suppose we want to calculate the integral of exp(-x^2) from negative infinity to positive infinity. of source we know this is sqrt(2\pi). suppose we don't, like Gauss. He invented the trick to multiply this integral with itself, but denoted as the integral of exp{-y^2) with the same limit. Now we have a two dimensional integral with integrand exp(-(x^2+y^2)). does this make things easier? Yes! Since we now can use the polar coordinate transform to integrate this out and get the result 2\pi. Take the square root we get the answer needed. How does a Turing machine discover this trick? I think after we understand Gauss' brain process on this we should be able to device a discrete machine to imitate the discovering process. But to start this understanding with a discrete algorithm approach seems to be very hard, and many people in the field of brain modelling agree (that this approach is very hard).