This week’s puzzle was a collaboration with high-schooler Connor Hill, the First Place Winner from this year’s Regeneron Science Talent Search. Connor’s project involved an original, computer-assisted proof to find the complete list of noble polyhedra, in which all the faces are indistinguishable and all the vertices are indistinguishable. The five Platonic solids are perhaps the best known examples of noble polyhedra, but Connor documented all of them, including two infinite sets and 146 exceptional cases, among which were 85 new examples!
Here’s the puzzle from Connor—unrelated to his work—which is about a frog that makes very particular hops:
A frog is hopping around a chessboard, always from the center of one square to the center of another square. Each square has side length 1, but the board itself is not necessarily 8-by-8. Instead, it’s N-by-N, where N is some large whole number.
Every jump the frog makes must be the same distance, which we’ll call L. The frog wants to make four jumps such that:
After the fourth jump, the frog has returned to its starting square.
The frog visits a total of four distinct squares along the way, including the square on which it starts (and also stops).
The path the frog takes is not a square loop.
The frog is never on a square that is diagonal (i.e., a bishop’s move away) or horizontal/vertical (i.e., a rook’s move away) from the starting square.
What is the smallest jumping distance L for which this is possible?
The frog's hops must make a rhombus. $L$ must be the hypotenuse of two different right triangles with integer legs.
The first positive integer that can be expressed as the sum of two squares in two ways is $50 = 7^2+1^2 = 5^2+5^2$. However, $5^2+5^2$ corresponds to an isoceles right triangle, and thus will be a bishop's move away from the starting square.
The second positive integer that can be expressed as the sum of two squares in two ways is $65 = 8^2+1^2 = 7^2+4^2$.
From Connor also comes some Extra Credit:
The frog is jumping around the board with the same minimum distance L you just found.
But this time, the frog also wants to be able to hop to every location on the chessboard. What is the minimum value of N for which this is possible?
