Partial answer that I'm saving for now
(For convenience, I will call $PCRON$ "the root" and $PRINCETOM$ "the square".
We can first deduce that the digit N
is not 6. It's a weird math thing that if you multiply two numbers that end in 6, the last number will also be 6. Since the last digit of the square is not N, then N is not six.
We can also do some quick tests to find the approximate range
of the square. $\sqrt(500,000,000)$ is approximately 22,360. Since the first digit of the root and the square match, we should go lower.
Let's try that:
$\sqrt(100,000,000)$ is $10,000$, so it looks like P is going to be 1.
We can then determine that C is
either 2 or 3. This is because the smallest possible 5-digit number that does not repeat digits and starts with 1 and 4 is 14,235, and $14,235*14,235$ is too big: $202,635,225$.