Going Places

Square wheels work fine if the road accommodates them — in this case, the road must be a series of catenaries suited to the size of the square. (A catenary is the shape that a cable assumes when suspended by its ends.)

Macalester College mathematician Stan Wagon designed a square-wheeled tricycle in 2004, and physics students at Texas A&M built a companion in 2007 (below).

Magic Space

knecht most-perfect square

Craig Knecht, whose “terraformed” magic squares we explored in 2013, has begun to experiment with applying “magic” properties to David Hilbert’s space-filling curve.

The Hilbert curve finds its way to every cell in the square above by following the pattern shown at the lower left. Knecht divided that path into eight-cell segments, as shown in (a), and then sought solutions in which each colored eight-cell panel produced the magic sum of 260 while each of the eight ordinal positions across the eight panels did so as well. For example, in (b), a large red digit 1 marks the “first” position in each panel; the hope was to find values for these eight cells that would sum to 260, and likewise for all the “second” cells, the “third” ones, and so on.

The result, shown in (c), is a “most-perfect” magic square: Each colored panel sums to 260, and every set of cells that are 8 spaces apart on the Hilbert curve also sum to 260.

The next step was to apply this idea in three dimensions, and recently Knecht made the breakthrough shown below — a 4×4×4 “most perfect” number cube. The 64 numbers in the cube can be broken into 36 2×2 subsquares in each dimension, as shown. In all 108 of these subsquares, the four constituent numbers total 130. And as with the two-dimensional square above, a Hilbert curve can be drawn through the cube that visits each cell once, and cells that are eight cells apart on this curve sum to 260.

One of Knecht’s correspondents pointed out that the cube is even magicker than he had supposed: The “wraparound” subsquares (for example, 5, 28, 44, and 53 on the top of the cube) also sum to 130, as does each set of four corners, making a total of 192 2×2 subsquares that sum to 130.

“So in summary … making the Hilbert space-filling curve path have this magic property of values 8 spaces summing to the magic constant + this 2×2 planar criteria produces a very interesting cube!”

knecht most-perfect cube

A Reflexive Rainbow

sallows color table

From Lee Sallows: The international color code is used to mark the values of electronic components such as resistors. It assigns a distinct color to each of the 10 decimal digits, as seen in the center column of the table at right: 0 = BLACK, 1 = BROWN, …, 9 = WHITE.

Lee’s table has an ingenious reflexive property. The letters in the left-hand column are associated with the values -1 to -9, and those in the right-hand column with the values 1 to 9.

Now spelling the name of each color produces a sum that matches the number represented by that color:

sallows color sums

“This is more remarkable that it may seem,” Lee writes, “because the numbers assigned to the letters are now restricted to single-digit values only.”

Lanchester’s Laws


In 1916 English engineer Frederick Lanchester set out to find a mathematical model to describe conflicts between two armies. In ancient times, he reasoned, each soldier engaged with one enemy at a time, so the number of soldiers who survived a battle was simply the difference in size between the two armies. But the advent of modern combat, including long-range weapons such as firearms, changes things. Suppose two armies, A and B, are fighting. A and B represent the number of soldiers in each army, and a and b represent the number of enemy fighters that each soldier can kill per unit time. Now the equations

dA/dt = -bB
dB/dt = -aA

show us the rate at which the size of each army is changing at a given instant. And these give us

bB2aA2 = C,

where C is a constant.

This is immediately revealing. It shows that the strength of an army depends more on its bare size than on the sophistication of its weapons. In order to meet an army twice your size you’d need weapons (or fighting skills) that are four times as effective.

Simple as they are, these ideas shed light on the historic choices of leaders such as Nelson, who sought to divide his enemies into small groups, and Lanchester himself illustrated his point by referring to the British and German navies then at war. Today his ideas (and their descendants) inform the rules behind tabletop and computer wargames.



Asteroid 46610 was named Bésixdouze in homage to Antoine de Saint-Exupéry’s character Le Petit Prince, who lived on Asteroid B-612.

B-612 is 46610 in hexadecimal.

(Thanks, Dan.)

The German Tank Problem


During World War II, as they mulled whether to attempt an invasion of the continent, the Allies needed to estimate the number of tanks Germany was producing. They asked their intelligence services to guess the number by spying on German factories and counting tanks on the battlefield, but these efforts produced contradictory estimates. Finally they resorted to statistical analysis.

