Last night, I was browsing one of the famous recreational mathematician Martin Gardner's books, and I came across a section called "Fallacies"—all about those weird mathematical "proofs" and paradoxes that seem to claim something absurd, like 2=1 or 3=0.
Today, I thought I'd put together my own collection of my favorite false "proofs" aka fallacies. Because these clearly aren't true, I invite you to find the errors in each of these arguments! I find that discovering errors and uncovering stones tells us something about the nature of mathematics: Of course, we can't have logical inconsistencies in our statements like politicians do—and often that leads to technicalities in the way we define things. But beyond that, sometimes exploring which specific cases do indeed satisfy a statement allows us to have a much fuller understanding of what's going on and see the general pattern (and the beauty) behind what we're doing in the first place!
This article should be pretty accessible to just about anyone!
So without further ado, here are five of my favorite fallacies: can you spot the mistakes?
We'll start fairly easy, and we'll range through all sorts of different fields from simple logic and counting to algebra to geometry! Just like one of my other favorite articles, "The Many, Many Ways to Cut a Cake", I'll place the puzzles in the main section of the article and hints (and sometimes solutions!) at the end. Let's start!
1) Let's start with a rhyme!
Source = Hexaflexagons and Other Mathematical Diversions by Martin Gardner
Martin Gardner's first puzzle was from a 19th-century British magazine, and he calls it "exceedingly elementary."
If you remember one of my early articles about an infinite hotel, we noted that we can somehow fit any number of hotel guests into the infinite halls of the hotel—even if all the rooms are full.
But what if we're looking at something a little.... smaller-scale? What if we try to fit 10 hotel travelers into 9 rooms—and each needs their own room? This hotel manager found an easy way to fit the 10 people, one each, into the 9 rooms, and it's all told in rhyme!
Ten weary, footsore travelers,
All in a woeful plight,
Sought shelter at a wayside inn
One dark and stormy night.
"Nine rooms, no more," the landlord said,
"Have I to offer you.
To each of eight a single bed,
But the ninth must serve for two."
A din arose. The troubled host
Could only scratch his head,
For of those tired men no two
Would occupy one bed.
The puzzled host was soon at ease—
He was a clever man—
And so to please his guests devised
This most ingenious plan.
In room marked A two men were placed,
The third was lodged in B,
The fourth to C was then assigned,
The fifth retired to D.
In E the sixth he tucked away,
In F the seventh man,
The eighth and ninth in G and H,
And then to A he ran,
Wherein the host, as I have said,
Had laid two travelers by;
Then taking one—the tenth and last—
He lodged him safe in I.
Nine single rooms—a room for each—
Were made to serve for ten;
And this it is that puzzles me
And many wiser men.
Did you spot the error? If not, remember you can check the end of the article!
2) A Classic Algebra "Proof"
Source = Hexaflexagons and Other Mathematical Diversions by Martin Gardner
If you've ever seen a fallacy before, you probably know the error that comes with this type of puzzle to be the classic fallacy.
For this one, let's assume that a is some positive integer greater than b. Then, because a is greater than b, we have that a = b + c for some positive integer c. (See if you can convince yourself that this is, in fact, what greater than means.)
First, we have:
Then, we multiply by a - b to get:
Rearrange by subtracting ca by both sides:
Next, we can factor this expression!
Finally, we can divide both sides by a - b - c:
But wait, that means that a is equal to a number we specifically said a was greater than! That can't be right...
3) 3+4 = 5?
Source: MindYourDecisions: "The Staircase Paradox"
Here's an interesting puzzle.
Say you build a 3-feet-tall staircase with two steps over a 4-foot span. How long is the perimeter of the staircase?
What if it has the same dimensions but four steps?
Well, in either case, the total length should be 7, as the total vertical spans of the steps add up to 3, and the total horizontal spans of the steps add up to 4, so the total length is 3+4=7.
In fact, no matter how many steps you have in your staircase, even if they're atom-sized, the total length will be 7!
But doesn't that approach the diagonal of the triangle if we take it all the way to infinite steps, and the length is constant, so it must still be 7?
And after all, by the Pythagorean Theorem, since 3^2 + 4^2 = 5^2, the length of that diagonal is 5!
So 3 + 4 = 5.
What went wrong here?
