Problem number eight. Casper has the following seven pieces. He uses some of these pieces to cover this grid completely without overlap. He uses as many different pieces as possible. How many pieces does Casper use? A, three, B, four, C, five, D, six, or E, seven. One thing that we will need to do to solve this problem is to carefully count how many small squares the grid has. Here's a close-up to make the counting a little easier. We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 pieces. Now let's take a look at the pieces that Casper has. The first one consists of one small square. The next one consists of two small squares. Next one is three, then four, then five, then 1, 2, 3, 4, 5, 6. And the last one consists of 1, 2, 3, 4, 5, 6, 7, 7 little squares. Now let's add these numbers together to see how many little squares the seven pieces have all together. 1 plus 2 is 3, plus 3 more is 6, plus 4 gives us 10, 10 plus 5 is 15, 15 plus 6 is 21, and 21 plus 7 is 28. Together the seven pieces consist of 28 squares, whereas the grid only has 17. So there is no way that we can use all seven pieces without overlapping to cover the grid. So answer E, 7, is definitely wrong. The difference between 28 and 17 is 11. We don't have a single piece that's 11 squares long. The longest one is 7. So we need to remove more than one piece. So we already can see that just six pieces will not work. We can find two pieces that add up to 11. We could remove the piece that's seven squares long and the piece that's four squares long, and the rest of them would add up to 17. We could also remove the piece that's six pieces long and the piece that's five pieces long. So we have to remove two pieces, and then we can use five pieces, which is the greatest number of different pieces that we can use. Let's look at an example here. If we used all the pieces except for the one that's four squares long and the one that's seven squares long, it would look something like this. This is one. The next piece is two squares long, then the one that's three squares long. We're skipping four, so the next one will be five squares long, 1, 2, 3, 4, 5, and then the one that's six squares long, 1, 2, 3, 4, 5, 6. So just as we figured out, it does work to use five pieces. That's the largest number of pieces that Casper can use, so the answer is C, five.

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