Tag Archives: plane geometry

String of Beads Puzzle

Floating Square Puzzle

This is another puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“A mysterious square has materialized in the middle of the MCG, hovering in mid-air. The heights above the ground of three of its corners are 13, 21 and 34 metres. The fourth corner is higher still. How high?”

See the Floating Square Puzzle

(Update 8/13/2023) Alternative Solution Continue reading →

Square In A Quarter Circle

Another puzzle by Presh Talwalkar.

“Thanks to John H. for the suggestion!

A square is inscribed in a quarter circle such that the outer vertices are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?

I am told this was given to 7th grade students (ages 12-13), and I think it is a very challenging problem for that age group. In fact I think it is a good problem for any geometry student.”

See the Square in Quarter Circle

A Triangle Puzzle

The Futility Closet website had the following problem:

“In isosceles triangle ABC, CD = AB and BE is perpendicular to AC. Show that CEB is a 3-4-5 right triangle.”

See a Triangle Puzzle

Right Angles in a Square

Here is another problem from the “Challenges” section of the Quantum magazine.

“Point L divides the diagonal AC of a square ABCD in the ratio 3:1, K is the midpoint of side AB. Prove that angle KLD is a right angle. (Y. Bogaturov)”

See Right Angles in a Square

Ant Connection Problem

This is a nice puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings.

“Two ants are on a cylindrical glass that is 5 centimetres in diameter. The ants are on opposite sides of the glass, 5 centimetres down from the glass’s rim. If both ants are on the outside of the glass, what is the shortest distance required for one ant to crawl to the other? What if one ant is on the outside of the glass and the other is on the inside?”

See the Ant Connection Problem.

Putnam Ellipse Areas Problem

This is a nifty problem from Presh Talwakar.

“This is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!

Let R be the region in the first quadrant bounded by the x-axis, the line y = x/2, and the ellipse x²/9 + y² = 1. Let R‘ be the region in the first quadrant bounded by the y-axis, the line y = mx and the ellipse. Find the value of m such that R and R‘ have the same area.”

See the Putnam Ellipse Areas Problem

Curious Sunbeam Problem

This is a Catriona Agg problem presented by itself, since it turned out to be the most challenging one I ever tried. Usually I can solve her problems in a few minutes or maybe hours, or sometimes days if they are especially challenging. But this problem has taken me weeks and I had to rely on a non-geometric argument. The problem is full of fascinating and unexpected relationships, but I couldn’t find a way to use them to prove the answer.

See the Curious Sunbeam Problem

(Update 5/5/2023) Alternative Solution Continue reading →

Geometric Puzzle Mystifiers

Here is yet another (belated) collection of beautiful geometric problems from Catriona Agg (née Shearer).

See Geometric Puzzle Mystifiers

Moon Quarters Problem

This is a straight-forward problem from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge.

“A circle has radius 1 cm and AB is a diameter. Two circular arcs of equal radius are drawn with centres A and B. These arcs meet on the circle as shown. Calculate the shaded area.”

There are several possible approaches and the SMC offers two examples.

See the Moon Quarters Problem

Meditations on Mathematics

A compendium of puzzles, problems, math inquiries, and math commentary.