Feel free to watch MindYourDecisions' video if you want more of an explanation of the problem:
In fact, if you want a comparable puzzle, here's a proof that π = 4:
What went wrong this time?
4) The Missing Square Problem
Source = MindYourDecisions "The Missing Square Illusion" (and inspired by a very similar problem by Martin Gardner)
Say you have a triangle, here with sides 5 and 13, and you dissect it into the following four shapes:
The crazy thing here is you can actually rearrange the shapes to make a triangle with the exact same dimensions but a missing square:
Clearly, the total area of the four shapes is constant and the area of the two big triangles should also be the same. Thus, we have that 1 square = 0 squares. That's absurd, so where did this square go?
In fact, this puzzle might look familiar if you've ever seen the viral Hershey's Bar problem aka the Infinite Chocolate Bar Trick! Because hey, if you can get one square from nothing in a triangle—you also can in a rectangle. Repeat this over and over again for infinite chocolate!
5) Is Logic Too Powerful?
Source = How to Bake π by Eugenia Cheng
Before I introduce the puzzle, I have to say something about its source. How to Bake π is by the amazing Eugenia Cheng, a British mathematician and a pianist. Her book is about the "mathematics of mathematics"—basically investigating why we view math the way we do—and it's incredibly accessible. In each chapter, Cheng picks another type of food and explains how to bake it—everything from a puff pastry to lasagna to custard—to illustrate her points about math through food.
The chapter I pull this puzzle from is Chapter 8: "What Mathematics Is" where Cheng essentially points out that logic itself has shortcomings. It's sometimes too slow, sometimes too methodical, sometimes too inflexible, sometimes too weak and sometimes too strong.
Today, we're looking at when it's too strong.
Say you can eat one slice of cake without getting too full.
Then, if you can eat x amount of cake, you can also eat x amount of cake and one teaspoon of cake without getting too full.
Logically then, you can eat any amount of cake without getting too full.
What went wrong?
Or rather, here's the puzzle I really want us to look at. Cheng names it "Fuji's Paradox."
You don't have to know much about stock markets to understand this, but just know that the markets of countries have interest rates, and it isn't in the interest of the market to lower their rate to the point where everybody knows it can't get any lower.
At the time, Japanese interest rates were the lowest in the world, and Cheng's acquaintance, Fuji, said that they could never hit zero because then everybody would know they could not fall further. (Negative interest rates are impossible.)
But, interest rates in Japan are only multiples of a quarter of a percent, so with that logic, the rate can't be 0.25% because it can only fall to zero (and Fuji reasoned it won't ever fall to zero.) In other words, if it's 0.25%, everybody will know it can't get any lower, so it can't be 0.25%.
And then, the rate can't be 0.5% by the same reasoning OR 0.75% OR 1% OR 1.25%... wait a second. This doesn't make any sense.
What went wrong?
(See the Solutions for a couple of other variants on the problem, one of which should be quite familiar. That section of this article is probably the most fascinating part of the whole thing, so if you don't read all of the Solutions, at least read that section!)
OTHERS:
There are so many of these fun puzzles out there I found them hard to pair down for this article! I was about to give you all the links when I realized this would make a really great series, so I'll leave you on a bit of a cliffhanger. Stay tuned in two weeks for part 2 for five more "false" proofs! And don't worry—I love the next five just as much as these, maybe even more!
CONCLUSION:
First, let me say: please share this article! It took lots of work to read and find and write about the best examples for you, and I'd definitely appreciate it if you'd spread the word.
I also left a couple of videos here of larger-scale math mistakes from the promotions of the mathematician Matt Parker's most recent book, Humble Pi. It's all about what can go wrong in the "real world" if we allow ourselves to make mathematical mistakes and logical errors, and these examples are so fascinating!
SOLUTIONS AND HINTS:
1) Let's start with a rhyme!
Hint: All I'll say here is did we really account for all ten men? Read closer and it'll come to you!
If you really want a solution, check out this ninth stanza from one of Martin Gardner's readers, John F. Mooney:
If we reflect on what he's done,
We'll see we're not insane.
Two men in A, he's counted one,
Not once, but once again.
2) A Classic Algebra "Proof"
Hint: What's that one rule in math you can never break? It has to do with division!
If you still need help, watch a video about another variation on the theme, courtesy of Math Easy Solutions!
(This will give it away, so make sure you're ready first.)
