In lieu of Readers’ Wildlife today, we have a puzzle, one posted (as the Guardian reports) by Russian Prime Minister, Mikhail Mishustin when he was visiting a science-oriented “sixth form” (what age of kids are these?) school. Matthew sent me a link.
Here’s the problem, and there’s a clue in both the photo below and in the Guardian article:
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.
Matthew couldn’t solve it and, as I haven’t had my coffee, I’m not even going to try.
Here’s, a picture of Mishustin posing the problem (and giving a bit of a solution):
Sixth form in my school was like 12th grade in public US schools.
I was really hoping I’d see the answer lol
It’s up now in the latest post
“Sixth form” in Britain refers to the final two years of school before entering university, so age 16-18. That’s junior and senior year of high school, in American terms.
The geometry problem is giving me flashbacks to my mathematics classes when I was around 14 or 15, which were largely learning how to do Euclidean proofs and constructions involving circles, triangles, circles inscribed inside triangles, triangles inscribed inside circles and so forth. We were allowed a compass and straight edge only. It was an almost entirely pointless pursuit, because I never used this kind of thing after the O-level exam, despite gaining a B.Sc. and a Ph.D. in mathematics and later teaching mathematics at university level. Still, my 15-year-old self would probably have been able to solve the Russian P.M.’s challenge!
Since we are allowed an unmarked ruler, this problem could be done by aligning the narrow edge (width) of the ruler on the diameter and then aligning the long edge at the red dot. But of course that is a mechanical method, and clearly not what is asked for.
Are you allowed to add your own marks to the straightedge?
If you can mark the straightedge, it’s easy.
You already have the length of the secant. The horizontal must be 3/5 of the secant so that the vertical is 4/5 of the secant. You can extend the lines of both the diameter and the secant beyond the circle until you get the proper ratio.
Voila! A 3, 4, 5 triangle.
Then you’d have a measuring tool.
You’re right; I would. That’s why I asked if you could mark the straightedge.
L
Allowed a straightedge, but not a compass? There is an entire theory here which is well worth teaching undergrads. It presages much else in algebra.
We were taught in high school how to do it with a compass of course.
Anyway,only a straightedge might be interesting.
Presumably the rules more precisely are that you’re given 2 or more points, in this instance 3. Lines joining points already constructed are constructible. And newly constructible points are any new intersections of constructible lines.
So is that line perpendicular to the horizontal line from the point above it constructible in that sense? Likely not.
But now given also the one circle and allowing points on it from intersections with lines as well, it’s claimed it is.
I’m late for dentist if I don’t rush, maybe just trying to avoid embarrassment!!
Looks unlikely without some additional ‘rules’, but I’m old and dumb.
Poncelet-Steiner theorem.
I am only posting this because my own top level post on the subject has disappeared. If it comes back, delete this comment.
The Poncelet-Steiner theorem states that, if you are given a circle whose centre is known, you can construct anything with just a ruler that you could construct with a ruler and compasses.
But I don’t see the center as a known in this problem. ??
I solved it but my solution takes around 10 steps, which for these types of problem is quite a few, so I don’t know if mine is the most efficient. I love the ditching of the compass in the formulation! Makes it much harder.
Aha, but that doesn’t mean I can’t use a pair of compasses. The problem is trivial, and, in fact, I don’t see how knowing where North is would make it any easier.
There’s a theorem called the Poncelet-Steiner Theorem that states that, provided you have a circle and you know its centre, you can construct anything with a straight edge that you can construct with a straight edge and compass. There are enough constructions on the wikipedia page to solve the problem, but there’s probably a trick you can do to make it more elegant.
But you don’t have a measuring device, so you can’t determine where the centre is. It’s somewhere along that diameter, obviously, but it’s not marked on the diagram, so the Poncelet-Steiner Theorem cannot help you.
Suppose I put the end of my straightedge on the right intersection of the diameter and the circle, run the straightedge to the dot, and mark the distance on my straightedge. Then I go that distance from the same intersection to the part of the circle that’s below the diameter, and mark that point on the circle. Then I draw a line connecting the dot to the point I just marked. That line should pass through the diameter at 90 degrees.
That’s equivalent to having a compass, and thus not allowed by the rules of the challenge.
I don’t consider an uncalibrated compass to be a “measuring device”. With that it is easy.
You can sort of use the straight edge as a compass.
Yes, I see that. Even without making marks, you could use the ends like a compass.
In straight edge and compasses constructions, the straight edge is assumed to be infinite in length.
You can always calibrate a pair of compasses by setting the distance between the points to the length of a particular line segment. They are definitely a measuring device.
