# An innovative algebraic approach

Usually, when faced with a word problem, I take the most obvious approach and call it done. But then, sometimes, I read of an alternative approach that makes me go “*Whoa.*” This is one of those times.

Here’s the problem:

One day, a person went to a horse racing area. Instead of counting the number of humans and horses, he counted 74 heads and 196 legs. How many humans and horses were there?

The standard approach is to note that (most) humans have one head and two legs, while (most) horses have one head and four legs. You can set up two equations, using $h$ for humans and $e$ for equines:

$h + e = 74$ $2h + 4e = 196$

And solve them simultaneously – dividing the second by two gives $h + 2e = 98$, and it’s clear that $e = 24$ and $h=50$.

User egreg on Math.StackExchange offered a different method, the one that made me go “*Whoa.*”. I’ll quote it here:

A hypercentaur is a creature with two heads and six legs; an anticentaur is a creature with no head and two negative legs.

Since 74 heads make for 37 hypercentaurs, with $\frac{74 \times 6}{2}=222$ legs, you have $\frac{222−196}{2}=13$ anticentaurs.

Since a hypercentaur is the same as a human on a horse, and an anticentaur is a human deprived of a horse, we have counted $37−13=24$ horses and $37+13=50$ humans.

What a bizarre, yet brilliant insight! By making up mythical animals, some of which have a negative number of legs, the answer pretty much drops out straight away! But how did egreg come up with it?

Well, I’m not party to the inner workings of other people’s brains, or else I’d be a killer poker player. However, I can have a good stab at explaining a plausible thought process: supposing there are equal numbers of horses and humans (37 each), how many legs would we be out by? We’d have 26 legs too many. That means we’ve got 13 horses too many - so there are 24 horses and 50 humans.

But I think we can all agree, the idea of an anticentaur is a much defter touch than simply halving the error in the number of legs.