Difficulty: Very Hard
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Want to try this without
help? If so, just don’t scroll down. Scroll lower for the explanation.
The sum of the numbers 1
through 9 is 45, so every row, column, and sub-grid of a valid Sudoku must sum to 45. Therefore, if you have the sum clues
for the gray squares, you can work out “shadow sum” clues for the white
squares, just by subtracting the gray-squares sums from 45. Therefore, the
given puzzle, above, is perfectly equivalent to the puzzle below:
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Now this is more like it!
With 26 gray squares, this is a normal Suduro puzzle,
with a manageable number of gray squares to solve for in stage 1 of the
solution. (This is still a level-“Hard”
puzzle, however, even after the transformation, so beware!) In theory, even
ordinary Suduro puzzles can benefit from this trick –
the same sums that tell you something about the gray squares also tell you
something about the white squares, and these can in turn
provide useful constraints on the other gray squares while you solve stage 1.
In practice, however, you always want to focus most attention on whatever color
of squares is relatively rare, making up about a third of the 81 squares in the
Suduro grid, and anything you learn about the
other-color squares during stage 1 is rarely useful, except in the fairly rare
cases where you must solve part of stage 2 to complete the solution of stage 1.
You might wonder why there
are not many Suduros with many more than a third of
the squares gray (or in the case of this particular trick Suduro,
with more than a third of the squares white. In theory, a Suduro
with half the squares gray and half white could be solved simultaneously for
both colors, which would surely be interesting. Of course, this would eliminate
any real challenge in stage 2, since once half the squares were solved in stage
1, there would presumably be more clues than you need in the stage-2 Sudoku problem, and this is one of the reasons for keeping
the count of gray squares low. In practice, though, there is a practical reason
why Suduros tend to have very close to one-third of
the squares gray: With much less than a third, the stage-2 Sudoku
problem tends to have more than one solution, so the solution given for the
puzzle might not be the solution you find. (Furthermore, it is overly difficult
to find any solution to a Sudoku that has multiple solutions, since a guess will
always be required at some stage of any puzzle with multiple solutions.) With much
more than a third of the squares gray, the stage-1 portion of the puzzle tends
to be much too difficult, and often to have multiple solutions – in general,
the lower the count of gray squares, the more likely it is that the stage-1
solution will be unique and will not require guesses at any stage of the
solution. Too few gray squares make stage 2 hard or non-unique, and too many
(too close to one-half, really) make stage 1 hard or non-unique. Therefore,
having close to one-third of the squares gray is the “Goldilocks ratio” for a
good Suduro – just right! (Having close to two-thirds
of the squares gray just means you need to solve for the white squares, instead
of the gray squares, so two-thirds, in a sense, is a second “Goldilocks ratio.”)
Still stumped? Here is the full answer.