How do I know how many cells are between 1 and 9?
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The clue alone rarely tells you the exact sandwich length - But it always narrows it. Because the filling uses distinct digits from 2 to 8, each length has a fixed minimum and maximum sum, and any clue outside a length's range rules that length out.
Minimum and Maximum Sums per Length
The smallest digits available are 2, 3, 4, … and the largest are 8, 7, 6, … - So each sandwich length has a hard range:
- 0 cells: sum 0 (the 0 clue)
- 1 cell: 2 to 8
- 2 cells: 5 (2+3) to 15 (7+8)
- 3 cells: 9 (2+3+4) to 21 (6+7+8)
- 4 cells: 14 to 26
- 5 cells: 20 to 30
- 6 cells: 27 to 33
- 7 cells: exactly 35
Notice that 1 and 34 are impossible clue values, and a clue of 2, 3, or 4 can only be a single cell.
From Lengths to Positions
Each allowed length translates into allowed position pairs for the 1 and 9. A length-2 sandwich means the 1 and 9 are exactly three cells apart, giving six possible placements in a nine-cell line; a length-6 sandwich allows only two. List the surviving pairs and strike out any that conflict with digits already placed, with box restrictions, or with the crossing line's own sandwich constraint. In most puzzles two or three lines collapse to a single arrangement quickly, and those placements cascade into their crossing lines.
Worked Example - The Clue of 8
Suppose a row's clue is 8. Three or more cells is impossible (minimum 9), so the sandwich is one or two cells long. One cell: that cell holds exactly the digit 8. Two cells: the only distinct pairs from 2-8 summing to 8 are {2,6} and {3,5} - Never {4,4}, since repeats are banned. So the 1 and 9 are either two or three cells apart, and the cells between them are limited to three small candidate sets. That is an enormous restriction from a single outside number.