Discrete Mathematics 43

Third proof. Suppose we are part of a finite group of people pointing at each other, and

everyone is more often pointed at than pointing. Let us instruct everyone to pay one dollar

each to the people to whom they point, for each instance of pointing; and let us assume that

we all have enough cash on hand to do this. The curious thing to notice is that, after the

payments, because everyone is more often pointed at than pointing, it follows that every

person will take in more money than they paid out. We made money! And we could do it

again and make more money, and again and again, as many times as we desire. We could

make millions of dollars simply by exchanging it like this. Since this is clearly impossible,

as the total dollar holdings of the group does not change as money is exchanged within it,

there can be no such pointing arrangement.

I find the third proof very clear, though I recognize that it is essentially similar to the first

proof, if one simply thinks of the pointing-at and pointing scores as measured in dollars.

Perhaps the reason it is so clear is that it replaces the abstract quantity-preserving argument

of the first proof with something much easier to grasp, namely, the fundamental fact that

we cannot get more money as a group simply by exchanging money within our group.

Such anthropomorphizing arguments or metaphors can often be surprisingly effective in

simplifying a mathematical idea. We leverage our innate human experience in order to

understand more easily what would otherwise be a complex mathematical matter. Our

human experience with the difficulty of getting money makes the final conclusion of the

third argument obvious.

5.2 Chocolate bar problem

Consider next the chocolate bar problem. Imagine a rectangular chocolate bar, the kind

having a pattern of small squares. We shall break the chocolate along these lines, in such a

way that in the end we have only those tiny squares as separate pieces.

There are a variety of ways that we might do this. For example, for the bar pictured above,

we could first make the three long breaks, making four 8 × 1 sticks, and then would break

off one square at a time from those sticks. This would make 3+ 4·7 = 31 breaks altogether.

Alternatively, we could first make all the short breaks, and then break off individual squares

from the resulting 1 × 4 sticks, resulting in 7 + 8 · 3 = 31 breaks.