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.

See Hjelmslev’s Theorem

Classic Geometry Paradox

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.

See the Classic Geometry Paradox


Additional Page Problem

This is a clever puzzle from the 1986 AIME problems.

“The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?”


See the Additional Page Problem for the solution.

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.

Learning Mathematics

In one of our periodic FaceTime calls I found out that my granddaughter in 6th grade was interested in learning algebra and had gotten a book to help her out.  Clearly this initiative to get a head start prior to the normal course curriculum excited me, so I wrote what I thought was an insightful essay on the meaning and purpose of algebra.  Needless to say it was an abysmal failure.

That got me to thinking deeply about what it meant to learn mathematics and in particular symbolic algebra.

See Learning Mathematics

Weight of Potatoes

The following is another puzzle from the Irishman Owen O’Shea.

“Suppose you buy 100 pounds of potatoes and you are told that 99 percent of the potatoes consist of water.

You bring the potatoes home and leave them outside to dehydrate until the amount of water in the potates is 98 percent.  What is the weight of the potatoes now?”

This problem takes a little concentration to get right and the solution is a bit surprising at first.


See Weight of Potatoes for solutions.

Locating the Loot

This is a straight-forward problem from Geoffrey Mott-Smith in 1954.

“A brown Terraplane car whizzed past the State Police booth, going 80 miles per hour. The trooper on duty phoned an alert to other stations on the road, then set out on his motorcycle in pursuit. He had gone only a short distance when the brown Terraplane hurtled past him, go­ing in the opposite direction. The car was later caught by a road block, and its occupants proved to be a gang of thieves who had just robbed a jewelry store.

Witnesses testified that the thieves had put their plunder in the car when they fled the scene of the crime. But it was no longer in the car when it was caught. Reports on the wild ride showed that the only time the car could have stopped was in doubling back past the State Police booth.

The trooper reported that the point at which the car passed him on its return was just 2 miles from his booth, and that it reached him just 7 minutes after it had first passed his booth. On both occasions it was apparently making its top speed of 80 miles per hour.

The investigators assumed that the car had made a stop and turned around while some members of the gang cached the loot by the roadside, or perhaps at the office of a “fence.” In an effort to locate the cache, they assumed that the car had maintained a uniform speed, and allowed 2 minutes as the probable loss of time in bringing the car to a halt, turning it, and regaining full speed.

On this assumption, what was the farthest point from the booth that would have to be covered by the search for the loot?”


See Locating the Loot for solutions.

Pythagorean Theorem Converse

One of the joys of getting old is that you forget things.   So one of the things I recall is that the converse of the Pythagorean Theorem is true, that is, if a triangle with short sides a and b and long side c is such that

a2 + b2 = c2,

then the triangle must be a right triangle with the angle between sides a and b being 90°.  But I didn’t recall how to prove it.  So I thought I would see if I could do it without looking up any sources.

See the Pythagorean Theorem Converse

Mystery Dice Question

This is a relatively simple probability question from Presh Talwalkar that becomes an excuse to describe a powerful tool.

“Amazon’s Mystery Dice Interview Question

You are given a normal die and a blank die. (Each die is six-sided and equally likely to show each face). Label the blank die using the numbers 0 to 6 so that when you roll the two die the sum shows each whole number from 1 to 12 with equal chance. You can use a number more than once, or not at all, so you could label the faces 1, 2, 3, 4, 4, 5. But you do have to label all six faces of the blank die.”

See the Mystery Dice Question

Spy Gift Problem

This is a recent Alex Bellos problem that supposedly can be solved by 12-year-olds!

“Today’s problems come from Axiom Maths, a charity that that takes high-attaining primary school children and provides them with maths enrichment during secondary school.

One of Axiom’s main activities is to organise ‘maths circles’, in which small groups of pupils get together to tackle fun problems. Such as the ones below, which are aimed at children aged 11/12, and form the basis for further explorations.

Really Secret Santa

A group of nine secret agents: 001, 002, 003, 004, 005, 006, 007, 008 and 009 have organised a Secret Santa. The instructions are coded, to keep the donors secret.

  • Agent 001 gives a present to the agent who gives a present to agent 002
  • Agent 002 gives a present to the agent who gives a present to agent 003
  • Agent 003 gives a present to the agent who gives a present to agent 004
  • and so on, until
  • Agent 009 gives a present to the agent who gives a present to agent 001

Which agent will agent 007 get her present from?”


See the Spy Gift Problem for solutions.