1. In how many ways can you spell the word FOOT in the grid below? You can start on any letter F, then on each step you can step one letter in any direction (up, down, left, right, or diagonal). LINK: https://latex.artofproblemsolving.com/1/a/b/1ab54d90fa697bd6fc4f71ee1860e48d88dfc83c.png
2. We call a number peachy if every digit in the number is either a 3 or next to a 3 For example, the numbers 333, 83, 303 and 3773 are all peachy, but the numbers 32523, 786, and 3999 are not peachy. How many positive 2-digit numbers are peachy?
3. I have 5 different mathematics textbooks and 4 different psychology textbooks. In how many ways can I place the 9 textbooks on a bookshelf, in a row, if there must be a psychology textbook exactly in the middle, and there must be a mathematics textbook at each end?
1. FOOT problem: There are 4 that have 1 entry point from the F, and you can't use the middle F, because if you get there, there is nothing to go to: 4(1*1*3) makes 12 (you have to move diagonally away, then vertical, then you are at the middle and can go diagonally or horizontal, making 3 possibilities.)
Starting from the middle outside: 4(2 + 10 = 12) = 48.
Staring from the top and bottom center: 2(2 + 3 + 3 + 2 = 10) = 20.
So there are 48 + 20 = 68 ways.