What sums should I memorize?
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Strong sandwich solvers do not recompute combinations - They recognise them. A short list of forced sums and length ranges, recalled instantly, covers the majority of deductions in real puzzles.
Forced Combinations
These clue/length pairs have exactly one solution - Free information the moment you spot them:
- 0 - Empty sandwich: 1 and 9 adjacent.
- 2, 3, 4 - One cell, holding exactly that digit (two distinct digits from 2-8 cannot sum below 5).
- 5 with two cells - {2,3}. 15 with two cells - {7,8}.
- 9 with three cells - {2,3,4}. 21 with three cells - {6,7,8}.
- 27 to 33 with six cells - The one digit left outside is 35 − clue (clue 33 leaves out 2; clue 27 leaves out 8).
- 35 - All of 2-8 inside; 1 and 9 at the ends.
The Length Ranges
Commit the per-length sum ranges to memory: 1 cell spans 2-8, 2 cells 5-15, 3 cells 9-21, 4 cells 14-26, 5 cells 20-30, 6 cells 27-33. They follow a simple pattern - The minimum adds the next smallest digit starting from 2, the maximum adds the next largest starting from 8 - So you can rebuild the table mentally if recall fails. These ranges are how you convert any clue into a shortlist of possible 1/9 position pairs, which is the core move of the whole variant.
A Useful Symmetry
Inside and outside mirror each other: the digits outside the sandwich always sum to 35 − clue, and an outside region of k cells obeys the same ranges as a sandwich of k cells. A clue of 30 therefore leaves outside digits summing to 5 - Either the single digit 5 (six-cell sandwich) or the pair {2,3} (five-cell sandwich, forcing the inside to be {4,5,6,7,8}). One memorized table covers both sides of every line.