## Math on TV: Halls Refresh

If you’ve watched any television recently, you may have noticed the following ad for Halls Refresh. I strongly encourage you to watch it, even if you’ve seen it before, because it’s basically fantastic.

A tremendous ad, to be sure. However, if you didn’t watch closely, you may be wondering what such a sensual commercial has to do with mathematics. Watch again if you missed it - it may help to watch it full screen, although the quality gets muddy.

Did you catch it the second time? When the camera cuts to the Asian kid sitting at his desk, right before he starts to charm Mrs. Hunter, you’ll notice that he has a poster on his wall filled with mathematics. There are 5 equations on the poster, but most are probably too difficult to make out from the Youtube copy. I was fortunate enough to see this ad on television, and after a few replays I made out 4 of the 5 equations. In order, they are as follows:

*x*/*x*=*x*^{x}.- (10+
*x*+*x*^{x})^{1/x}/*x*^{x/41/x}=*x*. - (
*x*^{x-1}-1)^{1/x}+ tan(π/(*x*+1)) =*x*. - This one I didn’t get, but I can tell you it was long and involved a logarithm. Bonus points to anyone who can fill in the blank here.
*x*^{x}-*x*^{3}= 4(*x*^{x-1}+*x*^{x-2}).

At first this poster made me a bit upset. Like other math jokes I’ve discussed before, throwing together mathematically complicated equations just for the sake of it seems lazy, when one could instead try to make some kind of joke. It’s not as if the equations above are famous, so initially it may seem like there’s nothing going on under the hood.

Upon further investigation, however, I discovered that somebody involved in the production had a sense of humor. Let’s see what happens when we look for solutions to each of the equations.

The first one is easy: since *x*/*x* = 1, we’re looking for a solution to *x*^{x} = 1. This happens, of course, when *x* = 1.

The equations that follow are too difficult to solve by hand, but this is where we let computers do the work for us. If you graph these equations, you’ll see that the solution to the 2nd is *x* = 2, the solution to the 3rd is *x* = 3, and the solution to the 5th is *x* = 5. Even though I wasn’t able to read that fourth equation, I’m fairly confident that the solution to it is *x* = 4.

It’s not a great joke, I know, but I appreciate the fact that there is a payoff for those who are willing to dig a little bit. It’s not perfect (for example, the third equation has a second solution near *x* = 0.33), but it’s certainly better than many attempts. Kudos to you, Halls Refresh. This almost makes up for your use of the stereotypical nerdy Asian dude who is good at math and likes dragons.

For more surreal advertising, there’s always this classic. No math involved, unfortunately, but I’m willing to let it slide.