Discover Boggler, July 2001


	THE DISCOVER BOGGLER July 2001: Child’s Play

This set of puzzles was inspired by another project I was working on at the time: a series of physical two-player strategy games from Binary Arts that includes a version of Dots and Boxes. As part of that project I read the book Dots and Boxes by Berlekamp, a deep mathematical analysis of the child's game that the author calls .
The games are existing classic mathematical games. My part of the project was to figure out a format for teaching winning strategies on the Binary Arts web site. Had to be entertaining, bite sized. My idea: instead of essay, present a series of problems, plus explanations. From there it was a small step to adapt the material for magazine puzzles. This is the first interesting step. Quite a bit more after that.
To fill out my quota of three puzzles, I added puzzles based on Battleship and Tic Tac Toe.

Ship Shapes. This puzzle is quite similar to my first Discover Boggler Pillow Talk, which ran in March 1990. Used the ship shapes from Hasbro's Battleship® game, but with a small 5x5 board instead of a big 10x10 board, in order to make it easier to construct problems with unique solutions.

Extensions. Is there a 5x5 board with just four hits or just four misses in which it is possible to deduce the positions of all five ships? I suspect the answer is no, but I'm not sure.

Readers. Scott Lenser and Nate burgess found this solution to the four hits problem by hand:

To prove that the solution is unique, first note that the upper left explosion must grow to the right as a 2-ship. Play around with the 5-ship and you'll soon find that there's only one place it can go.

Mark Miller went one further and wrote a program to find all possible solutions. He confirmed that there are no solutions to the four misses problem, but one additional solution to the four hits problem, closely related to the first solution:

You can download Mark's C program here.

Trick-tac-toe.

Many variations. Some in Zillions of Games, a bit too hard to explain. The draws on a 4x4 board was original.

Blocked In. Elwyn Berlekamp, Strategy Games from Binary Arts. Berlekamp's book. Also covered in more detail in Scientific American.