We tend to be particularly intrigued by puzzles that seem to have insufficient information to reach a logical conclusion. I’ve given several examples of this class of puzzle (My favorite is still the one about a hole being drilled through a sphere. The hole is one inch long. What is the volume of material left in the sphere?) However, sometimes we come across a puzzle that works the other way around.
Here is a puzzle that seems to be overly complex, but has a nifty solution. The charm is that the solution immediately implies a host of other observations, many of which are relevant to issues in physics even though the puzzle itself seems to have nothing to do with the real world.
Here it is. Lay out a rectangle formed of a grid of equal squares. For simplicity, start with a rectangle of 7 x 5 small equal-sized squares. Think of pan of hard candy that has been scored to break into smaller squares. A one-dimensional example is the old Three Musketeers candy bar, which could be broken into three parts along two inscribed indentations.
The question is simple. How many straight cuts (breaks) does it take to separate all the squares?
Try it a couple of different ways. The only constraint is that each cut must be made on a single piece. You can’t line up separate pieces and cut them all at once. Each cut is through a single piece.
Now after you have solved that problem empirically, can you derive a simple analytical solution? Does it generalize to an N x M sized rectangle?
Interestingly enough, this problem illustrates several issues involving parity and other operations. Understanding just how it works can lead to understanding other problems that seem at first glance to be totally unrelated.
There are several ways to solve this problem. One of the easiest is to use the common technique of starting at the end and working backward.
In the next column (Friday), I will discuss solutions and implications of this puzzle.
For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.
[tags]solution,sherman deforest,puzzle,decision theory,analytical[/tags]