by Simon Singh
Homer’s mathematical brain receives another temporary boost in “$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)” (1993). At the start of that episode, Henry Kissinger is (somewhat inexplicably) touring Homer’s workplace, the Springfield Nuclear Power Plant. Unfortunately, the former U.S. secretary of state drops his trademark spectacles into the toilet while visiting one of the power plant’s washrooms. Too timid to fish them out, and too embarrassed to tell anyone about his missing glasses, Kissinger then mutters to himself: “No one must know I dropped them in the toilet. Not I, the man who drafted the Paris Peace Accords.”
A short while later, Homer visits the same washroom and discovers the glasses in the toilet bowl. Of course, he cannot resist putting them on, whereupon the glasses seem to endow him with the powers of Kissinger’s brain. While still in the washroom, Homer even starts regurgitating a mathematical formula:
“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”
At first, this sounds like a straightforward proclamation of the Pythagorean theorem, but in fact it is wrong in several ways. The actual theorem states:
“The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.”
The most obvious difference is that Homer’s statement concerns isosceles triangles, whereas the Pythagorean theorem relates to right triangles. You may remember from school that an isosceles triangle has two equal sides, whereas a right triangle has no restriction on the lengths of its sides, as long as one corner is a right angle.
There are two more problems in Homer’s statement. First, he talks about the “square roots” of lengths, whereas the Pythagorean theorem relies on the “squares” of lengths. Second, the Pythagorean theorem relates the hypotenuse (the longest side) of the right triangle to the other two sides, whereas Homer relates “any two sides” of the isosceles triangle to “the remaining side.” “Any two sides” could be the two equal sides or just one of the equal sides and the unequal side.
The diagrams and equations below summarize and highlight the differences between Homer’s statement and the Pythagorean theorem. Homer has taken a standard piece of mathematics and given it a twist, thereby creating a modification of the Pythagorean theorem, namely Simpson’s conjecture. The difference between a theorem and a conjecture is that the former has been proven to be true, whereas the latter is neither proven nor disproven . . . yet.
Simpson’s conjecture concerns all isosceles triangles, so if we try to prove it then we should need to show that it holds true for an infinity of triangles. However, if instead we try to disprove Simpson’s conjecture, then we would need to find just one triangle that defies the conjecture. As disproving seems easier than proving, let us see if we can find a one counterexample that destroys the conjecture.
Let us consider an isosceles triangle with two sides of length 9 and a base of length 4. Does the sum of the square roots of any two sides of this isosceles triangle equal the square root of the remaining side?
√9 + √9 = √4 implies that 3 + 3 = 2, which is wrong
√9 + √4 = √9 implies that 3 + 2 = 3, which is also wrong
In both cases, the square roots simply do not add up, so the conjecture is clearly false.
This is not Homer’s finest hour, obviously, yet perhaps we should not judge him too harshly, particularly as he was under the influence of Kissinger’s spectacles. Indeed, if anybody is to blame, it must be the writers.
Josh Weinstein, who shared lead writer credit with Bill Oakley on the episode, told me how the scene had developed and why it contained such a nonsensical conjecture: “That joke developed backward, because we needed Mr. Burns, Homer’s boss, to think that Homer is smart. We thought, ‘So how is he going to think that Homer is smart? Oh, it would be funny if he found a pair of glasses in the toilet. Who would the glasses belong to? Oh, Henry Kissinger!’ We like Henry Kissinger (and Nixon-era stuff) and he seemed like somebody who would be friends with Mr. Burns.”
The script then needed a line in order for Homer to demonstrate his newly acquired confidence in his own intelligence. At this point, the writing team got to work, and one of the more mathematical writers realized that Homer’s situation had strong parallels with one of the final scenes in The Wizard of Oz (1939).16 As Dorothy follows the yellow brick road to Oz, she is accompanied by the Cowardly Lion, who is searching for courage, the Tin Man, who is searching for a heart, and the Scarecrow, who is searching for a brain. It is said that the Scarecrow represents a typical down-to-earth decent Kansas farmer, who would probably have had tremendous common sense, but would have lacked any formal education. When they eventually find the Wizard, he is unable to give the Scarecrow a brain, but he does reward him with a diploma, at which point the Scarecrow blurts out: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”
Thus, Homer was quoting a line originally delivered by the Scarecrow in The Wizard of Oz. The Simpson conjecture is really the Scarecrow conjecture. The writers of The Simpsons were using the same mathematical pseudo-conjecture, because Homer’s discovery of Kissinger’s glasses and the Scarecrow receiving his diploma had the same effect on the characters involved, inasmuch as afterward both Homer and the Scarecrow were much more confident about their intellectual ability.
