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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.

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Size considerations (starting from the left)

  • $PRINCETOM<1,000,000,000\Rightarrow PCRON<\sqrt{1,000,000,000}=31,623\Rightarrow P\leq3$. If $P=2$ or $P=3$, then the nine-digit number $PCRON^2$ begins with something at least 4, contradiction. So $P=1$.

  • $PRINCETOM\leq198,765,432\Rightarrow PCRON\leq\sqrt{198,765,432}=14,098\Rightarrow C\leq4$. All the digits are distinct and nonzero, so in fact $C\leq3$ and $C\neq1$, i.e. $C$ must be 2 or 3.

  • $12,345\leq PCRON\leq13,987\Rightarrow PRINCETOM\geq152,399,025\Rightarrow R\geq5$.

Modular considerations (starting from the right)

  • $N^2\equiv M$ modulo 10, and $M\neq N$, so $N$ must be one of $2,3,7,8$ with $M$ being respectively one of $4,9$. (We know $M\neq1$ which means $N\neq9$.)

  • $(ON)^2\equiv OM$ modulo 100, i.e. $20*O*N+N^2\equiv 10*O+M$. 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

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

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  • 1
    $\begingroup$ Hat-tip to Brandon J, who saw some of these ideas before I did, but didn't make as strong deductions as he could have :-) $\endgroup$ – Rand al'Thor Jun 8 at 18:15
  • $\begingroup$ E is missing....that’s ok..actually if you saw the clue in the puzzle, it is only 3 step simple process. I will reveal it when I post the answer. $\endgroup$ – Uvc Jun 8 at 18:21
  • $\begingroup$ @Uvc Oops, added E. Presumably the clue in the puzzle is "highest possible number fulfilling"? I didn't understand why you wrote that, since there's only one possibility so no need to choose the highest. $\endgroup$ – Rand al'Thor Jun 8 at 18:25
  • $\begingroup$ Yes..that’s right..otherwise there are 3 more solutions..this clue makes it a cinch to figure out the number because of restricted choices $\endgroup$ – Uvc Jun 8 at 18:32
  • $\begingroup$ 3 more solutions refer to other squares involving same 5 digits on the right..not for this one..but that clue leads to quicker resolution. $\endgroup$ – Uvc Jun 8 at 19:32
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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: $202,635,225$.

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