Create a 5x5 checkerboard pattern using 5 of the 8 pieces below. You must decide which 5 pieces to choose. You may not use the same piece multiple times. The letters are just names we'll use to refer to the pieces in the hint and answer. You may rotate the pieces, but there is no pattern on the other side, so you can't flip them over. The Hint and Solution tell you how to solve this puzzle in your head (or using scratch paper) without creating the physical puzzle pieces.

The 5x5 pattern will have 12 squares of one color and 13 squares of the other color. The five pieces you choose must match this 12/13 relationship. The following chart will help you decide which pieces to choose:

6 of the 8 pieces have one extra white square. The only way to balance the black and white squares is to choose the X as one of your pieces. If you also choose the T, you will have 13 black squares. Otherwise, you’ll have 13 white squares. If you have more white squares than black, you will have white squares in all four corners. This forces the X piece to the center of the board; anywhere else will prevent you from filling a corner with a white square.

A little trial-and-error experimenting should quickly convince you that there is no way to fill the 5x5 grid with the X in the center. The available shapes can’t fill all the corners. Therefore, the pattern needs to have more black squares than white. That means you’ll need to choose the T as well as the X.