# Hjelmslev’s Theorem

I came across this remarkable result in Futility Closet:

“On each of these two black lines is a trio of red points marked by the same distances.  The midpoints of segments drawn between corresponding points are collinear.

(Discovered by Danish mathematician Johannes Hjelmslev.)”

This result seems amazing and mysterious.  I wondered if I could think of a proof.  I found a simple approach that did not use plane geometry.  And suddenly, like a magic trick exposed, the result seemed obvious.

Coming across this classic geometric paradox recently in Futility Closet motivated me to write down its solution in detail.

“Where did the empty square come from?”

In any case, this is the canonical example for why I avoid visual geometric proofs—you can be so easily fooled.  Real proofs require plane or analytic geometry arguments.

# Box Code Puzzle

This is an intriguing puzzle from Futility Closet.

“In Robert Chambers’ 1906 novel The Tracer of Lost Persons, Mr. Keen copies the figure below from a mysterious photograph. He is trying to help Captain Harren find a young woman with whom he has become obsessed.

‘It’s the strangest cipher I ever encountered,’ he says at length. ‘The strangest I ever heard of. I have seen hundreds of ciphers—hundreds—secret codes of the State Department, secret military codes, elaborate Oriental ciphers, symbols used in commercial transactions, symbols used by criminals and every species of malefactor. And every one of them can be solved with time and patience and a little knowledge of the subject. But this … this is too simple.’

The message reveals the name of the young woman whom Captain Harren has been seeking. What is it?”

As is usual with these types of puzzles, I felt foolish that I couldn’t see the immediate, simple interpretation of the boxes—after a great deal of effort.  So I solved it using the usual cryptographic methods that rely heavily on logic and letter frequencies, though the message is a bit short for that.

See Box Code Puzzle for solutions.

# “Fermat’s Last Theorem” Puzzle

Here is a mind-numbing logic puzzle from Futility Closet.

“A puzzle by H.A. Thurston, from the April 1947 issue of Eureka, the journal of recreational mathematics published at Cambridge University:

Five people make the following statements:—

Which of these statements are true and which false?  It will be found on trial that there is only one possibility.  Thus, prove or disprove Fermat’s last theorem.”

Normally I would forgo something this complicated, but I thought I would give it a try.  I was surprised that I was able to solve it, though it took some tedious work.  (Hint: truth tables.  See the “Pointing Fingers” post regarding truth tables.)

One important note.  The author is a bit cavalier about the use of “Either …, or …”.  In common parlance this means “either P is true or Q is true, but not both” (exclusive “or”: XOR), whereas in logic “or” means “either P is true or Q is true, or possibly both” (inclusive “or”: OR).  I assumed all “Either …, or …” and “or” expressions were the logical inclusive “or”, which turned out to be the case.

See the Fermat’s Last Theorem Puzzle

# Sizing Up

This is another fairly simple puzzle from Futility Closet from a while ago (2014).

“Two lines divide this equilateral triangle into four sections. The shaded sections have the same area. What is the measure of the obtuse angle between the lines?”

See Sizing Up for solutions.

# Existence Proofs II

Futility Closet has another example of an existence proof like their previous one taken from Peter Winkler’s 2021 Mathematical Puzzles (see my post “Existence Proofs”):

“Four bugs live on the four vertices of a regular tetrahedron. One day each bug decides to go for a little walk on the tetrahedron’s surface. After the walk, two of the bugs have returned to their homes, but the other two find that they have switched vertices. Prove that there was some moment when all four bugs lay on the same plane.”

# Existence Proofs

Here is a seemingly simple problem from Futility Closet.

“A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?”

Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.

But the real significance of the problem is that it is an example of an “existence proof”, which in mathematics refers to a proof that asserts the existence of a solution to a problem, but does not (or cannot) produce the solution itself.  These proofs are second in delight only to the “impossible proofs” which prove that something is impossible, such as trisecting an angle solely with ruler and compass.

Here is another classic example (whose origin I don’t recall).  Consider the temperatures of the earth around the equator.  At any given instant of time there must be at least two antipodal points that have the same temperature.  (Antipodal points are the opposite ends of a diameter through the center of the earth.)

See Existence Proofs (revised)

(Update 10/2/2021) I fixed a minor typo: “tail” should have been “head”

# Bottema’s Theorem

This seemingly magical result from Futility Closet defies proof at first.  Go to the Wolfram demo by Jay Warendorff and then …

“Grab point B above and drag it to a new location. Surprisingly, M, the midpoint of RS, doesn’t move.

This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.

It was discovered by Dutch mathematician Oene Bottema.”

As we shall see, Bottema’s Theorem has shown up in other guises as well.