James Tanton posted the following interesting problem on his Twitter account:
A square of area A, perimeter P. For which values 0< r <1 is there a line across the square that chops off rA of the area and rP of the perimeter as shown? (r = 1/2, yes; r = 1/3, no.)
See the Perimeter-Area Problem.
James Tanton has provided further elaborations on the polygons and the sum of perpendicular distances from interior points. Again I approached the solutions with a mix of areas and vectors. It is rather impressive to see the number of variations that can be rung on the Viviani Theorem theme. See Polygon Altitude Problems II
I found this collection of related problems by James Tanton on Twitter. Even though all these problems do not involve perpendiculars, they have a common solution approach – a sort of theme and variations idea. In a later tweet Tanton refers to a Viviani Theorem associated with these types of problems. I did not recall that theorem explicitly or by name. I also have not looked it up yet, in order to solve these problems on my own. I am guessing there is a more classical Euclidean geometry proof, but I like my vector approach for its clarity. I also throw in a bit a calculus at the end for fun. See Polygon Altitude Problems I
This is a collection of simple but elegant puzzles, mostly from a British high school math teacher Catriona Shearer, for which I thought I would show solutions (solutions for a number of them had not been posted yet on Twitter at the time of writing). See the Geometric Puzzle Medley.
Apparently Catriona Shearer creates these problems herself, which shows an especially gifted talent. Ben Olin, of Math with Bad Drawings fame, had an interesting interview with Ms. Shearer. The reason for the interest in her work becomes evident the more of her geometry problems one sees. They are especially elegant and minimalist, and often have simple solutions, as exemplified by the “5 Problem” or “Shear Beauty” problem illustrated here. Words, such as “beauty” and “elegance”, are often bandied about concerning various mathematical subjects, but as with any discussion of esthetics, the efforts at explanation usually fall flat. Shearer’s problems are one of the best examples of these ideas I have ever seen. If you contemplate her problems and even solve them, you will understand the meaning of these descriptions.
One of the key aspects of mathematics is often its “hidden-ness” (some would say “opacity” or “incomprehension”). Her problems appear to have insufficient information to solve. But as you look at the usually regular figures, you see that there are inherent rigid constraints that soon yield specific information that leads to a solution. This discovery is akin to the sensation of discovering Newton’s mathematical laws underlying physical reality. It is the essence of one of the joys of mathematics.
This problem comes from the defunct Wall Street Journal Varsity Math Week collection.
“The coach then shows the team the diagram to the left and asks: What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10 and 17?”
I changed the numbers a bit to make my calculations easier, but left the problem otherwise unchanged. When I checked the Varsity Math Week solution, I saw they used a simplifying formula that I could not remember. I also believed their solution left out a justification for the maximal area. Besides an intuitive solution for this, I also included a calculus version. See the Triangular Boundary Problem.
Yet another Futility Closet puzzle.
“Point E lies on segment AB, and point C lies on segment FG. The area of parallelogram ABCD is 20 square units. What’s the area of parallelogram EFGD?”
I had an alternative solution that I thought was a bit simpler and clearer. See the Parallelogram Puzzle.
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.
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.
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.
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.