They did this by studying the serial numbers on captured and destroyed German tanks. Suppose German tanks are numbered sequentially 1, 2, 3, …, B, where B is the total number of tanks that we seek to know. And suppose that we have five captured tanks whose serial numbers are 21, 35, 42, 60, and 89. It turns out that

\displaystyle B = \frac{(N+1)M}{N} - 1,

where N is the sample size (here, 5) and M is the highest sampled number (here, 89). In this example, the formula tells us that B = 105.8, so we’d estimate that 106 tanks had been produced at that time.

In the event, Allied statisticians reportedly estimated that the Germans had produced 246 tanks per month between June 1940 and September 1942. Intelligence estimates had put the total at about 1,400. When the Allies captured German production records after the war, they found that they had produced 245 tanks per month during those three years, almost precisely what the statisticians had predicted, and less than 20 percent of the intelligence estimate.

(Thanks, Ryan.)

Living Large


In his 1984 book Scaling, Duke University physiologist Knut Schmidt-Nielsen points out a pleasing coincidence:

A 30-gram mouse that breathes at a rate of 150 times per minute will breathe about 200 million times during its 3-year life; a 5-ton elephant that breathes at the rate of 6 times per minute will take approximately the same number of breaths during its 40-year lifespan. The heart of the mouse, ticking away at 600 beats per minute, will give the mouse some 800 million heartbeats in its lifetime. The elephant, with its heart beating 30 times per minute, is awarded the same number of heartbeats during its life.

In fact, most mammals have roughly the same number of heartbeats per lifetime, about 109. Small mammals have high metabolic rates and short lives; large ones have low rates and long lives. Humans are lucky: “We live several times as long as our body size suggests we should.”

Current Affairs


Is it possible to sail on a river on a windless day? In Why Cats Land on Their Feet (2012), Mark Levi points out that the answer is yes, at least in principle. If the keel is turned broadly against the current, then this will carry the boat downstream, drawing the sail through the still air. Now the roles of the sail and the keel are reversed: The keel catches the motion of the river, acting as a sail, and the boat follows the course established by the sail, which acts as a keel. “It’s just like regular sailing,” Levi writes, “except upside down.”

That’s from the point of view of an observer on shore. In the boat’s reference frame, the water is still and a wind is blowing upstream. From this perspective the boat is sailing conventionally — the sail is catching the wind and the keel slices through the water.

“This is a neat symmetry,” Levi notes. “The sail and the keel exchange roles, depending on your reference frame! So the sail and the keel have completely equal rights in that respect.”

Braess’ Paradox

Image: Wikimedia Commons

The diagram above represents a road network where T is the number of travelers. The leg from START to A takes T/100 = t minutes to traverse, as does the leg from B to END. The legs from START to B and from A to END each take a constant 45 minutes.

Now suppose that 4000 drivers want to travel from START to END. The northern and the southern routes are equally efficient, so the drivers will split into two groups, and each will arrive at END in 2000/100 + 45 = 65 minutes.

But now suppose that planners, hoping to improve matters, add a shortcut between A and B with a travel time of 0 minutes. Now all the drivers will take the route from START to A, since in the worst case it will take 4000/100 = 40 minutes, rather than the guaranteed 45 minutes taken by the leg from START to B. From A every driver will take the shortcut to B, for the same reason: Even in the worst case, the trip from B to END is 5 minutes faster than the trip from A to END.

As a result, every driver’s trip now takes 4000/100 + 4000/100 = 80 minutes, which is 15 minutes longer than in the original state of affairs. No individual driver has an incentive to change his behavior, since now the two original routes (northern and southern) each take 4000/100 + 45 = 85 minutes. If the 4000 drivers as a body could agree never to use the shortcut, they’d all be better off. But without a way to enforce this, all are stuck with longer commutes.

The principle was discovered by German mathematician Dietrich Braess in 1968. It’s known as Braess’ paradox.

Math Notes

The American Mathematical Monthly of January 1959 notes an “interesting Pythagorean triangle” discovered by Victor Thébault: If the two perpendicular sides of a right triangle measure 88209 and 90288, then the hypotenuse is 126225.

In other words, if you sum the squares of 88209 and its reverse, the result is a perfect square.