[strangestchild]

All good fun problems - and some less well-known as well as the old favourites. That said, calling these the "12 most controversial facts in mathematics" is like calling the truth of the moon landings the most controversial fact in astronautics. But I suppose "The Most Unintuitive Mathematical Results That Laymen Can Be Made To Understand" is not quite so catchy.

[gordaco]

Agree, this is mostly controversial for non-math people. Although the Monty Hall problem supposedly caused a lot of mail from people, some even math teachers, who got it wrong but didn't realize and/or admit it.

[DerekL]

People often get the Monty Hall problem wrong because it is often stated ambiguously, and the listener fills in the details in his own way.

Here's the question as posed in Marilyn Vos Savant's column [http://marilynvossavant.com/game-show-problem/]: "Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, 'Do you want to pick door #2?' Is it to your advantage to switch your choice of doors?"

But what exactly is the procedure followed by the host? I can think of three:

A. If you pick a goat, then the host shows you the other goat. If you pick the car, then the host picks one of the goats at random to show it to you.

B. The host picks one of the other doors at random and shows it to you. So you might pick a goat, then he shows you the car, and gives you a chance to switch, but you already know that the remaining doors have goats, so there's no point.

C. If you pick the car, then the host shows you one of the goats picked at random. If you pick the goat, then the host says "You picked the goat!", and you don't get a chance to switch.

People assume that the procedure is A, but I don't see how the other procedures are excluded by the description.

[T-hawk]

There's even a more dastardly procedure, where the host doesn't follow fixed rules but has free rein to outwit and trick the contestant. If Monty thinks the contestant can be induced into switching away from the car, he can try an offer, but doesn't have to. So he basically follows C (already the most disadvantageous for the contestant), but can also throw in a game-theory curveball of occasionally offering the switch even when the contestant is already wrong. Your described rule leaks information (offering the switch is a telltale that the contestant was already right), which the host can discredit by occasionally behaving otherwise. And of course if Monty is playing the psychology of the contestant, there's no rigid mathematical answer at all, like a poker bluff.

And by most accounts, the actual "Let's Make A Deal" show did let Monty do whatever he liked, so Vos Savant's presentation of the problem following rigid rules didn't really have a basis in reality.

[thebear]

That was my first thought as well. Puzzling, vexing, yes, but there is nothing "controversial" here. Moreover, I'm not sure if the expression "controversial fact" even makes sense, linguistically speaking. Isn't it true that if something is a fact, it may be "disputed," but it can, by definition, not be "controversial?"

[B-Con]

To be fair, in the introduction they do clearly point out that these are not technically "controversial" and just have a tendency to evoke controversial/confused reactions from laymen.