Can you solve this puzzle?

Text from the image:

And Let’s Consider Another Quotient. It takes shape as men’s minds go deep into space. One center of these calculations is Martin-Denver, where successful professional enjoy full scope from basic research to vehicle design. You will find range, reward, and stature here…

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:

- 992
- 507
- 496
- 1116
- 1116

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.

Brilliant!

Brilliant! I can’t tell you how many countless hours and wasted paper it took me to come to essentially the same conclusion. I might quible with the whole “9408 issue” but I got to send a shout out and props to you!

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