Can You Spot The Errors? Five Fascinating "Faulty" Puzzles to Try

Updated: Dec 16, 2020

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 onethe tenth and last

He lodged him safe in I.

Nine single roomsa 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: