Welcome back to GLeaM! It seems like my last post was pretty well-received, so we're back at it again with a part 2!
If you missed the last post, let me give a brief introduction. One of my favorite types of
puzzles are those where you're supposed to discover an error in a proof—one of those bizarre arguments that ends in something completely absurd like 2=1 or "squares have three sides". The goal is to discover where that mistake is and where everything went so wrong. These false "proofs" are often called fallacies, and I invite you to scour them for errors. Consider it a contest—you vs the puzzle—and you want to come out on top.
To me, finding errors and discovering contradictions reveals something innate about the field of math itself. After all, we can't have all sorts of errors in a field based entirely on logic, so we have to avoid the gray areas! Many outsiders would argue that this rigid structure makes math super bland and boring, but it's actually what pushes us to new creative heights. We're inspired to tackle new angles and new directions to find new avenues around possibly wishy-washy statements, and that's what leads to the most brilliant sparks of intuition in math across the world.
So without further ado, here are five more of my favorite fallacies.
For some of the more introductory examples to fallacies, you'll want to read my last article, but this concept should explain itself just as well with the examples here too!
We'll range through a bunch of different topics, and you'll find hints (and occasionally solutions!) at the bottom of the article. Let's start!
1) The Root of All Errors
Source = MindYourDecisions: "Prove" 2 = 0 Using Square Roots
(Similar to an idea in Hexaflexagons and Other Mathematical Diversions by Martin Gardner)
Ok, maybe I went a little bit far with the pun in this title, but this error is certainly the "root" of a lot of others. This one requires a bit of knowledge of complex numbers, but essentially all you need to know is that i is the square root of -1. (Feel free to skip forward if that still seems foreign.)
See if you can figure out what might be wrong with this proof:
Let's start with something fairly obvious:
1 is the square root of 1, so we can replace it:
1 is the product of -1 and -1, so we can replace again:
i is the square root of -1, so we're multiplying i and i together.
i times i is -1:
Let's add:
Wait a second, that's impossible! Did you find the mistake?
If not, remember all hints and solutions are at the end of the article!
2) Probability and Presents
Source = Hexaflexagons and Other Mathematical Diversions by Martin Gardner
There are all sorts of fallacies in probability. Here, I'm going to take Martin Gardner's puzzle originally phrased in terms of neckties and tweak it a bit, but most of it is quoted pretty directly:
Say you and your friend Alice have just got presents for Christmas. You begin to argue over which of you received the more expensive present. You and Alice finally agree to settle this by visiting the store where both presents were bought and checking their value. Whoever wins (and has the most expensive present) must give their present to the loser as a consolation.
This is how you reason: "The chances that I will win the argument or lose it are equal. If I win, I will be poorer by the value of the present. But if I lose, I am sure to gain a more expensive present. Therefore, the contest is clearly to my advantage (because on average, I'll end up being richer).
To make this line of reasoning more clear, say your present is $30. Half the time, you'll lose $30, but the other half of the time you'll gain MORE than $30.
But, Alice can reason in the same way! No bet can put both sides at an advantage, so what's the matter here?
3) Paradox of Perpendicular Bisectors
Source = MindYourDecisions: "Prove" 90 Equals 100 (and PROMYS!)
For those of you who love geometry, I certainly have a problem for you!
As a bit of a backstory, it turns out that I'd actually seen this problem before: Back in March when I was applying to math summer programs, including the one called PROMYS, I solved this problem for a problem set. Flash forward to this series and I came across it again in more of the popular sphere, and I thought it would be a perfect problem to show you, so here goes!
Take a quadrilateral, ABCD. Two of its opposite sides, AD and BC, are the same length. Angle BAD is 90 degrees, and angle ABC is 100 degrees.
Draw in the perpendicular bisectors of side CD and side AB. Take their intersection and call it point E.
Because these two line segments are perpendicular bisectors, E is equidistant from point A and point B, and E is also equidistant from point C and point D. Thus, EA = EB and EC = ED.
(This is just a basic fact of perpendicular bisectors. To convince yourself that this is true, let's label the point where AB meets the perpendicular bisector, X. (This is just the midpoint of AB.) The triangles AEX and BEX are congruent because AX = BX by definition and XE is a side of both triangles. Also, the angle between these two sides is 90 degrees for both. Thus, EA = EB)
If we consider this part of the diagram,
you'll notice that triangles AED and BEC are congruent because they have three equal sides. This means that angle DAE is equal to angle EBC.
