# Intuitive way to get 10 dimensions in string theory?

To get the 26 dimensions is sort of intuitive (in a handwavey sort of way). Basically we solve:

$$(D-2)\frac{1}{2}(1+2+3+4+...)=-1$$

Where $$1+2+3+..$$ times $$\frac{1}{2}\hbar$$ are the ground energy for the different modes of a string which each have $$D-2$$ degrees of freedom to vibrate in. This sums to $$-\frac{1}{12}$$. This all has to sum to $$-1$$ in order that the graviton is massless. (Although because it all sums to -1 it means the lowest state, the tachyon has mass $$-1$$).

What is the equivalent equation for superstring theory? I notice if you take all the half integers because it now includes fermions:

$$\frac{1}{2}+1+\frac{3}{2}+2+\frac{5}{2}+...=-\frac{1}{6}$$

Then if you add these to the the previous sum $$-\frac{1}{12}-\frac{1}{6}=-\frac{1}{4}$$. Which if you put this in you get $$D=10$$. But of course, this is just a random guess.

Just wondered if there's a more-or-less intuitive explanation which can be explained in a paragraph of less.