# Counting Tanks

A great example of the application of simple math to real-world problems is provided in a recent Numberphile video on YouTube by the British mathematician James Grime.

It is taken from a real story from World War II where the British were trying to estimate the number of tanks the Germans were producing each month.  The spies came up with an estimate of about 1500 tanks per month, whereas the mathematicians estimated the number to be closer to 250 tanks per month.  How the mathematicians did this is explained by Grime.  (I added an explanation of one step that whizzed by in Grime’s presentation and that I didn’t understand at first.)

See Counting Tanks.

# Parallel Lines Problem

This is an interesting problem from the collection Five Hundred Mathematical Challenges.

“Problem 251.  Let ABCD be a square, F be the midpoint of DC, and E be any point on AB such that AE > EB.  Determine N on BC such that DE || FN.  Prove that EN is tangent to the inscribed circle of the square.”

See the Parallel Lines Problem.

# Aunt Hillary and the Anteater

In this moment when the collective actions of humans seem to be hurtling towards several cataclysms (burning up the planet, ending the American Experiment), I am reminded of a powerful image that invaded my psyche some 45 years ago.  It was from Douglas Hofstadter’s magnum opus, Gödel, Escher, Bach (1979) and concerned his investigation of what became popularized as “emergent behavior” and “self-organization.”  This was in the early days of chaos theory and Holland’s emerging complexity theory.  Conway’s artificial life cellular automaton, the Game of Life, was the screen saver on countless computer terminals and burgeoning personal computers.  It was also the time when neural nets were beginning to capture the imagination of machine learning researchers among the artificial intelligence community.

Hofstadter’s aim was to explore these ideas as they related to understanding the brain and he used the vehicle of an ant colony.

Also see the excerpt Ant Fugue (2.5 MB) and the Atlantic article “What the Microsoft Outage Reveals”.

# Danica McKellar Interview

I recently watched a remarkable and surprising interview on Numberphile of the actress Danica McKellar.  She began as a child actress and recently has played romantic roles on the Hallmark series of TV movies.  So it comes as a surprise to find she was an undergraduate math major, and with a female colleague and the help of their advisor one summer proved a theorem involving math and metallurgy.

She was terrifically articulate and enthusiastic during the interview and made a number of telling observations that would be helpful for any student, especially girls, who might find they like math in grade school and might even consider majoring in it in college.

Her big emphasis was on self-esteem and the importance of reaching girls to support any interest they might have in math against the negative forces they often experience.  To this end she has written a series of math books, each tailored to a different grade-level of students.  But she feels the middle school years are especially critical to support girls in math.

Perhaps the thing I liked best in the interview (maybe because it echoed my own views) was her attitude towards math—she loved solving puzzles.  In fact, just doing math was its own reward.  She politely deflected all efforts by Brady, the interviewer, to ask if she regretted not pursuing a professional career more devoted to math.  Not to knock Brady’s fine interview but I found this line of questioning supporting the current utilitarian view of education (What good is it, how can it be of use to help me earn a living) to be a bit condescending.  And there was even a whiff of what the most important thing to do with math and what the professors are training students for is math research.   Since most math majors and even math Ph.D.s are not going to find employment in academia, math as an ancillary component to one’s career is the most one could imagine in this context.  But above and beyond that is the real truth: math is fun for its own sake, and Danica McKellar championed that in spades.  I don’t think I have seen anyone unabashedly extol the joys of doing math as well as she did.

I heartily recommend this interview to anyone, especially young girls, who might have a secret affection for mathematics but fear some stigma that might attach to them if they admitted it.

# Horses to Qi

This is a challenging problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website.

“Now a good horse and an inferior horse set out from Chang’an to Qi.  Qi is 3000 li from Chang’an.  The good horse travels 193 li on the first day and daily increases by 13 li; the inferior horse travels 97 li on the first day and daily decreases by ½ li.  The good horse reaches Qi first and turns back to meet the inferior horse.  Tell: how many days until they meet and how far has each traveled?”

The solution involves common fractions, which the Chinese were already adept at using by 100 BC.

See Horses to Qi for a solution.

# Making Arrows

This is an interesting problem from 180 BC China.

“In one day, a person can make 30 arrows or fletch [put the feathers on] 20 arrows.  How many arrows can this person both make and fletch in a day?”

It turns out the solution to this problem led me into the history of numerator/denominator (aka common) fractions, a subject I had been finding difficult to track down.

See Making Arrows for a solution.

# 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.

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.