While I have received some helpful suggestions by email for puzzles which I will likely use in the future, I am going to start the puzzles off with a simple exercise in mental mathematics (I actually invented it one night when I was having a hard timing falling asleep... as a kid I never understood what people were talking about with counting sheep, but I did find doing arithmetic in my head helped. My default exercise was to simply multiply numbers continuously by two and see how large I could get them, but every so often I would try something else too). The exercise goes like so:
Let x and y be two integers from 0 to 9. Iterate x and y in the following manner: Let z = x*y. Then the tens digit of z becomes the new x and the ones digit becomes the new y (note, if the numbers multiply to equal a single digit number, the new x becomes 0). What starting values of x and y allow one to achieve the highest number of iterations before both x and y become 0?
Just to clarify, I will do an example: x = 2, y = 6. 2*6 = 12, so the new x = 1 and y = 2. 1*2 = 2, so the new x = 0 and the new y = 2. 0*2 = 0, so you are finished after 3 iterations.
Note: Please remember to jot a few notes about your reasoning or methods along with any solution you send me, as I am curious what people will come up with. Seeing as how I completely made this exercise up as an aid to fall asleep, I have no idea what the optimal approach would be.