Problem of the Week #1: Monday Jan. 9th, 2023 After some mathematical soul searching, I figured I should get back into making math problems. I'll try to post weekly problems for fun. As these are the "where my curiosity takes me" sort of problems, I make no promises regarding their difficulty or topic, but I'll try to give a few different parts to each problem going from easy to hard. I also make no promises regarding whether I've personally solved any of them or if they are solvable at all because I think that's part of the fun of the exploration. If you would like to take part in the discussions for these problems, or if you would like to post math problems of your own making, please register for an account here. All you need is an email address, a username, a password, and 2mins. Please refrain from posting the full solutions to any parts of the problems in the comments to avoid spoiling other people's fun. Thank you, and enjoy! ^_^
a) How many ways are there to completely tile (without overlap or leftovers) a 2×7 strip using 2 2×2 squares and 6 1×1 squares?
b) How many ways are there to completely tile a 2×7 strip using any number 2×2 and 1×1 squares?
c) How many ways are there to completely tile a 2xn strip using any number of 2×2 and 1×1 squares?
d) How many ways are there to completely tile a 3xn strip using any number of 3×3, 2×2, or 1×1 squares?
e) How many ways are there to completely tile an mxn rectangle using any number or sizes of whole number squares?
f) If you’d like, please share a problem you made that’s inspired by this problem.