It can be solved with a piece of string and a weight. The circle is held vertical so that the diameter line is horizontal. Pin the string to any point on the upper half of the circumference, and where it crosses the diameter line it will define a perpendicular line from that point to the diameter line. One could sort of do the same with the straight edge ruler, using it instead of a weighted string.
These various tricks, with a compass, straight edge, or string, were once essential skills for carpenters and masons.
How do you determine that the diameter is horizontal?
It’s not obvious to me at first look, but my first thought is that the solution is derived from the same underlying relationships as is the compass technique that solves the problem by drawing pairs of arcs above and below a line from 2 separate points on the line. I’ll have to think about it when I’ve got a few minutes.
I was curious and, after some fruitless attempts, I searched the solution on the internet. Easy to understand, but rather difficult to figure out, I would say.
For those who want to find the solution, I confirm that:
– no compass (or pair of compasses) should be used
– the center of the circle is not known
– you cannot add your marks to the unmarked ruler
I’ve just thought of an easy way, provided it’s legal to draw a tangent to a circle at a point. If so:
1. Extend the diameter out to the right.
2.Draw a tangent from the red point and extend it to the right so that it crosses the extended diameter.
3. Draw a line from the intersection that just touches the circle below the diameter i.e. another tangent. Mark this point.
The line connecting the two points is the perpendicular.
That was my approach, but I am not sure that the assumption that one can draw an accurate tangent is legitimate.
I googled the answer and interestingly, the final stage of the proper solution works in the same way as I described above. The difference is that it first does a construction that that gives you two points, one of which is the red one, so you don’t have to draw a tangent.
Contralutions! I used the tangent to the red point, and covered a sheet of paper with similar triangles. i think that I found an an alternative to your initial solution, but far more complicated. i could see no alternative to using tangents.
By using Thales theorem, you can construct a triangle whose altitude is perpendicular to the diameter. Once you have any perpendicular, it is easy to construct another one through the red point using a technique similar to the tangent one.
This is what I did and I don’t understand why it is wrong.
That sounds really good to me.
Your new rule “… provided it’s legal to draw a tangent to a circle going through any given point on or outside the given circle”,
which is your biggest ingenuity here.
I’d said
“But now
—–>given also the one circle and allowing points on it from intersections with lines as well, it’s claimed it is.
I’m late for dentist if I don’t rush, maybe just trying to avoid embarrassment!!
—>Looks unlikely without some additional ‘rules’,
but I’m old and dumb.”
It never occurred to me about tangents! That’s the additional rule, unless someone can come up with a solution without that (and I tried to ignore what the dentist was doing to me by thinking about this problem!)
I’ll bet lots of people would find it much easier to solve if the rules had been spelt out explicitly, since they are sort of also hints.
It could be difficult to prove, even if true, that NOT allowing lines which are tangents to a circle given to start, that this asked for construction is impossible. But maybe the theory already exists (in some obscure monastery??) of given points, lines, circles, and only allowing lines joining points given or constructed as intersections and not allowing tangents. Probably only after Descartes so you can reduce it to numbers, the ones which are coordinates.
Well, I’m wrong again in just above last paragraph conjecture, since the rather complicated but damn ingenious solution in the Guardian looks fine. But it uses some nice old Greek facts about triangles inside a circle having a right angle, and about the common intersection of the three perpendicular drops, so is really quite complicated.
And Jeremy’s solution certainly does satisfy the way the problem was posed:
“Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.”.
And his is vastly simpler than the Guardian’s !
You really should be given exactly what the rules allow you to do.
e.g. using gravity isn’t ‘really’ geometry, but it also certainly isn’t a measuring device as far as I’m concerned!
Sorry, me again. But messing with the Guardian solution, it strikes me that they seem to have a defect in that they appear themselves to also NEED AN ADDITIONAL RULE. Their solution depends in #2 on choosing ‘randomly’ another point on the circle to draw the second right angled triangle. Maybe there’s a way around this, but otherwise, just like the extra ‘drawing tangents’ rule of Jeremy, they have another rule of ‘drawing that random line’.
They of course could have said that given is the circle, plus a diameter, plus two points on the upper hemicircle, to draw a perp down from one of them.
Jeremy’s solution, to repeat, is way simpler, IMHO!
As I said, tell us the damn rules, not just some vague and insufficient “no measuring devices”. Say it’s pure geometry, implying no plumb bobs and gravity. Just say what rules you are allowed in a perfectly precise way.
Still, both are clever. I prefer Jeremy’s, though maybe once you said tangents are allowed, you’d make it too much simpler. But saying instead you can draw some random line seems further from the classical straightedge and compass rule specification—which leads to standard add, multiply, divide, subtract and the non-standard ‘take square roots’ for a description of all numbers which are coordinates of all constructible points.
