# Four of a Kind

From Futility Closet we have another intriguing problem with what turns out to be a simple and elegant solution.

“If squares are drawn on the sides of a triangle and external to it, then the areas of the triangles formed between the squares each equal the area of the triangle itself.”

I originally assumed that the center triangle was a right triangle as suggested by the picture. But then I realized there was a solution that did not depend on that. See Four of a Kind.

# Polygon Areas Problem

This is another problem from Futility Closet, though Futility Closet provides a “solution” of sorts. They provide a set of steps without explaining where they came from. So I thought I would fill in the gap. The problem is to find the area of an irregular polygon, none of whose sides cross one another, if we are given the coordinates of the vertices of the polygon.

See Polygon Areas Problem for a solution.

# The Four Travelers Problem

This is another Futility Closet puzzle.

“Four straight roads cross a plain. No two are parallel, and no three meet in a point. On each road is a traveler who moves at some constant speed. If Blue and Red meet each other at their crossroad, and each of them meets Yellow and Green at their respective crossroads, will Yellow and Green necessarily meet at their own crossroad?”

I was not able to understand the solution given at first, so I tried to solve the problem on my own. Once I did, I was able to see what the Futility Closet solution was getting at. Certainly diagrams were needed to make sense of it all, and that is what I provided. See the Four Travelers Problem.

# Corner Reflectors

I came across the following entry in the Futility Closet website that cried out for justification. “An arrangement of three mutually perpendicular planes, like those in the corner of a cube, have a pleasing property: They’ll reflect a ray of light back in the direction that it came from.” So the question is, why is this reflection property true? See Corner Reflectors.

# The Two Errand Boys

This is another puzzle from the Futility Closet that was originally from Henry Dudeney’s Canterbury Puzzles.

“A country baker sent off his boy with a message to the butcher in the next village, and at the same time the butcher sent his boy to the baker. One ran faster than the other, and they were seen to pass at a spot 720 yards from the baker’s shop. Each stopped ten minutes at his destination and then started on the return journey, when it was found that they passed each other at a spot 400 yards from the butcher’s. How far apart are the two tradesmen’s shops? Of course each boy went at a uniform pace throughout.”

See the Two Errand Boys for solutions.

# Keyhole Problem

This is another problem from the Futility Closet website. It turned out to be pretty simple. The idea is to show the length of BC remains the same no matter where A is chosen on its arc of C1.

(Update 7/1/2020) There is more to this problem than I realized, thanks to a revisit prompted by a question from Deb Jyoti Mitra.  See the revised Keyhole Problem.

# Two Circles Puzzle

Another good source of problems is the Futility Closet site. This puzzle involved finding the line of maximal length passing through the intersection of two circles. I solved it before looking at the Futility Closet solution. Their solution of course was short, sweet, and elegant. Mine was more like the old adage of cracking a walnut with a sledge hammer. Still, I thought there were some unexplained parts to the elegant solution that justified the effort on mine. At least my solution provided an interesting, though convoluted, alternative. See the Two Circles Puzzle.