Puzzle 1

Fit all 12 pentominoes on the two large rectangular grids on Attachment 2.
Fitting all the pentominoes on a 6 x 10 or a 5 x 12 grid can take a lot of patience to solve; however, there are many different solutions. Here is one solution for each. It’s probably not the one you came up with.

Puzzle 2

1. Are there any other rectangular grids on which you can fit all five pentominoes?
   
   There are two others, both have an area of 60 square units, 4 x 15 and 3 x 20. The first has many solutions. The second is very difficult and has only two known solutions.

One solution on a 4 x 15 grid:
One of only two solutions on a 3 x 20 grid:

2. Is there any rectangle with an area of 60 squares on which you cannot fit 12 pentominoes?
   
   There are two other rectangles with an area of 60 squares, 2 x 30 and 1 x 60. If you think about it for a few minutes, you’ll realize that you can’t possibly fit all the pentominoes on either rectangle. This is because several pentominoes are 3 squares across in both directions.

Puzzle 3

Is it possible to place all five tetrominoes on a 4 x 5 grid?

You would think it’s possible, since the total area of the five tetrominoes is 20 square units. Surprisingly the answer is no. No matter how you try to rearrange the pieces, you’ll always have one square sticking out and one blank square on the grid. Here are two samples.

Puzzle 4

1. Place all four tetrominoes and just one pentomino on a 5 x 5 square grid.
   
   Actually this one is fairly easy to find solutions for. Just arrange your tetrominoes so that there are five connected empty squares on the 5 x 5 grid. This will fit one of the pentomino pieces. If you want a much harder challenge, start by placing a pentomino on the 5 x 5 grid and try to fit the tetrominoes around it. You’ll probably have to move the pentomino before you find a solution.
   
   

   Here are three sample solutions:

2. Are there any pentominoes that you can’t use in this way? How do you know?
   
   If you think about this, you’ll realize that you can’t solve this problem using a straight pentomino. If you use it along the edge of the grid, you’ve changed the rest of the grid to a 5 x 4 grid. If you use it in another place, you also create a situation in which the tetrominoes won’t fit.

Puzzle 5

Divide your pentominoes into two sets of six. Then try to fit all the pentominoes onto two 5 x 6 grids.

This is especially challenging because there is only one way you can divide the pentominoes to form two 5 x 6 grids. There are many ways to place six different pentominoes on one 5 x 6 grid, but the ones left over won’t fit on the same sized grid except for one special case.

If you want to solve a simpler puzzle, here are the two sets to start with:

Now use your pieces to see if you can find two 5 x 6 arrangements. Here is one solution: