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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.
$PRINCETOM<1,000,000,000\Rightarrow PCRON<\sqrt{1,000,000,000}=31,623\Rightarrow P\leq3$. 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.
$12,345\leq PCRON\leq13,987\Rightarrow PRINCETOM\geq152,399,025\Rightarrow R\geq5$.
$N^2\equiv M$ 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 $M=9$.
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 $12543^2=157326849$.
$P=1,C=2,N=3,O=4,R=5,E=6,I=7,T=8,M=9$.
(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: $202,635,225$.
