: Good question. Let's try it!
After a minute, the students realized that they had already figured this problem out -- there were an odd number of squares, so no, they couldn't cover this chessboard with dominoes.
One last thing.
With the remaining time (not much of it), Dr Nielsen had the students create an addition table on the board, adding numbers from 1 to 5.
After they had finished the table, he pointed out that they could make an addition table for even and odd numbers.
And for those who knew multiplication, they could do a multiplication table for even and odd numbers.
And then we broke for cookies.
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