02 Nov '14 23:02

Given that a few people seem interested in solving more problems in

elementary mathematics, here's another one (which I recall with nostalgia):

Let a convex polygon with N sides (N>3) have all its diagonals constructed,

dividing it into interior regions. What's the maximum number M of interior regions?

Advice: Recreational mathematics should be fun, so I don't want anyone to

waste many hours of time on this problem. If you think about the problem

in the right way, you could solve it within a few minutes. But there seem to

be various tempting ways to go astray. You could start by playing around

with specific cases when N is small (if N=4, M=4) to get an intuitive sense of

how quickly M increases when N does. You could conjecture a solution and

attempt to prove it through mathematical induction. You could ignore this.

As usual, I would ask that people rely only upon their own brains and not

attempt to look up the solution elsewhere. I used to explain the solution

in a lecture (which is unavailable as a video, alas, on YouTube), and I expect

that the problem and its solution have been published in various places.

Good luck to everyone who cares enough to make an honest effort.

elementary mathematics, here's another one (which I recall with nostalgia):

Let a convex polygon with N sides (N>3) have all its diagonals constructed,

dividing it into interior regions. What's the maximum number M of interior regions?

Advice: Recreational mathematics should be fun, so I don't want anyone to

waste many hours of time on this problem. If you think about the problem

in the right way, you could solve it within a few minutes. But there seem to

be various tempting ways to go astray. You could start by playing around

with specific cases when N is small (if N=4, M=4) to get an intuitive sense of

how quickly M increases when N does. You could conjecture a solution and

attempt to prove it through mathematical induction. You could ignore this.

As usual, I would ask that people rely only upon their own brains and not

attempt to look up the solution elsewhere. I used to explain the solution

in a lecture (which is unavailable as a video, alas, on YouTube), and I expect

that the problem and its solution have been published in various places.

Good luck to everyone who cares enough to make an honest effort.