# Moving Constants

If you mark two points on a circle, A and B, and a third point T, then angle ATB remains constant as T moves along the segment between A and B. (If you mark a point S in the circle’s other segment then you get another constant angle, ASB, and ASB = 180 – ATB.)

If two circles intersect at A and B and we move T as before along the segment opposite the second circle, and we extend TA and TB to P and Q on the second circle, then the length of chord PQ remains constant as T moves.

(From David Wells, The Penguin Dictionary of Curious and Interesting Geometry, 1992.)

# The Falling Chain

Here are two identical rope ladders with slanting rungs. One falls to the floor, the other onto a table. The ladders are released at the same time and fall freely, but the one on the left falls faster, as if the table is “sucking” it downward. Why does this happen?

# The Hollowood Function

California high school student Derek Hollowood created this function after considering recurrence relations:

$h(x)=\frac{10^{x+1}-9x-10}{81}$
h(-10) = 0.987654321
h(-9)  =  0.87654321
h(-8)  =   0.7654321
h(-7)  =    0.654321
h(-6)  =     0.54321
h(-5)  =      0.4321
h(-4)  =       0.321
h(-3)  =        0.21
h(-2)  =         0.1
h(-1)  =           0
h(0)   =           0
h(1)   =           1
h(2)   =           12
h(3)   =           123
h(4)   =           1234
h(5)   =           12345
h(6)   =           123456
h(7)   =           1234567
h(8)   =           12345678
h(9)   =           123456789


(Thanks to Chris Smith for the tip.)

# 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

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!”

# A Reflexive Rainbow

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:

“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.

# Fitting

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

(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.”