I haven’t read the above posts, but here are two ‘cheaty’ ways. I say ‘cheaty’ because while the problem calls for an ‘unmarked’ ruler, these use the geometry of the ruler itself as a measure.
1. More cheaty way: lay the ruler down with the short edge on the diameter line. Draw a straight line along the long edge from the diameter line to the circumference. If your ruler has a squared-off tip (which it almost certainly does), this line will be at right angle to the diameter.
2. Less cheaty, actual geometry way, but still uses the ruler itself:
2a. Lay the ruler down with one long edge on the diameter line (i.e. running parallel to it). Draw a straight line on the other edge from one side of the circumference to the other. If your ruler has parallel sides (which it almost certainly does), this line will be parallel to the diameter
2b. Repeat above process using your new line until you have a set of parallel lines crossing your circle both above and below the original diameter line.
2c. Count the same number of lines ‘away’ from the diameter on top and bottom. Put your ruler on the left-hand spots where they cross the circumference, and draw your straight up and down line. This will be perpendicular to the diameter line.
Right, I think either of these work. Your #1 is what I was trying to say in the entry 4. above.
Actually now that I think about it, you can use the second method *even if* the ruler doesn’t have parallel long *or* parallel short sides, so long as both long sides are straight lines and the ruler is big enough so you can draw across the circle (but not too big such that you can’t draw a chord by putting one tip down on the edge of the circle). In that case, you have to place one point of the ruler at the point where the diameter and circumference line meets. Draw your first line, then ‘flip’ the ruler to the other side of the diameter, then draw another line. The perpendicular drawn between the two left-hand (or right-hand) points made by your new lines and the circumference should be at right angles to the diameter. So you should be able to use my second method with any quadilateral ruler…also any triangular ruler (with the ‘big enough’ caveat)
Actually, I would regard way number 1 as the geometry way. Unfortunately it must be (ahem) ruled out because the straight edge is considered to be a single infinitely long edge. In any case, a right angle on your ruler would be a measuring device.
Way number 2 – if I understand it correctly – is not really a geometry way. It’s more a numerical method in that your perpendicular will only be an approximation unless you choose a ruler whose width divides the distance from the point to the diameter exactly (the perpendicular has to pass through the point). Even then, you’re using the width of your ruler as a measuring device.
You’re using the constancy and two-dimensionality of the real-life ruler to help you solve the problem. But as my follow-on message points out, any quadrilateral can be used as the ruler; it doesn’t require any of the sides of the ruler be parallel or the same length. You are using the ‘flip’ operation to create a geometrical relationship where there wasn’t one before.
But yes I agree that the problem is supposed to be worked out assuming a one-dimensional ‘ruler’ that gives you nothing except the ability to draw straight lines. My method would work in practical real world space, but not math-only space. 🙂
Even if you don’t have parallel edges and you flip the rule, you are still using it as a measuring device, so even in the real world, your solution breaks the rules.
My thought was similar until your step 2c. Rather than count the number of lines. Use the unmarked ruler to draw a straight line (chord) from the where the diameter intersects the circle on the left with the red dot. Now repeat the process of drawing parallel lines with this line like you did for the diameter. There will be at least 2 or more intersections formed with the parallels of the diameter and the parallels of the chord. Those points can be used to draw the perpendicular line.
#2. Well done.
Can you draw a line tangent to a point on the circle? If so, it’s just a few steps.
All I will say is, you have to think outside the circle. (I’ve seen the solution on youtube.)
The origami way (no ruler needed): fold the chord line onto itself so that the fold line passes through the red dot and unfold. Then darken the fold line with your pencil.
(This is axiom 4 of the Huzita-Justin origami axioms; see https://langorigami.com/article/huzita-justin-axioms/.)
A bit hard to accomplish on a chalkboard (or computer screen), of course.
The solution is clever. I looked it up.
Since it says we’re not allowed measuring devices, I’d do what carpenters and artists do – plumb a line using a piece of string with a weight at one end and a tack at the other. Then fold the string in half and you have the point of intersection on the diameter. Then draw in your vertical line. But maybe that is not allowed.
Gravitational geometry!
What will these carpenters and artists do once they are living on tiny asteroids with no gravity?
Let the red point be P and the left and right points of the diameter A and B
Extend AB on both sides
Choose a point C on the top left half of the circle
Draw a tangent through C till it intersects the line AB on the left at point O
Draw a tangent from O to the bottom left of the circle, let this point be D
Draw CD
Draw a line from C thru P all the way till it touches AB at point E
Draw a line from E to D to and label the intersection of this line and the circle as Q
Now PQ is the line you need
Thusly:
https://imgur.com/a/JdFPg6n
https://imgur.com/a/JdFPg6n
I mentally got through the first two steps anyway, which would have gotten me a right angle line to the center of the diameter…I think.