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The sandwich clue is 0. Where are the 1 and the 9?
Level 5 · Outside clues and specials
How to solve sandwich sudoku
A clue outside the grid is the sum of the digits between the 1 and the 9 in that row or column.
In sandwich sudoku, numbers outside the grid give a clue for each row or column. The clue is the sum of the digits sandwiched between the 1 and the 9 in that line.
The clue tells you how far apart the 1 and the 9 can be. Small clues keep them close together, and big clues push them toward the ends.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Clues outside the grid give the sum of the digits sandwiched between the 1 and the 9.
- Sandwich sudoku.
Worth memorizing
- 0 means the 1 and 9 are next to each other.
- 35 means the 1 and 9 are at the two ends, since 2 + 3 + … + 8 = 35.
- Small clues mean few cells between: 2 or 5 is too small for three digits.
Solving tips
Start with 0 and 35
A clue of 0 means the 1 and the 9 are next to each other. A clue of 35 means they sit at the two ends, since 2 + 3 + 4 + 5 + 6 + 7 + 8 = 35.
Small clues mean one cell
Only 2 to 8 can sit in the sandwich, and two of them add to at least 2 + 3 = 5. A clue of 2, 3 or 4 means exactly one cell sits between the 1 and the 9.
Work out the crust
The 1 and the 9 add to 10, so the digits outside the sandwich add to 45 - 10 - the clue. A clue of 20 leaves 15 for the cells outside.
List the combinations
Treat the sandwich like a killer cage that can slide along the row. A clue of 5 in two cells is 2 + 3, and a clue of 9 in three cells is 2 + 3 + 4.
A worked example: a sandwich clue of 33
We name cells by row and column, so row 1, column 9 is r1c9. Row 1 has a sandwich clue of 33 on its left. Column 9 already has a 2 lower down in the grid.
- Look up 33 in the table below. Only six cells between the 1 and the 9 can make it, and only one set works: 3 + 4 + 5 + 6 + 7 + 8 = 33.
- Six cells between them puts the 1 and the 9 seven columns apart. They sit in columns 1 and 8, or in columns 2 and 9.
- Check the crust. The digits outside the sandwich add to 45 - 10 - 33 = 2, so the one leftover cell is 2. It's r1c9 if the sandwich runs from column 1 to column 8, or r1c1 if it runs from column 2 to column 9.
- Column 9 already has a 2, so r1c9 can't be 2. The 2 goes in r1c1, and the 1 and the 9 sit in r1c2 and r1c9 in some order.
- The six cells from r1c3 to r1c8 hold 3 to 8. Mark them as a set and let the columns sort out the order.
Practice
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The sandwich clue is 35. Where are the 1 and the 9?
Reference table
| Cells between | Smallest sum | Largest sum | Sums with only one set |
|---|---|---|---|
| 0 | 0 | 0 | 0 (the 1 and 9 touch) |
| 1 | 2 | 8 | every sum from 2 to 8 |
| 2 | 5 | 15 | 5 (2 + 3), 6 (2 + 4), 14 (6 + 8), 15 (7 + 8) |
| 3 | 9 | 21 | 9 (2 + 3 + 4), 10 (2 + 3 + 5), 20 (5 + 7 + 8), 21 (6 + 7 + 8) |
| 4 | 14 | 26 | 14 (2 + 3 + 4 + 5), 15 (2 + 3 + 4 + 6), 25 (4 + 6 + 7 + 8), 26 (5 + 6 + 7 + 8) |
| 5 | 20 | 30 | 20 (2 + 3 + 4 + 5 + 6), 21 (2 + 3 + 4 + 5 + 7), 29 (3 + 5 + 6 + 7 + 8), 30 (4 + 5 + 6 + 7 + 8) |
| 6 | 27 | 33 | every sum from 27 to 33 |
| 7 | 35 | 35 | every sum from 35 to 35 |
Going deeper
Clues that fix the length
Many clues can only be made by one number of cells. From the table, 0 means zero cells, 2 to 4 mean one cell, 31 to 33 mean six cells, and 35 means seven. No clue can be 1 or 34. A 1 can't sit in the sandwich, and six cells top out at 33.
Other clues allow two or three lengths. 5 to 8 can be one or two cells, and 16 to 19 can be three or four. The widest are 14 and 15 (two, three or four cells) and 20 and 21 (three, four or five cells).
Once you know the length, you know how far apart the 1 and the 9 are. A sandwich of k cells puts them k + 1 columns apart. Five cells leave three positions for the sandwich, six leave two, and seven leave one.
Using the crust
The digits outside the sandwich (the crust) add to 35 minus the clue. A small crust is as useful as a small sandwich.
A clue of 30 leaves a crust of 5. That's a lone 5 with six cells between the 1 and 9, or 2 and 3 with five cells between. Either way, you learn which digits sit at the ends of the row.
Placing 1s and 9s across the grid
Every row and column has exactly one 1 and one 9, so sandwich clues work together. Place the 1 and 9 in a box and several rows and columns gain fixed ends at once. Corner-mark the possible 1 and 9 spots in each box early.
A cell that can't be the 1 or the 9 for its row's sandwich might still be one for its column's. Check both clues for each candidate before you rule it out.
Common mistakes
Adding the 1 and 9 into the sum
The clue adds only the digits between them. The 1 and the 9 are never part of the total.
Using 1 or 9 inside the sandwich
Only 2 to 8 can sit between the 1 and the 9. A two-cell sandwich of 10 can't be 1 + 9. Its options are 2 + 8, 3 + 7 and 4 + 6.
Locking in one length too soon
A clue of 14 can be two cells (6 + 8), three cells, or four cells (2 + 3 + 4 + 5). Check every possible length before you commit to one.
In mixed-rule puzzles
Sandwich clues pair well with rules that place 1s and 9s. A German whisper line pushes 1s and 9s onto cells with few neighbors that fit, and a two-cell killer cage of 3 or 17 puts a 1 or a 9 in a known pair. On a thermometer, a 1 can only sit on the bulb and a 9 only at the far end, which limits where each sandwich can start. X-sums clues often share an edge with sandwich clues. When both sit beside one row, solve the one with fewer options first.
What the video shows
- The clue is the sum of the digits sandwiched between the 1 and the 9.
- Here two cells sit between the 1 and the 9.
- 3 + 7 = 10, the clue.
Questions solvers ask
How does sandwich sudoku work?
Each outside clue is the sum of the digits between the 1 and the 9 in that row or column. Normal sudoku rules still apply.
What does 0 mean in sandwich sudoku?
It means the 1 and the 9 are next to each other, with nothing between them.
What is the biggest sandwich sudoku clue?
35. That happens when the 1 and the 9 sit at the two ends and every digit from 2 to 8 is between them.
Does it matter whether the 1 or the 9 comes first?
No. Either can come first. The clue only counts the digits between them.
What does a sandwich clue of 2 mean?
One cell sits between the 1 and the 9, and it holds a 2. Clues of 3 and 4 work the same way with a single 3 or 4.
Can a sandwich sudoku clue be 34?
No. Seven cells between the 1 and the 9 always total 35, and six cells total 33 at most. A clue of 1 can't happen either.
How do you find sandwich combinations?
List the digits 2 to 8 and find every set that adds to the clue, for each possible number of cells. The table on this page shows the range for each length and which sums have only one set.
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