# Sum Of Squares Puzzle

Yet another interesting problem from Presh Talwalkar.

“Two side-by-side squares are inscribed in a semicircle.  If the semicircle has a radius of 10, can you solve for the total area of the two squares? If no, demonstrate why not. If yes, calculate the answer.”

This puzzle shares the characteristics of all good problems where the information provided seems insufficient.

See the Sum of Squares Puzzle.

# Lopsided Hexagon Problem

Here is another good problem from Five Hundred Mathematical Challenges:

“Problem 100.  A hexagon inscribed in a circle has three consecutive sides of length a and three consecutive sides of length b. Determine the radius of the circle.”

This problem made me think of the Putnam Octagon Problem.  Again my approach might be considered a bit pedestrian.  500 Math Challenges had a slightly slicker solution.

See the Lop-sided Hexagon Problem

# Geometric Puzzle Munificence

Having fallen under the spell of Catriona Shearer’s geometric puzzles again, I thought I would present the latest group assembled by Ben Orlin, which he dubs “Felt Tip Geometry”, along with a bonus of two more recent ones that caught my fancy as being fine examples of Shearer’s laconic style. Orlin added his own names to the four he assembled and I added names to my two, again ordered from easier to harder.

(Update 4/16/2020)  Ben Orlin has another set of Catriona Shearer puzzles 11 Geometry Puzzles That Drive Mathematicians to Madness which I will leave you to see and enjoy. But I wanted to emphasize some observations he included that I think are spot on. Continue reading

# A Tidy Theorem

This is another fairly simple puzzle from Futility Closet.

“If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (q = p + r).

(A corollary of Ptolemy’s theorem.)”

# Amazing Triangle Problem

Here is another simply amazing problem from Five Hundred Mathematical Challenges:

Problem 154. Show that three solutions, (x1,.y1), (x2,.y2), (x3, y3), of the four solutions of the simultaneous equations
____________(x – h)² + (y – k)² = 4(h² + k²)
______________________xy = hk
are vertices of an equilateral triangle. Give a geometrical interpretation.”

Again, I don’t see how anyone could have discovered this property involving a circle, a hyperbola, and an equilateral triangle. It seems plausible when h.=.k, but it is not at all obvious for h..k. For some reason, I had difficulty getting a start on a solution, until the obvious approach dawned on me. I don’t know why it took me so long.

See the Amazing Triangle Problem.

# Magic Hexagons

This is truly an amazing result from Five Hundred Mathematical Challenges.

Problem 119. Two unequal regular hexagons ABCDEF and CGHJKL touch each other at C and are so situated that F, C, and J are collinear.

Show that

(i) the circumcircle of BCG bisects FJ (at O say);
(ii) ΔBOG is equilateral.”

I wonder how anyone ever discovered this.

See the Magic Hexagons

# Tipsy 3-4-5 Triangle

Presh Talwalkar had another interesting problem.

“A triangle is drawn inside a square with sides 4, 3, and 5, as shown. What is the length of the square’s side?”

The problem looks simple at first, but it takes some care to avoid some hideous quartic equations.