Also, if we consider this part of the diagram,
you'll notice that triangles AEX and BEX are congruent (Here, X is once again the midpoint of AB) because they have three equal sides (and even a 90-degree angle) in common. This means that angle EAB is equal to angle EBA.
Thus, angle DAE + angle EAB = angle EBC + angle EBA.
But, angle DAE + angle EAB = 90 degrees, and angle EBC + angle EBA = 100 degrees, so
90 = 100.
This is absurd. What went wrong?
What's even crazier is that the fact we chose 100 degrees as our angle is kind of arbitrary. If we replace that with any other obtuse value, we have that every obtuse angle has measure 90 degrees, meaning every obtuse angle is right?? How is that even possible?
4) All numbers are... interesting?
Source = Hexaflexagons and Other Mathematical Diversions by Martin Gardner
Don't worry, this isn't a philosophical debate.
We're going to be looking at another situation where logic can be too strong. Remember, in the last article, we looked at a couple situations where applying logic seems to go astray, showing that if we can eat some amount of cake without getting full, we can eat any amount of cake, and talking all about the famous unexpected hanging paradox. This is such a fascinating topic to discuss, and I'd encourage you to check out the discussion in the appendix at the end of that article, but for now, here's an interesting argument for you along the same lines.
Let's say any number that has a kind of unique property is "interesting".
1 is interesting because it's the only number for which multiplying by any number spits the same number back out. (It's the identity: 1 * x = x for any x.)
2 is interesting because it's the smallest prime.
3 is interesting because it's the first odd prime.
1729 is interesting because it's the smallest number that can be expressed as the sum of two cubes two ways: 12^3 + 1^3 = 10^3 + 9^3 = 1729. (The great Indian mathematician Srinivasa Ramanujan loved this one.)
602,200,000,000,000,000,000,000 is interesting because it's an approximation for a very important concept in chemistry. (Look up Avogadro's Number.)
And so on...
There are so many interesting properties, but every number can't be that special right. Like what about 38901398123489 or 129087102349707? That's just me typing down random digits.
There's no way to presume any number has some sort of property that is entirely unique to it. In fact, we'd be hard-pressed to come up with these unique properties for the first 100 numbers.
Well, here's a proof that every number is interesting (and not "dull"), courtesy of Martin Gardner:
If there are dull numbers, we can then divide all numbers into two sets—interesting and dull. In the set of dull numbers, there will be one number that is the smallest. Since it is the smallest uninteresting number, well that's an interesting property, so it becomes, ipso facto, an interesting number. We must, therefore, remove it from the dull set and place it in the other. But now there will be another smallest uninteresting number. Repeating this process will make any dull number interesting.
How would you resolve this paradox? Why doesn't this argument make a lot of sense?
You could brush this aside with a "logic doesn't apply here" but think deeper. What about this generates such a bizarre conclusion?
5) Repeated Exponents
Source = MindYourDecisions: "Prove" 4 = 2
One thing mathematicians love more than anything is to take things to infinity. Let's try making exponent towers that go to infinity and beyond!!
Say you were asked to find an answer x to the following problem:
Since this goes to infinity, we can replace the expansion at the top of the tower (the power the lowest x is raised to) with 2 as well. After all, both the overall expression and the power contain infinite x's! This gives:
And then:
Which tells us:
Guess what? We can do the exact same thing with 4.
Say you were asked to find an answer x to the following problem:
Similarly, we have:
And then:
And finally:
So,
What??
Something went wrong here, what is it? (Check out the solutions when you're ready!)
CONCLUSION:
Thanks for reading this week's article! I actually mentioned some really exciting videos and books about real-life math mistakes in the conclusion of last week's article if you'd like to check that out, but I will say that logical errors and straight-up calculation mistakes happen all the time across so many fields of math. Hopefully, this article gave you more of an eye for finding them and a little more confidence when it comes to approaching situations that don't exactly tell you what you'd expect!
This article took a surprisingly long time to write, so I'd really appreciate it if you'd share it with someone you think would enjoy it. Thanks for all your support, and see you in two weeks!
