This is another Brainteaser from the Quantum math magazine .
“How can a polygonal line BDEFG be drawn in a triangle ABC so that the five triangles obtained have the same area?”
I found this problem rather challenging, especially when I first tried to solve it analytically (using hyperbolas). Eventually I arrived at a procedure that would accomplish the result. The Quantum “solution,” however, was tantamount to just saying divide the triangle into triangles of equal area—without providing a method! That is, no solution at all.
See the Equitable Slice Problem
Here is a problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“Three circles with the same radius r all pass through a point H. Prove that the circle passing through the points where the pairs of circles intersect (that is, points A, B, and C) also has the same radius r.”
Indeed, I found this quite challenging. It took me several weeks to work out my approach and details.
See Three Equal Circles
This is a nifty little problem from the Quantum math magazine.
“Two ants stand at opposite corners of a 1-meter square. A barrier was placed between them in the form of half a 1-meter square attached along the diagonal of the first square, as shown in the picture. One ant wants to walk to the other. How long is the shortest path?”
See the Barrier Minimal Path Problem
This is another delightful Brainteaser from the Quantum math magazine.
“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.
Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”
I especially liked this problem since I was able to find a solution different from the one given by Quantum. Who knows how many other variations there might be.
See the Circle-Halving Zigzag Problem
This is a nice Brainteaser from the Quantum math magazine.
“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”
See the Circle Projection Problem.
Here is another problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).
“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”
See Quadrangle in Parallelogram
Here is another Brainteaser from the Quantum magazine.
“Prove that the area of the red portion of the star is exactly half the area of the whole star. (N. Avilov)”
This is a relatively simple problem, but I wanted to include it because of its cartoon. Its implied gentle post-Soviet humor reminded me of that strange decade in US-Russian affairs between the end of the Cold War and the rise of Putin in the 21st century. The strangeness was brought home when we had our annual security checks of our classified document storage. Being mostly anti-submarine warfare (ASW) material the main concern was that it would not fall into the hands of the Soviets. But with the “demise” of the Soviet Union in 1989 no one cared any more about the classification. After decades of painfully securing these documents we could not suddenly turn them loose and throw them into the public trash. So we kept them secure anyway. You can imagine how we old cold-warriors feel about the current regime.
That is not to say that I didn’t welcome the thaw. Russian literature, both classical and even “Soviet realism”, as well as Russian cinema, is some of the world’s best. And Russian mathematicians have always been superior, and especially adept at communicating with novices. The collaboration of the American mathematicians and Kvant contributors in Quantum produced excellent results during the thaw. It is unfortunate that it could not survive the rise of Putin and his oligarchs.
See the Red Star
In looking through some old files I came across a math magazine I had bought in 1998. It was called Quantum and was published by the National Science Teachers Association in collaboration with the Russian magazine Kvant during the period 1990 to 2001 (coinciding with the Russian thaw, which in the following age of Putin seems eons ago). Fortunately, they are all online now. Besides some fascinating math articles the magazine contains a column of “Brainteasers.” Here is one of them:
“Alice used to walk to school every morning, and it took 20 minutes for her from door to door. Once on her way she remembered she was going to show the latest issue of Quantum to her classmates but had forgotten it at home. She knew that if she continued walking to school at the same speed, she’d be there 8 minutes before the bell, and if she went back home for the magazine she’d arrive at school 10 minutes late. What fraction of the way to school had she walked at that moment in time? (S. Dvorianinov)”
This is fairly straight-forward, but other problems in the magazine are a bit more challenging.
See Calculating on the Way