This is a riff on a classic problem, given in Challenging Problems in Algebra.
“N. Bank and S. Bank are, respectively, the north and south banks of a river with a uniform width of one mile. Town A is 3 miles north of N. Bank, town B is 5 miles south of S. Bank and 15 miles east of A. If crossing at the river banks is only at right angles to the banks, find the length of the shortest path from A to B.
Challenge. If the rate of land travel is uniformly 8 mph, and the rowing rate on the river is 1 2/3 mph (in still water) with a west to east current of 1 1/3 mph, find the shortest time it takes to go from A to B. [The path across the river must still be perpendicular to the banks.]” See the River Crossing.
Here is another imaginative geometry problem from Catriona Shearer’s twitter account.
“What fraction of the largest square is shaded?”
See the Cascading Squares Problem.
The issue 7 of the Chalkdust mathematics magazine had an interesting geometric problem presented by Matthew Scroggs.
“In the diagram, ABDC is a square. Angles ACE and BDE are both 75°. Is triangle ABE equilateral? Why/why not?”
I had a solution, but alas, the Scroggs’s solution was far more elegant. See the Chalkdust Triangle Problem.
This problem posted by Presh Talwalkar offers a variety of solutions, but I didn’t quite see my favorite approach for such problems. So I thought I would add it to the mix.
“Thanks to Nikhil Patro from India for suggesting this! What is the sum of the corner angles in a regular 5-sided star? What is a + b + c + d + e = ? Here’s a bonus problem: if the star is not regular, what is a + b + c + d + e = ?”
See Star Sum of Angles
This was a nice geometric problem from Poo-Sung Park @puzzlist posted at the Twitter site #GeometryProblem.
“Geometry Problem 65: Given one square leaning on another, what is the ratio of the triangular areas A:B?”
See the Leaning Squares.
This is another interesting problem from Catriona Shearer. She shows the following figure with a regular hexagon and rectangle.
“The area of the regular hexagon is 30. What’s the area of the rectangle?”
See the Hexagon-Rectangle Problem.
This problem from Futility Closet proved quite challenging.
“University of Illinois mathematician John Wetzel called this one of his favorite problems in geometry. Call a plane arc special if it has length 1 and lies on one side of the line through its end points. Prove that any special arc can be contained in an isosceles right triangle of hypotenuse 1.”
My attempts were futile (maybe that is where the title of the website comes from). Maybe this qualifies for another Coffin Problem. But I did have one little comment about the Futility Closet solution. See Containing an Arc.
These are three “Coffin” Problems posed by Nakul Dawra on his Youtube site GoldPlatedGoof. (Nakul is extraordinarily entertaining and mesmerizing.) The origin of the name is explained, but basically they are problems that have easy or even trivial solutions—once you see the solution. But just contemplating the problem, they seem impossible. The idea was to kill the chances of the pupil taking an (oral) exam with these problems. I was able to solve the first two problems (after a while), but I could not figure out the third. See the Three Coffin Problems.
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.
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.