Only a tiny fraction of viewers would have noticed that Homer was recycling the Scarecrow conjecture. These viewers could best be described as occupying the overlap in the Venn diagram that has obsessive fans of The Wizard of Oz in one set and mathematicians in the other set. This overlap includes James Yick, Anahita Rafiee, and Charles Beasley, students in the Department of Mathematics and Computer Science at Augusta State University in Georgia, who have scrutinized the original scene from The Wizard of Oz. In particular, they have challenged the theory that the Scarecrow was supposed to quote the Pythagorean theorem, and that the actor playing the Scarecrow, Ray Bolger, accidentally made an error that was not spotted until it was too late. Instead, these mathematicians have argued that the scriptwriters of The Wizard of Oz deliberately distorted the Pythagorean theorem. They state: “We feel it was an act of deliberate sabotage because of the speed at which the actor states his lines, suggesting a lot of practice, and the three obvious errors in the wording of the lines . . . Were [the writers] trying to make a point about their view of the real value of diplomas? Were they trying to make a statement about the lack of real knowledge in the population of viewers at large, implying that we are all ‘scarecrows’ as their little inside joke?”
Regardless of its origins and the motivations behind it, the Scarecrow conjecture is undoubtedly false, but it did inspire the trio of mathematicians at Augusta State to investigate the opposite of the Scarecrow conjecture, known as the crow conjecture, which states:
“The sum of the square roots of any two sides of an isosceles triangle is never equal to the square root of the remaining side.”
So, is Yick, Rafiee, and Beasley’s crow conjecture true? We can test it by checking the two equations. Starting with equation (1), we can restate it and then rearrange it slightly:
√a + √a
≠
√b
2√a
≠
√b
4a
≠
b
a
≠
¼b
This final equation states that it can never be true that the lengths a are only one-quarter of the base b. Indeed, this must be the case, because a must be bigger than ½b, otherwise the three sides of the triangle will not touch each other. A quick look at the triangle above should make this obvious.
Having demonstrated that equation (1) is valid, let’s check equation (2):
√a + √b
≠
√a
√b
≠
0
&nb
sp; b
≠
0
In other words, equation (2) states that the base of an isosceles triangle cannot have zero length. This is indeed true, otherwise we would have a triangle with only two sides! These sides would overlap, so arguably we would have a triangle with only one side!
Therefore, we can be sure that it is never possible for the sum of the square roots of any two sides of an isosceles triangle to be equal to the square root of the remaining side. It is not a deeply profound discovery, but the crow conjecture can now be elevated to the status of the crow theorem.
Simpson’s conjecture turned out to be nothing more than a restatement of the Scarecrow conjecture, which in any case turned out to be false. There is, however, some consolation for the Simpson family, as several important—and valid—concepts in mathematics bear their name.
For example, Simpson’s paradox is arguably one of the most baffling paradoxes in mathematics. It was popularized and investigated by Edward H. Simpson, who developed an interest in statistics while working at Bletchley Park, the secret British code-breaking headquarters during the Second World War.
One of the best illustrations of Simpson’s paradox concerns the American Civil Rights Act of 1964, a historic piece of legislation aimed at tackling discrimination. In particular, the paradox emerges if we scrutinize in detail the voting records of the Democrats and Republicans when the act came before the U.S. House of Representatives.
In the northern states, 94 percent of Democrats voted for the act, compared with only 85 percent of Republicans. Hence, in the north, a higher percentage of Democrats than Republicans voted for the act.
In the southern states, 7 percent of Democrats voted for the act, compared with 0 percent of Republicans. So, also in the south, a higher percentage of Democrats than Republicans voted for the act.
The obvious conclusion is that Democrats showed more support for the Civil Rights Act than Republicans. However, if the numbers are combined for both southern and northern states, then 80 percent of Republicans voted for the act compared with only 61 percent of Democrats.
In other words, I am stating that Democrats outvoted Republicans in the north and south separately in support of the act, but Republicans outvoted Democrats in the north and south combined! This sounds ludicrous, yet these facts are undeniable. This is Simpson’s paradox.
In order to make sense of the paradox, instead of dealing in percentages it will help to look at the actual voting numbers. From the northern states, those voting for the act consisted of 145 out of 154 Democrats (94 percent), alongside 138 out of 162 Republicans (85 percent). From the southern states, those for voting for the act consisted of 7 out of 94 Democrats (7 percent), alongside zero out of 10 Republicans (0 percent). As stated already, Democratic support for the act seems to be stronger than Republican support in both the northern and southern states. However, the trend reverses nationally, because 152 out of 248 Democrats (61 percent) voted for the Act, compared with 138 out of 172 Republicans (80 percent).
