The solution to this morning’s puzzle

September 20, 2021 • 2:39 pm

I haven’t yet looked at the comments about the puzzle I reported this morning that had been proposed by Russian Prime Minister Mikhail Mishustin when visiting a science-oriented “sixth form” class.

Here’s the puzzle again:

Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.

In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)

The beauty of this question is the seemingly outrageous restriction not to allow measuring devices, which means that you cannot use a compass or a marked ruler. All you are allowed is an unmarked ruler to draw straight lines.


The Guardian has now published the four-step solution (there may be others); go to the preceding link to see it. Below you can see the PM drawing the solution (he must know his math).

14 thoughts on “The solution to this morning’s puzzle

  1. This all assumes that the original diameter of the circle was drawn correctly and is a true diameter, exactly bisecting the circle. Otherwise what is done is a mess.

    1. It’s a mathematical geometry problem. Of course the original diameter is assumed to have been drawn correctly. And in the picture of the blackboard, the lines are assumed to be straight.

  2. “Below you can see the PM drawing the solution (he must know his math)” – Given the results from past Russian elections, I’m not sure their math(s) is the same as ours… Given that it is officially acknowledged that United Russia got less than 50% of the votes in the latest one, I can’t help wondering what the real figure was!

    1. I should admit that I stupidly couldn’t see why the green dotted line didn’t meet the criteria of “a way to draw a line from the point that hits the diameter at a right angle” in the first place…

      1. Anyone connected to the present Russian government is probably crooked, so maybe they look straight when viewed through his eyes!

        See the previous on this for my criticism of this clever Guardian solution, which does need the rule to be able to draw a random line for the second triangle, and a much more complicated solution than our Jeremy Perriera’s, whose extra rule of allowing tangents gives a simpler more elegant solution.

  3. Oh man, I love that solution. No tricks, just pure Euclidean geometry. Takes me back to 10th grade geometry class – my all time favorite math class.

  4. My 15-year-old self is thinking (vide my comment on yesterday’s post in which PCC(E) set us the challenge) that he could have figured that out, if only all that laboriously-learned Euclidean geometry hadn’t been displaced by high-school algebra and calculus, and several years of university-level mathematics in which geometric proofs did not feature at all.

    1. Yeah. One of my solution ideas actually got me as far as the selecting another arbitrary point on the circle and constructing a 2nd triangle from that point, and even drawing the larger triangle by extending the legs of the first two triangles up. But then I couldn’t make anything of it because I completely forgot that all of the altitudes of a triangle intersect at a single point.

      I figured out early on the part about how to translate the point down to get the final solution, by constructing the 4th triangle bisected by the diameter line and with legs that intersect the circle at the given point and at the solution point, but I couldn’t figure out how to get the 2nd point necessary to draw the first line of that triangle. Orthocenter, I’ll have to remember that.

  5. Those who liked the puzzle might like this:

    Draw three non-overlapping circles on a plane. Neither size (as long as they are all different) nor position matters.

    For each pair of circles, draw two lines tangent to the sides of both circles. They will cross somewhere. Mark that point.

    The three points always lie on the same line.

    Exercise: prove it.

  6. I would cheat and use the unmarked ruler as a T square. Not exact math but close enough for all practical purposes. That’s the beauty of being an engineer rather than a mathematician.

    1. I was thinking nearly the same thing, except with a framing square. I wonder if the square wouldn’t be more accurate in practice – drawing all those lines and triangles doesn’t seem conducive to precision.

      1. Absolutely it would be more accurate to draw the problem and solution with proper drafting tools and techniques. The solution would be reduced to drawing a single perpendicular line, starting at the given point on the circle, directly. Drawing the given solution out, even with proper drafting tools and techniques, would be much less precise because of the cumulative errors associated with manual alignment of tools over multiple steps compared to just one step.

        But for a good draftsperson it wouldn’t really matter. Sharp lead, good tools and good skills and the difference would be negligible to the eye.

  7. I’m waiting for a U.S. cabinet-level or higher official, of their own volition, to similarly pose STEM-related, intellectual curiosity-motivated questions to the younger set. (Or would that be “acting Russian” or some other such thing?)

    I’m putting on my To Do List inquiring what the purpose of a Russian prime minister is vis-a-vis a president (Putin). The rough equivalent of a U.S. vice-president? (John Nance Garner euphemistically said the VP position wasn’t “worth a bucket of warm spit.” I don’t understand that perspective. It’s a heartbeat away from the presidency. Ask Lincoln’s, Garfield’s, McKinley’s and JFK’s successors.)

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