This kind of fallacy comes up more often than you think, especially when you're working with computers, so it's great to be aware of it.
3) 3 + 4 = 5?
Essentially, we're approaching the area of the triangle or of the circle without approaching the perimeter. You can't approach the value of a perimeter by holding it constant at 7 or at 4.
And even though we're theoretically taking this process all the way to infinity, we're still zigzagging around the line (the diagonal of the triangle or the curve of the circle) by some tiny amount as we reach larger and larger amounts of steps (or zigzags), and what's more: we're completing more and more zigzags! The number of zigzags gets larger and larger (approaching infinity) even as the distance from the line at any given zigzag gets smaller and smaller (approaching zero). Therefore, we're not approaching the value of the perimeter at all!
I think ViHart might do a better job of explaining this than me, so watch her video here:
4) The Missing Square Problem
Hint: Are the slopes of the inclines of these triangles really the same? Do they really fit together when rearranged as exactly as shown? Try to quantify and measure everything in the diagram and see where that takes you!
If you're still stuck, check out this article: https://mindyourdecisions.com/blog/2013/05/13/the-missing-square-problem/#.VULLz_lVhBc
Here's the solution for the Chocolate Bar problem too: https://www.youtube.com/watch?v=NmEkL0yHQaI
5) Is Logic Too Powerful?
Fuji's Paradox is actually a restatement of the Unexpected Hanging Paradox:
"A prisoner is told that he will be hanged on some day between Monday and Friday, but that he will not know on which day the hanging will occur before it happens. He cannot be hanged on Friday, because if he were still alive on Thursday, he would know that the hanging will occur on Friday, but he has been told he will not know the day of his hanging in advance. He cannot be hanged Thursday for the same reason, and the same argument shows that he cannot be hanged on any other day. Nevertheless, the executioner unexpectedly arrives on some day other than Friday, surprising the prisoner."
This is quite a common paradox, but it still does not have an agreed-upon solution in the math community. We might have to throw up our hands and say that logic simply doesn't apply here, but that's not satisfying at all!
The way I'd like to resolve this problem is saying that though the prisoner knows the executioner shouldn't show up on Friday (as that would contradict the claim), there's no way he can extend that argument to the days before Friday as he doesn't know that the executioner will reason that far ahead with him. He only knows his own reasoning capabilities, not the executioner's.
However, there are lots of holes in that argument as well because it doesn't take the "[The prisoner] will not know on which day the hanging will occur before it happens." statement at face value, and math and logic takes everything at face value.
There are lots of interesting questions here. Can you come up with an answer that satisfies you enough for the Unexpected Hanging Paradox? What about for Fuji's Paradox? Can logic not be applied to problems with gray areas—like that of the "you can eat however much cake you want"—I mean, are we flawed by looking at the amounts of cakes as "yes I can eat it" and "no I can't" without some sort of intermediate idea between "yes" and "no"?
Ponder yourself, and tell me: what do you think? Comments are open! :)
If you're curious, here's a video on the Unexpected Hanging Paradox:
And, so you're not completely unsatisfied, here's a "proof" for a logical paradox that also takes logic too far, claiming everybody on the planet has the same age, but it does have a resolvable answer as to why the proof fails. This does require a bit of the terminology about induction, which does take some background knowledge, but it's essentially the same domino kind of reasoning we've been doing in the last two problems. (If you want to read a whole extra article that includes an explanation of induction, check out my series on the Four-Color Theorem here.)
A Random Note about Typesetting (if you're interested):
So far, though I love the platform where GLeaM is hosted, Wix, for so many other reasons, Wix does not support LaTeX or other ways of mathematical typesetting. This is why you often see me typing ^ instead of true exponents, or / instead of true fractions. If there are extended sets of equations, I find it easier on the eye to insert images of equations that I've typed up in LaTeX or other software elsewhere, but of course, it would be much more ideal if I could simply type the math in! Bear with me as I continue to look for solutions to this—I'll continue to make sure it doesn't sacrifice any of my article content or quality, but I thought I'd address this if you noticed the typesetting. Of course, if you have any ideas, feel free to contact me or leave a comment!
And finally... thanks for reading! Have a great week everybody!
Cover Image Courtesy of Owen Chikazawa
(It's an impossible cube because we're looking at paradoxes!)