Jigsaw Pieces

Shayan from Dubai, said:

$12$ pieces are used- $3$ different types of corner pieces, side pieces could be $4$ different types, and $5$ for the middle.

Unfortunately Shayan didn't give any further explanation.

Jimmy and Meg, from Mrs Luke's class, started out by working out how many different ways there were of making jigsaw pieces only by removing buttons, and not adding any.

There is one piece with no holes in.



There are four pieces with one hole in, because there are four sides so there are four ways we can take a hole out of one side to make one piece. If the holes are in the middles of the sides, so the piece is symmetric, then we can rotate the two yellow pieces so they are the same and we can rotate the two blue pieces so they are the same.



There are six pieces with two holes in. First we take a hole out of the left side, and then we can take a hole out of any other of the three sides. That makes three pieces. Then we take a hole out of the top side and then we can take a hole out of the right side or the bottom side, but not the left side because we already made that piece earlier. That makes five pieces. Then we take a hole out of the right side and the bottom side. That makes six pieces. We have taken a hole out of each side three times altogether.

If the holes are in the middles of the sides, we can rotate the two purple pieces so they are the same and the two green pieces so they are the same. We can flip over the purple pieces so they are the same as the green pieces.




There are four pieces with three holes in. We make each piece by leaving one side alone and making holes in the other three sides, and because there are four sides we can do this four different ways. If the holes are in the middles of the sides, we can rotate the two yellow pieces so they are the same and the two black pieces so they are the same.



There is one piece with four holes in.



We made 16 pieces. If we do not count things we can rotate to be the same, we made 9 pieces. If we do not count things we can rotate or flip to be the same, we made 8 pieces.

Well done, Meg and Jimmy. This is fantastic - I like the system you've used to get all the pieces with holes.

Jenny, from Mrs Luke's older class, spotted a way to use Jimmy and Meg's work to see how many pieces there were in total. Jenny again uses a very systematic way of working. Even if you didn't get this far, you may be able to understand some of Jenny's working and see which of her shapes you did find yourself.

I looked at the edges that didn't have buttons taken out of them. I could either leave them alone or add a button to them. I could not take a button out because Jimmy and Meg had already counted all the ways to do that. If there was one free edge, I could make two different pieces (one with a button and one without a button).

If there were two free edges, I could make four different pieces. I could make one with no buttons, one with two buttons, one with a button on one edge and one with a button on the other edge, like this:



If there were three free edges, I could make eight different pieces. I could pick one side and make that flat, and then I could make four different pieces with the other two free sides, like I did in my picture. Then I could make the first side I picked have a button, and then there would be another four different pieces I could make with the other two free sides. That makes eight.

If I had four free edges, I could make sixteen different pieces because that's the same as the number of pieces Jimmy and Meg made taking different buttons out, and I am doing the same thing as them only putting buttons in.

For each number of buttons taken out, I worked out the total number of pieces I could make. This was the number of ways of taking that number of buttons out times by the number of ways of adding new buttons when you have taken that number of buttons out. This is because for each way you take the buttons out, you can make all the different new pieces by adding buttons.

This did not work for when there were four buttons taken out. (You're on the right track! Perhaps if you think of "Number of ways of adding new buttons" as "Number of things you can do either adding buttons or leaving things alone" you'll see why.)

Number of buttons out	Number of ways of taking the buttons out	Number of ways of adding new buttons	Total number of pieces
0	1	16	16
1	4	8	32
2	6	4	24
3	4	2	8
4	1	0	1

In total there were 81 different pieces.