SOLUTIONS AND HINTS:
1) The Root of All Errors
What step did the proof seem to go wrong at?
To me, the answer to this one is a little unsatisfying, but it's still super important.
Here's the square root product rule.
When is it true?
It turns out this rule is only for when a and b are nonnegative numbers!
Complex numbers don't "belong" with our typical real numbers, so when we add them into the fold, we have to carefully examine what rules we're using and where and if they go wrong.
Think carefully about this. Just because this rule was only written for when a and b are nonnegative numbers doesn't mean it can't also be true in other situations. What happens when one of a and b is negative? What happens when both are negative?
Why do we allow it to be true in certain situations but not in others? What assumptions can we make about how this rule works? And, most importantly, how does this resolve the problem we got?
Did you consider all of that? If you did, you're thinking like a true mathematician! Math is just as much about asking questions as anything else, so see if you can form a full understanding of this situation here, and feel free to ask any questions in the comments!
If you'd like to see MindYourDecisions' original video, here you go:
2) Probability and Presents
I'm going to leave you on a cliffhanger for just this once. I want you to explore this problem and come up with your own answers.
So, go ahead and hit the comments or the forum, and see if you can give an answer to this gem of a puzzle all the way from 1958.
I'll give you one hint: Try making a table of prices (say from $1 to $10 as an example) that both you and Alice can have. If you look at all 100 combinations, is it still true that you're at an advantage (that, on average, you're likely to gain more money than you would lose?) Maybe the problem here lies in the fact that you're thinking of the price of your present as fixed, when you don't know the price of your present either!
3) Paradox of Perpendicular Bisectors
Here's your hint. This diagram is not to scale. Try drawing your own diagram to confirm the positions of all the points. Maybe E lies somewhere it shouldn't? How would that resolve the paradox?
Be super, super careful because diagrams can be misleading.
If you're still stuck, feel free to watch MindYourDecisions's video:
There are a couple super interesting properties of the diagram I found when I was investigating this setup for PROMYS, so let me know if you're interested and I can totally put up something about that!
4) All numbers are... interesting?
I feel like solutions to this problem are pretty much wide-open. Lots of people resolve this in different ways, and I'd love to know what you think.
For now, I'd like to refer you to the discussion of "Is logic too strong?" in the solutions section of the last article, as it raises some interesting points about the applicability of math to all sorts of situations.
I know that this problem itself isn't too well-defined. After all, each one of us might come away with a completely different idea of "interesting." But, the essence of this problem lies more in the idea that we could somehow—by default—say that every number has a property that the numbers themselves don't really seem to have. In general, does this mean that there's an aspect of our logic that we have to reconsider? I'd love to hear your thoughts.
5) Repeated Exponents
We assumed that the square root of 2 actually worked in all those situations.
Us plugging in 2 or 4 into the extended exponent tower to get that x^2 = 2 or x^4 = 4 was fine, and even solving for the square root of 2 was fine. However, this only told us that if there was a solution, it would have to be the square root of 2. It did not actually tell us that the square root of 2 was a solution!
As an analogy, if you've ever seen geometric sequences, you'll know that 1/(1 - (1/2)) = 2 is one way to find the sum of the sequence 1 + (1/2) + (1/4) + ... = 2, but you might also know that if you were to apply the equation to a situation like 1 + 2 + 4 + ... , you'd get 1/(1-2) = -1, which is completely absurd! The formula only tells you that if that sequence were to converge, i.e. if it were to equal a finite value, it would be -1, not that -1 is actually the value of the series. This is often the case in math: You have to check and make sure your answer is reasonable to confirm that it actually is the answer.
More relevant to us, let's look at the function:
You'll have to take my word for it, but this function can take values between about 0.37 and about 2.72. 4 is not a possible value, so the equation
has no solutions at all, and therefore, our supposed solution makes no sense.
As you move further in math, solutions that seem right but don't actually work become more and more frequent, and you've got to keep an eye out for them! (They're called extraneous solutions.)
MindYourDecisions has a little bit more of a complete explanation of what's going on here, so check it out!
And finally... that's it! Have a great week everybody! Feel free to contact me at catherine@gleammath.com, or just use the comment section or the forum. See you soon!
Cover Image Courtesy of Owen Chikazawa
(It's another impossible shape—a Penrose triangle—because we're looking at paradoxes!)
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