What’s unusual about this magic square?
It works just as well upside down:
From Royal V. Heath, Scripta Mathematica, March-June 1951. See Topsy Turvy.
Science & Math
Small Business
To help interest young students in chemistry, James Tour of Rice University devised “NanoPutians,” organic molecules that take the form of stick figures. The body is a series of carbon atoms that join two benzene rings; the arms and legs are acetylene units, each terminating in an alkyl group; and the head is a 1,3-dioxolane ring.
This gets even better — by using microwave irradiation, Tour found a way to vary the heads, creating a range of NanoProfessionals:
The synthesis is detailed on the Wikipedia page.
Lights Out
I pass on to Eclipses. When the Moon (see above) gets between the Earth (see below) and the Sun (do what you like), the resulting phenomenon is called an Eclipse of the Sun. When the Sun gets between the Earth and the Moon there will be the devil to pay. It will be called the Eclipse of the Earth and is likely to be total.
— H.F. Ellis, So This Is Science!, 1932
Equal Opportunity
Can two dice be weighted so that the probability of each of the numbers 2, 3, …, 12 is the same?
Cause for Alarm
Sherlock Holmes is walking through the valley of Reichenbach Fall. On a clifftop overhead, Moriarty has perched a boulder. When he pushes it, it will have a 90 percent chance of killing Holmes. Just as he is about to send it over the edge, Watson arrives at the clifftop. Watson can’t see Holmes, so he’s not able to push the boulder safely clear, but he reasons that it’s better to push the boulder in a random direction than to let Moriarty aim it carefully. So he pushes the boulder off the cliff in such a way that Holmes’ chance of dying is reduced to 10 percent.
Unfortunately the boulder crushes Holmes anyway. Watson’s push decreased the chance of Holmes’ death, but it also caused it.
What are we to make of this? Generally speaking, it seems true to say that Pre-emptive pushing prevents death by crushing. That is, Watson’s push was of the sort that made it less likely that Holmes would die — if the scenario were re-enacted many times, with the boulder pushed sometimes by Watson, sometimes by Moriarty, Watson-type pushes would result in fewer deaths. But it also seems true to say that Watson’s pushing the rock caused Holmes to die. But cause and prevent are antonyms. How can both of these statements be true?
(Christopher Read Hitchcock, “The Mishap at Reichenbach Fall: Singular vs. General Causation,” Philosophical Studies, June 1995.)
Half and Half
A bisecting arc is one that bisects the area of a given region. “What is the shortest bisecting arc of a circle?” Murray Klamkin asked D.J. Newman. Newman supposed that it was a diameter. “What is the shortest bisecting arc of a square?” Newman answered that it was an altitude through the center. Finally Klamkin asked, “And what is the shortest bisecting arc of an equilateral triangle?”
“By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector,” Klamkin writes. “But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer.”
Was he right?
Proizvolov’s Identity
List the first 2N positive integers (here let N = 4):
1, 2, 3, 4, 5, 6, 7, 8
Divide them arbitrarily into two groups of N numbers:
1, 4, 6, 7
2, 3, 5, 8
Arrange one group in ascending order, the other in descending order:
1, 4, 6, 7
8, 5, 3, 2
Now the sum of the absolute differences of these pairs will always equal N2:
| 1 – 8 | + | 4 – 5 | + | 6 – 3 | + | 7 – 2 | = 16 = N2
(Presented by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads.)
Self-Replicating Resistors
From Lee Sallows:
In an electrical network, if resistors x and y are placed in series their total resistance is x + y; if they’re placed in parallel it’s 1/(1/x + 1/y).
This offers an intriguing opportunity for self-reference. Each of the networks above contains four resistors with values 1, 2, 3, and 4, and the total resistances of the networks themselves are 1, 2, 3, and 4. So any one of the numbered resistors in these networks can be replaced by one of the networks themselves.
The challenge was posed by Sallows and Stan Wagon as a Macalester College “problem of the week”; these examples were discovered by Brian Trial, an automotive electronics engineer from Ferndale, Mich. Sallows points out that any such solution has a dual that results from changing series connections to parallel, and vice versa, and then replacing all resistors values by their reciprocals.
This leads to a further idea: The two sets of resistors below are “co-replicating” — the four networks on the left can be used to replace the four resistors in any of the networks on the right, and vice versa.
(Thanks, Lee.)
Stage Business
The goal of the Shakespeare programming language is to create code that reads like a Shakespearean play: Variables are “characters” that interact through dialogue, constants are represented by nouns and adjectives, and if/then statements are phrased as questions. (Insulting Macbeth assigns him a negative value.) Act and scene numbers serve as GOTO labels, and characters can tell one another to “remember” or “recall” values. The phrases “Open your heart” and “Speak your mind” output a variable’s numerical value and the corresponding ASCII character, respectively.
This program prints the phrase HELLO WORLD:
Romeo, a young man with a remarkable patience.
Juliet, a likewise young woman of remarkable grace.
Ophelia, a remarkable woman much in dispute with Hamlet.
Hamlet, the flatterer of Andersen Insulting A/S.
Act I: Hamlet's insults and flattery.
Scene I: The insulting of Romeo.
[Enter Hamlet and Romeo]
Hamlet:
You lying stupid fatherless big smelly half-witted coward! You are as
stupid as the difference between a handsome rich brave hero and thyself!
Speak your mind!
You are as brave as the sum of your fat little stuffed misused dusty
old rotten codpiece and a beautiful fair warm peaceful sunny summer's
day. You are as healthy as the difference between the sum of the
sweetest reddest rose and my father and yourself! Speak your mind!
You are as cowardly as the sum of yourself and the difference
between a big mighty proud kingdom and a horse. Speak your mind.
Speak your mind!
[Exit Romeo]
Scene II: The praising of Juliet.
[Enter Juliet]
Hamlet:
Thou art as sweet as the sum of the sum of Romeo and his horse and his
black cat! Speak thy mind!
[Exit Juliet]
Scene III: The praising of Ophelia.
[Enter Ophelia]
Hamlet:
Thou art as lovely as the product of a large rural town and my amazing
bottomless embroidered purse. Speak thy mind!
Thou art as loving as the product of the bluest clearest sweetest sky
and the sum of a squirrel and a white horse. Thou art as beautiful as
the difference between Juliet and thyself. Speak thy mind!
[Exeunt Ophelia and Hamlet]
Act II: Behind Hamlet's back.
Scene I: Romeo and Juliet's conversation.
[Enter Romeo and Juliet]
Romeo:
Speak your mind. You are as worried as the sum of yourself and the
difference between my small smooth hamster and my nose. Speak your
mind!
Juliet:
Speak YOUR mind! You are as bad as Hamlet! You are as small as the
difference between the square of the difference between my little pony
and your big hairy hound and the cube of your sorry little
codpiece. Speak your mind!
[Exit Romeo]
Scene II: Juliet and Ophelia's conversation.
[Enter Ophelia]
Juliet:
Thou art as good as the quotient between Romeo and the sum of a small
furry animal and a leech. Speak your mind!
Ophelia:
Thou art as disgusting as the quotient between Romeo and twice the
difference between a mistletoe and an oozing infected blister! Speak
your mind!
[Exeunt]
Because it’s written as a play, a program can be performed by human actors, but the drama lacks a certain narrative drive:
See Output.
Quarters
From Lancelot Hogben’s Mathematics in the Making, an appealingly memorizable table of basic trigonometric values:
See Alison’s Triangle.
(Thanks, Tom.)