Northern Voting Record
Southern Voting Record
National Voting Record
Democrats
145/154
94%
7/94
7%
152/248
61%
Republicans
138/162
85%
0/10
0%
138/172
80%
So, how do we resolve this example of Simpson’s paradox? There are four points about the data that shed light on the mystery. First, if we are comparing Republican and Democratic voting records, then we have to look at the overall data—the combined national totals—which leads to the conclusion that Republicans were more supportive of the Civil Rights Act than Democrats. That has to be the bottom line.
Second, although we might want to look for a difference in the Republican and Democrat voting records, the really striking difference is between the northern and southern representatives, regardless of political party. Support in the north is at roughly 90 percent, whereas support in the south plummets to just 7 percent. If we focus on one variable (e.g., Democrat v. Republican), while paying less attention to a more important variable (e.g., north v. south), then the latter is often referred to as a lurking variable.
Third, percentages can be helpful for making comparisons in some situations, but when we started off looking at only percentages we failed to take into account the actual numbers of votes, and therefore we failed to see the significance of particular results. For example, the 0 percent result for southern Republicans sounds damning, but there were only 10 Republican representatives from the south; if just one southern Republican had voted for the act, then Republican support in the south would have increased from 0 percent to 10 percent and overtaken Democratic support, which was only 7 percent.
Finally, the most important part of the data is the voting record of the southern Democrats. The key point is that there was much less support for the act in the southern states than in the northern states, and the southern states elected predominantly Democrats. This large level of weak support from southern Democrats dragged down the Democratic average, and this was ultimately responsible for reversing the trend when we look at the totality of the data.
Importantly, the voting records for the 1964 Civil Rights Act are not a rare statistical quirk. This sort of reversal in interpreting data, Simpson’s paradox, causes confusion in many other situations, ranging from sporting statistics to medical data.
Before finishing this chapter, I should point out that there are further Simpsons in the world of mathematics. For example, the name Simpson is also mathematically immortalized in Simpson’s rule, a technique in calculus that can be used to estimate the area under any curve. It was named after the British mathematician Thomas Simpson (1710–61), who at the age of fifteen became a mathematics teacher in Nuneaton, England. Eight years later, according to the historian Niccolò Guicciardini, he made one of those mistakes that could happen to any of us when he “had to flee to Derby in 1733 after he or his assistant had frightened a girl by dressing up as a devil during an astrology session.”
And, of course, there is the Carlson-Simpson theorem, which needs no explanation, except to state that it implies the coloring Hales-Jewett theorem and is used in the Furstenberg-Katznelson argument. But I am sure that you do not need me to tell you that.
And finally, there is the unforgettable Bart’s theorem.17
Examination III
UNIVERSITY SENIOR PAPER
Joke 1
Q: Why do computer scientists get Halloween and Christmas mixed up?
2 points
A: Because Oct. 31 = Dec. 25.
Joke 2
If the Teletubbies are a product of time and money, then:
4 points
Teletubbies = Time × Money
But, Time = Money
Teletubbies = Money × Money
Teletubbies = Money2
Money is the root of all evil
Money = √Evil
Money2 = Evil
Teletubbies = Evil
Joke 3
Q: How hard is counting in binary?
2 points
A: It is as easy as 01 10 11.
Joke 4
Q: Why should you not mix alcohol and calculus?
2 points
A: Because you should not drink and derive.
Joke 5
Student: “ What’s your favorite thing about mathematics?”
2 points
Professor: “ Knot theory.”
Student: “ Yeah, me neither.”
Joke 6
When the Ark eventually lands after the Flood, Noah releases all the animals and makes a proclamation: “Go forth and multiply.”
Several months later, Noah is delighted to see that all the creatures are breeding, except a pair of snakes, who remain childless. Noah asks: “What’s the problem?” The snakes have a simple requ
est of Noah: “Please cut down some trees and let us live there.”
Noah obliges, leaves them alone for a few weeks and then returns. Sure enough, there are lots of baby snakes. Noah asks why it was important to cut down the trees, and the snakes reply: “We’re adders, and we need logs to multiply.”
4 points
Joke 7
Q: If
then solve the following:
4 points
A:
TOTAL - 20 POINTS
CHAPTER 11
Freeze-Frame Mathematics
The Flintstones, first broadcast in 1960, was a major prime-time success for the ABC network, with 166 episodes aired across six seasons. However, there would not be another major prime-time animated sitcom until 1989, when The Simpsons started its run of over five hundred episodes. By proving that an animated sitcom could appeal to both young and old, The Simpsons inspired other shows, such as Family Guy and South Park. Matt Groening and his team of writers also proved that comedies did not necessarily require a laugh track, which paved the way for shows such as Ricky Gervais’s The Office.
