Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$ Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid?
I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.

If you are willing to rely on the problem setter to make sure it is a natural, it has to  be close to $29.5^2=841+29+.25=870.25$  The one's digit of the stuff under the square root sign is $1$, so it is either $869$ or $871$.  You can either calculate and check, or note that two of the factors are below $30$ and only one above, which should convince you it is $869$.
