Given:

P, R, I, N, C, E, T, O, M are all distinct digits varying from 1 to 9.

PCRON, PRINCETOM are two concatenated numbers.

PCRON is the highest possible number you can have fulfilling the following relation:

$$(PCRON)^2=PRINCETOM.$$

Please give your deductive reasoning to figure out the Pan digital Square.

Only few calculations will be needed.

Size considerations (starting from the left)

• 
  

  $PRINCETOM<1,000,000,000Rightarrow PCRON<sqrt1,000,000,000=31,623Rightarrow Pleq3$. If $P=2$ or $P=3$, then the nine-digit number $PCRON^2$ begins with something at least 4, contradiction. So $P=1$.

  
  




• 
  

  $PRINCETOMleq198,765,432Rightarrow PCRONleqsqrt198,765,432=14,098Rightarrow Cleq4$. All the digits are distinct and nonzero, so in fact $Cleq3$ and $Cneq1$, i.e. $C$ must be 2 or 3.

  
  




• 
  

  $12,345leq PCRONleq13,987Rightarrow PRINCETOMgeq152,399,025Rightarrow Rgeq5$.

  
  

Modular considerations (starting from the right)

• 
  

  $N^2equiv M$ modulo 10, and $Mneq N$, so $N$ must be one of $2,3,7,8$ with $M$ being respectively one of $4,9$. (We know $Mneq1$ which means $Nneq9$.)

  
  




• 
  

  $(ON)^2equiv OM$ modulo 100, i.e. $20*O*N+N^2equiv 10*O+M$. So $ON$ must be one of $23,27,43,63,83$, which means $N$ must be 3 or 7 and for sure $M=9$.

  
  

Case checking

So far we have:

$P=1$, $M=9$, $C$ is 2 or 3, $N$ is 3 or 7, $R$ is 5 or 6 or 7 or 8, $O$ is 2 or 4 or 6.

Let's try

$C=3$, then $N=7$, so $O=2$, and $PCRON$ is one of $13527,13627,13827$. The squares of all three of these numbers have repeated digits, so it's impossible.

So

$C=2$, which means $O$ is 4 or 6 and $N=3$, so $PCRON$ is one of $12543,12643,12743,12843,12563,12763,12863$. Only one of these has a square with no repeated digits, namely $12543^2=157326849$.

Final answer

$P=1,C=2,N=3,O=4,R=5,E=6,I=7,T=8,M=9$.

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: $202,635,225$.