A Puzzle by Midnight Math

mar2014_princeNew Puzzle

A prince is surveying his kingdom. No matter where he stands, if he walks 100 paces forward in a straight line, turns to the left 90 degrees and walks 100 paces forward again in a straight line, again turns 90 degrees to the left and walks 100 more paces in a straight line, he arrives where he began walking. Where is he? (Bonus points, who is he?)

Correct Answerers
Tom Pandolfo
Greg Edelston

mar2014_pizzaAnswer: “Each The general form is as follows, with n being the number of cuts available and S being the maximum number of resulting slices:
n + (n – 1) + (n – 2) + … + 2 + 1 +1 = n(n + 1)/2 + 1

This is because each chord can intersect with, at most, the number of chords that have already been made. If a chord intersects with 3 other chords, then it must pass through (and thereby bisect) four existing slices of the pizza. Thus, the 1st cut will increase the maximum number of divisions of the pizza by 1, the 2nd will increase it by 2, the 3rd by 3, and so on. Starting with one slice (the whole pizza), this results in a maximum of 56 slices for 10 cuts.”
Proof by Tom Pandolfo.

midnightmathSend in your solutions (with proofs) to midnight.math@outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.