## Ken's POTW

Books and Cannonballs
Two more from Sam Loyd, then I'll put my book back on its shelf:
1. You have nine large books, labeled 1,2,3,4,5,6,7,8,9 on a bookshelf with two shelves. Placing volumes 6,7,2,9 on the top shelf and 1,3,4,5,8 on the bottom shelf, you create a fraction (6729/13458) exactly equal to 1/2. Is it possible to find other arrangements, using all nine volumes, that will make fractions equivalent to 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, and 1/9? [Tricks like inverting the books aren't needed, but I would be happy to post any solutions found in this way.]

2. James is an adept cannonball stacker. In each of the cases below, he is asked to split one pyramid of cannonballs into two smaller ones. Assuming James can complete each request, what are the minimum (and perhaps other representative) sizes of each pyramid?
• Split a pyramid with a square base into two smaller pyramids, each with a square base.
• Split a pyramid with a square base into two smaller pyramids, each with a triangular base.
• Split a pyramid with a triangular base into two smaller pyramids, each with a triangular base. [A simple answer can be found if the smaller pyramids are identical. Try to also find an answer with them unequal.]
• Split a pyramid with a triangular base into two smaller pyramids, each with a square base.

Source: Mathematical Puzzles of Sam Loyd, Volume Two, edited by Martin Gardner, Dover Publications, New York, 1960, #74, #78.

Solution
Mail to Ken