Railroad Tie Problem

This is a surprisingly challenging puzzle from the Mathematics 2020 calendar.

“The sketch is of equally spaced railroad ties drawn in a one point perspective. Two of the ties are perceived to the eye to be 25 feet and 20 feet respectively. What is the perceived length x of the third tie?”

Even though the ties are equally-spaced and of equal length in reality, from the point of view of perspective they are successively closer together and diminishing in length. The trick is to figure out what that compression factor is. I had to review my post on the Perspective Map to get some clues.

See the Railroad Tie Problem

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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?”

Answer.

See the Perpetual Meetings Problem for solutions.

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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?”

Answer.

See the Pinwheel Area Problem. for solutions.

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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.)”

See A Tidy Theorem

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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.

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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?”

Answer.

See the Train Buffs for a solution.

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One Year Anniversary

It is hard to believe a whole year has passed since I started this blog. What is even more surprising is that by February I thought I was about done. I had more or less uploaded the math curiosities and problems I had been thinking about over the years and had presented most of the math essays I had written. There are of course only a finite number of math problems in the world, so I thought I was about done. But much to my surprise I kept finding one more thing that interested me, either an essay or math problem. So here I am. We will have to see what the next year brings.

What to say on this anniversary? I think I will give a retrospective about how the website has been received this past year. This is a challenge, since virtually all my visitors have been silent (which means I haven’t faced criticism, but then I generally haven’t received the necessary correctives either). There are ways to glean some information about visitors and I extracted what I could from the simple plugin I use to count visitors to different posts. I hear that Google analytics provides lots of details, such as the country of origin of a visitor (which would be interesting) and the like, but I have avoided Google and the other social networks as much as possible. After all, I am only an old curmudgeon with old-school notions of privacy.

See One Year Anniversary

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