This is a nice little puzzle from the late Nick Berry’s Datagenetics Blog.
“A quick little puzzle this week. (I tried to track down the original source, but reached a dead-end with a web search as the site that hosted it, a blogspot page under the name fivetriangles appears password protected, and no longer maintained). …
There are three identical triangles with aligned bases (in the original problem, it is stated they are equilateral, but I don’t think that really matters; Any congruent triangles will do, and I’m going to use isosceles triangles in my solving). If we say that one triangle has the area A, what is the area of the two shaded regions?”
See the Three Triangles Puzzle.
This is a provocative puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2013.
“In the picture the top curve is a semicircle and the bottom curve is a quarter circle. Which has greater area, the red square or the blue rectangle?”
See the Spiral Areas Puzzle
Here is another Brainteaser from the Quantum magazine.
“King Arthur ordered a pattern for his quarter-circle shield. He wanted it to be painted in three colors: yellow, the color of kindness; red, the color of courage: and blue the color of wisdom. When the artist brought in his work, the king’s armor-bearer said there was more courage than wisdom on the shield. But the artist managed to prove that the proportions of both virtues were equal. Can you tell how? (A. Savin)”
This is another relatively simple problem, though it may look a bit daunting at first.
See Wisdom of Old
This is a fairly straight-forward problem from Presh Talwalkar.
“A triangle is divided by 8 parallel lines that are equally spaced, as shown below. Starting from the top small triangle, color each alternate stripe in blue and color the remaining stripes in red. If the blue stripes have a total area of 145, what is the total area of the red stripes?”
See the Triangle Stripes Problem
This is a Maths Item of the Month (MIOM) problem that seems opaque at first. (“The Maths Item of the Month is a monthly problem aimed at teachers and students of GCSE and A level Mathematics.”)
“Two fixed circles, C1 and C2, intersect at A and B. P is on C1. PA and PB produced meet C2 at A’ and B’ respectively. How does the length of the chord A’B’ change as P moves?”
Just start noticing relationships and the answer falls out nicely.
(MIOM problems often appear on MathsMonday and are also produced by Mathematics Education Innovation (MEI).)
See the Triangle Projection Problem
This is a nice problem from the UKMT Senior Mathematics Challenge for 2022:
“Five line segments of length 2, 2, 2, 1 and 3 connect two corners of a square as shown in the diagram. What is the shaded area?
A 8____B 9____C 10____D 11____E 12”
The pleasure of solving this problem may be lessened if one is under a time crunch, as is the case with all these timed tests.
See the Broken Diagonal Problem
This is an interesting problem from the Scottish Mathematics Council (SMC) 2014 Senior Math Challenge .
“Two straight sections of a road, each running from east to west, and located as shown, are to be joined smoothly by a new roadway consisting of arcs of two circles of equal radius. The existing roads are to be tangents at the joins and the arcs themselves are to have a common tangent where they meet. Find the length of the radius of these arcs.”
See the Road Construction Problem
This turned out to be a challenging puzzle from the 1980 Canadian Math Society’s magazine, Crux Mathematicorum.
“Proposed by Leon Bankoff, Los Angeles, California.
Professor Euclide Paracelso Bombasto Umbugio has once again retired to his tour d’ivoire where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder’s Calculus of Extension. His goal is to prove Neuberg’s Theorem:
If D, E, F are the centers of squares described externally on the sides of a triangle ABC, then the midpoints of these sides are the centers of squares described internally on the sides of triangle DEF. [The accompanying diagram shows only one internally described square.]
Help the dedicated professor emerge from his self-imposed confinement and enjoy the thrill of hyperventilation by showing how to solve his problem using only highschool, synthetic, Euclidean, ‘plain’ geometry.”
Alas, my plane geometry capability was inadequate to solve the puzzle that way, so I had to resort to the sledge hammer of analytic geometry, trigonometry, and complex variables.
See Neuberg’s Theorem
Here is another elegant Quantum math magazine Brainteaser problem.
“Two squares are inscribed in a semicircle as shown in the figure at left. Prove that the area of the big square is four times that of the small one.”
See More Squares in Semicircle
James Tanton provides another imaginative problem on Twitter.
“I am at point A and want to walk to point B via some point, any point, P on the circle. What point P should I choose so that my journey A → P → B is as short as possible?”
Hint: I got ideas for a solution from two of my posts, “Square Root Minimum” and “Maximum Product”.
See Minimum Path Via Circle