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Sandwich Sudoku Clue Reference

Work out where the 1 and 9 can sit from a sandwich clue — and see which two totals are impossible, and why they are impossible for the same reason.

Two totals are impossible, and for the same reason

A sandwich clue is the sum of the digits lying strictly between the 1 and the 9. Every line of a sudoku holds each digit once, so the digits that can sit in the middle are exactly 2, 3, 4, 5, 6, 7, 8 — and the achievable totals are their subset sums, running from 0 to 35.

Of the 36 values in that range, 34 occur and 2 cannot: 1 and 34. A total of 1 would need a single digit 1 as the filling — but the 1 is one of the slices of bread. A total of 34 would mean leaving exactly one point out of the maximum, which again needs a digit 1 to omit. They are mirror images of each other, 1 and 35 − 1, and both exist because the digit the clue is anchored to can never be inside it.

Checked two ways that share no code: by enumerating all 128 subsets of the middle digits, and by brute force over all 362,880 possible lines. Neither impossible total appears once in the brute-force pass.

Eight clues pin the gap, and they are all at the extremes

Most totals can be built from several different numbers of digits, so they leave the 1 and 9 free to sit at various distances. Eight cannot:

ClueDigits betweenPlacements left of 72
0 exactly 0 16
2 exactly 1 14
3 exactly 1 14
4 exactly 1 14
31 exactly 6 4
32 exactly 6 4
33 exactly 6 4
35 exactly 7 2

They cluster at both ends and there are none in the middle at all. A 35 is the strongest clue in the game: it leaves 2 of the 72 ways to place the 1 and 9, because the only way to fit all seven other digits between them is to put them at the two ends. Compare the weakest, a 14, which still allows 30.

A 0 is worth a second look. It sounds extreme — nothing between them at all — but it still leaves 16 placements, because "adjacent" can happen anywhere along the line in either order. It is a strong statement about the relationship and a weak one about the position, which is the opposite of how it reads.

Every clue, and what it is worth

ClueDigits betweenPlacements left
0 0 16
2 1 14
3 1 14
4 1 14
5 1 or 2 26
6 1 or 2 26
7 1 or 2 26
8 1 or 2 26
9 2 or 3 22
10 2 or 3 22
11 2 or 3 22
12 2 or 3 22
13 2 or 3 22
14 2 or 3 or 4 30
15 2 or 3 or 4 30
16 3 or 4 18
17 3 or 4 18
18 3 or 4 18
19 3 or 4 18
20 3 or 4 or 5 24
21 3 or 4 or 5 24
22 4 or 5 14
23 4 or 5 14
24 4 or 5 14
25 4 or 5 14
26 4 or 5 14
27 5 or 6 10
28 5 or 6 10
29 5 or 6 10
30 5 or 6 10
31 6 4
32 6 4
33 6 4
35 7 2

The grid rules of Sandwich Sudoku are the ordinary ones — every row, column and box holds one to nine. What the sandwich clues add is a layer of arithmetic, and the solving becomes about sums rather than positions. Our other sudoku pages cover the seventeen-clue minimum and the letter variant, which are different questions entirely.

How to use

  1. Enter a sandwich clue to see every place the 1 and 9 could sit in that line.
  2. Or type a line of digits to read off its sandwich sum.
  3. Totals of 0 and 35 pin the 1 and 9 almost completely.
  4. Totals of 1 and 34 are impossible and the tool will say so.
  5. Use the table to spot which clues on your grid are worth attacking first.

Frequently asked questions

What is a sandwich clue in sudoku?

The sum of all the digits lying strictly between the 1 and the 9 in that row or column. The 1 and 9 themselves are the bread and are not counted, and neither is anything outside them. A clue of 0 means the two are next to each other; a clue of 35 means everything else is between them.

Which sandwich totals are impossible?

Exactly two: 1 and 34. Nothing else in the range 0 to 35 is ruled out. We checked it twice over, once by enumerating all 128 subsets of the digits that could be in the middle and once by brute force over all 362,880 possible lines, and the two methods agree exactly.

Why are 1 and 34 the impossible ones?

For the same reason, seen from opposite ends. A total of 1 would need a single digit 1 sitting in the middle, but the 1 is one of the slices of bread and cannot also be the filling. A total of 34 would mean leaving exactly one point out of the maximum 35, which again needs a digit 1 to omit. They are mirror images: 1, and 35 minus 1.

Which sandwich clues are the most useful?

A 35 is the strongest clue in the game. Of the 72 ways the 1 and 9 could be placed in a nine-cell line it leaves exactly 2, because the only way to fit all seven other digits between them is to put them at the two ends. A 31, 32 or 33 leaves 4. The weakest are 14 and 15, which still allow 30.

Do all sandwich clues fix the distance between the 1 and 9?

Only eight of them do: 0, 2, 3 and 4 at the low end, and 31, 32, 33 and 35 at the high end. Every other total can be reached with several different numbers of digits in between, so it leaves the gap open. The forcing clues cluster at both extremes and there are none in the middle at all.

What does a sandwich clue of 0 tell you?

That the 1 and the 9 are adjacent, in either order. That still leaves sixteen possible placements in a nine-cell line, so it is a strong clue about the relationship and a weak one about the position — the opposite of what people often assume, since 0 looks like the most extreme value.

How is this different from ordinary sudoku?

The grid rules are identical — every row, column and box holds one to nine. Sandwich clues are an extra layer of arithmetic on top, and the solving is about arithmetic rather than position. Our other sudoku pages cover the seventeen-clue minimum and the letter variant, which are different questions entirely.

Does this send anything anywhere?

No. Everything is computed in your browser and nothing is uploaded.

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