# Perpetual Meetings Problem

The following problem from Five Hundred Mathematical Challenges was a challenge indeed, even though it appeared to be a standard travel puzzle.

Problem 118. Andy leaves at noon and drives at constant speed back and forth from town A to town B. Bob also leaves at noon, driving at 40 km per hour back and forth from town B to town A on the same highway as Andy. Andy arrives at town B twenty minutes after first passing Bob, whereas Bob arrives at town A forty-five minutes after first passing Andy. At what time do Any and Bob pass each other for the nth time?”

See the Perpetual Meetings Problem for solutions.

# Pinwheel Area Problem

Here is another engaging problem from Presh Talwalkar.

___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”

See the Pinwheel Area Problem. for solutions.

# Quintic Nightmare

Here is another challenging problem from the 2004 Pi in the Sky Canadian magazine for high school students.

Problem 4. Find the real solutions of the system
________________ (x + y)^5 = z,
________________ (y + z)^5 = x,
________________ (z + x)^5 = y.”

See the Quintic Nightmare for a solution.

# 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.

# The Train Buffs

Here is another train puzzle, this time from J. A. H. Hunter’s Entertaining Mathematical Teasers:

“Mike had made the [train] trip many times. ‘That’s the morning express from Tulla we’re passing,’ he said. ‘It left Tulla one hour after we pulled out from Brent, but we’re just 25% faster.’ ‘That’s right, and we’re also passing Cove, two-thirds the distance between Brent and Tulla,’ Martin agreed. ‘So we’re both right on schedule.’ Obviously a couple of train buffs! Assuming constant speeds and no stops, how long would it be before they reached Tulla?”

See the Train Buffs for a solution.

I have been subverted again by a recent post by Ben Orlin, “Geometry Puzzles for a Winter’s Day,” which is another collection of Catriona Shearer’s geometric puzzles, this time her favorites for the month of November 2019 (which Orlin seems to have named himself). I often visit Orlin’s blog, “Math with Bad Drawings”, so it is hard to kick my addiction to Shearer’s puzzles if he keeps presenting collections. Her production volume is amazing, especially as she is able to maintain the quality that makes her problems so special.

The Stained Glass puzzle generated some discussion about needed constraints to ensure a solution. Essentially, it was agreed to make explicit that the drawing had vertical and horizontal symmetry in the shapes, that is, flipping it horizontally or vertically kept the same shapes, though some of the colors might be swapped.

# 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

# Factory Location Problem

This is a somewhat elegant problem from the 1987 Discover magazine’s Brain Bogglers by Michael Stueben:

“Each dot in the figure at left represents a factory. On which of the city’s 63 intersections should a warehouse be built to make the total distance between it and all the factors as short as possible? (A much simpler solution than counting and totaling the distances is available.)”

Note that the distance is the taxicab distance I discussed in my article South Dakota Travel Problem rather than the distance along straight lines between the warehouse and factories.

See the Factory Location Problem for solutions.