Sphinx's Riddle
You are facing the Sphinx guarding the exit portal of the pyramid. On the stone wall, there are 9 stone tablets arranged in
square with inscribed runes you can't read (but we designate them as
to
for simplicity as shown), and behind each tablet is a distinct digit from 1 to 9.
The Sphinx then asks you to answer all the hidden numbers correctly without flipping any tablet, or you'll be eaten alive, before giving you 3 clues:
Clue #1 : The sum of three numbers in each row, column, and diagonal involving the center equals to 12.
Clue #2 : The product of three numbers in each row and column not involving the center is over 107.
Clue #3 : The number in the upper left corner ( ) can divide the number in the lower left corner ( ).
What is the value of 9-digit integer ?
The answer is 493516827.
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1 solution
From the first clue, we can assume that 12 = A + E + I = B + E + H = C + E +G = D + E + F.
Thus, adding all 4 equations: 48 = (A + B + C + D + E + F + G + H + I ) + 3E.
We know that the 9 letters stand for the distinct 9 digits, so the sum of (A + B + C + D + E + F + G + H + I ) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
Thus, 48 = 45 + 3E; E = 1. And the sum of 2 digits on the opposites equals to 11, and these pairs are 2+9, 3+8, 4+7, & 5+6.
From the second clue, the product not involving 1 (center) is always greater than 107. So for digit 2, 107/2 = 53.5. That means the product of other two digits aligned with 2 must be over 53.5, and only 7 × 8 applies (9 must be opposite 2 with 1 in between, as explained in the first clue).
That also means the digit 2 can't stay at any corner because at that point it will need other four digits to exceed the product, so 2 must be on the side or above or under 1 and is between numbers 7 and 8 on the corners. Then the numbers 3 and 4, which are opposite 8 and 7 respectively, will also be on the opposite corners with number 9 between 3 & 4 (it's opposite 2). Note that 9 × 3 × 4 = 108 > 107 correlating to the second clue.
Now we have to fill 5 and 6 in the gaps:
3 __ 7
4 __ 8
Again the product of these digits are still over 107, so if 5 is between 3 and 7, then 3 × 5 × 7 = 105, which is not applicable. So 6 must be between 3 and 7, and 5 must be between 4 and 8.
Finally, we know that the corners consist of numbers 3, 4, 7, and 8, and from the last clue, A can divide G. In this case, only 4 can divide 8, so A=4; G=8.
Going after all the values with these restraints, we will get: A=4; B=9; C=3; D=5; E=1; F=6; G=8; H=2; and I=7.