Can you solve this puzzle?
Text from the image:
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CLUE: The sum of the digits of the divisor leaves a remainder of seven when divided by nine, and the sum of the digits of the quotient leaves a remainder of three when divided by nine.
Some clues to help you out, based on *my* solution:
- The color of the chess pieces is not important.
- The type of chess piece is not important.
- None of the numbers represented have leading zeros.
- The “clues” are required to solve the puzzle.
Ready for my solution? Ok, here goes.
First, let’s rewrite the base equation as A / B = C. A is seven digits, so it is in the range 1000000 <= A <= 9999999. Likewise, 100<= B <= 999 and 1400 <= C <= 9499.
Upon inspection of the first line of the long division (pawn pawn pawn pawn minus knight pawn knight) it can be certain that the first two digits of A are 1,0. And the first digit knight-pawn-knight is a 9 (no other combination would result in a difference with a zero in the first two places).
Thus 1000000 <= A <= 1099999. At this point we can refine the ranges for B and C : The largest value of A divided by the smallest B gives the upper range of C, etc…
So C is at most: 1099999 / 100 = 11000. Previously we’d shown C <= 9499, so this doesn’t help.
C is at least 1000000 / 999 = 1001. Again no help.
B is at most 1099999 / 1400 = 785. Progress.
B is at least 1000000 / 9499 = 106.
However, we know that 4 * B is a three digit number (blue bishop – red king – blue bishop). So if 4 * B < 1000, B must be less than 250. Revising again we restrict the ranges to
1000000 <= A <= 1099999, 106 <= B <= 249, 4416 <= C <= 9499
At this point we can apply the rule about the quotient (C). In the range 4416 to 9499 with the second digit a 4, only these values meet the rule: 8409, 8400, 7401, 6402, 5403. From here is a simple matter to further restrict the ranges for B, e.g.
for C = 8409, 119 <= B <= 130. However, applying the clue for B shows that only 124 is a valid value for B in that range.
It is then a simple matter to calculate A for B=124, and C=8409 : 1042716.
Then do the long division and see if the number of digits line up. For each line in the figure they are:
Success! Crank through the other values for B (8400, 7401, 6402, 5403) and you’ll see this is the only answer with the correct number of digits on each line.