Episode 3: 1=2?

As my Functional Analysis professor says,

"Math is full of really cool things.
The only problem is you need some knowledge
to understand what's going on."

I hope that I can find enough things that are easily understandable to make these Math Episodes interesting. I hope that I don't completely lose all of you with what I say. Well, here's hoping!


I am going to prove to you that 1=2. Now, I know what you are probably thinking, "That's impossible!" But I'm going to do it! And completely blow your mind.*

Let's let a=1 and b=1. So, since a and b are equal

b²=ab.

We can also see that

a²=a².

Now, if we subtract the first equation from the second one, we see

b²-a²=ab-a².

Now, if we factor both sides of the equation, we see

(a-b)(a+b)=a(a-b).

So now, divide both sides by (a-b),

(a+b)=a.

But we know that a=1 and b=1, so if we plug in, we see

(1+1)=1

2=1!

I think the world just exploded.




*Note: If you see where this proof fails, please let me know! Otherwise, I'll tell you all on next week's episode!





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Solution to Cannibal Island:

Ok, I'm not going to just give you the answer, that wouldn't be any fun. But, I will walk you through it and try and show you how to get there!

Now, some people (trust me, not too many) can see pretty quickly that you can save at least half of the suckers on the island. How so, you ask?

The first guy, Steve, says the color of the second guy, Doug's, hat, "Blue!". Then Doug thinks to himself, "Oh ok, that's my hat color," and says, "Blue!" too. Then the third guy tells the fourth guy his hat color, and the fourth guys shouts his out. This continues on so that every other person doesn't get eaten by the cannibals.

But that's only half of the people saved! Can we do better?

In fact, we can. A lot better.

Remember how there was an initial strategy session in the cage before the game began? Well everyone got to thinking, and their mathematical skills came out (I mean, wouldn't yours if you were about to get eaten by Cannibals?) They decided on a strategy and then pulled straws (do Cannibals have straws?) to see who went first. Poor Steve.

So the first guy, Steve, counts all the blue hats in front of him. One-by-one, all 999 of them. If there is an even number of blue hats, he says, "Blue." If there is an odd number of blue hats, he says, "Red." Then he gets eaten (the Cannibals knew the strategy, so they would just put the opposite on his head).

But he did a great deed to the rest of the mathematicians (oh yeah, everyone's a mathematician; wouldn't you want to be one right before you die? It sounds so cool: Mathematician.) The second guy, Doug, now counts all the blue hats in front of him. He has all the information he needs from Steve. If the number of blue hats is still even, then he must have a red hat on his head, and thus says "Red". If the number of blue hats is now odd, then he must have a blue hat, and thus says "Blue". Doug is one lucky guy.

But this goes on and on. Everyone can hear the people behind them guessing, and everyone knew what Steve's guess was. They just keep calculating and keep realizing that they must either have a red hat or a blue hat.

All 999 people can be saved.

Isn't that pretty cool?

Silly cannibals :)