Sandwich Sudoku: rules, sums and free puzzles online

In sandwich sudoku, the numbers outside the grid tell you the sum of the digits sandwiched between the 1 and the 9 in that row or column. Play sandwich sudoku online free here at four ranks.

The 1 and the 9 are the bread; everything between them is the filling. Working out how long the filling is, and where the bread goes, is the whole game.

The rules

• Normal sudoku rules apply: 1 to 9 once in every row, column and 3×3 box.
• A clue outside the grid is the sum of the digits between the 1 and the 9 in that row or column, whichever order they come in.
• The 1 and the 9 themselves are not counted. A clue of 0 means they are next to each other.
• At Theorem and Conjecture rank a few clues are left out. A missing clue tells you nothing; it does not mean 0.

How to start

Know the sandwich sums

The filling uses distinct digits from 2 to 8, so each length allows only a band of totals:

So a clue of 2, 3 or 4 means exactly one cell between the 1 and 9, holding that digit. A clue of 35 means all seven middle digits, so the 1 and 9 sit at the two ends. Sums of 1 and 34 can never appear.

Count, then place

Most clues fit more than one length. A clue of 33 must be six cells (every digit 3 to 8, with the 2 outside the sandwich), while a clue of 15 could be two cells (7 + 8), three (such as 2 + 5 + 8) or four (such as 2 + 3 + 4 + 6), but never five, because five cells sum to at least 20. Compare row and column clues that cross the same 1s and 9s to narrow them down.

Track the 1s and 9s

Every row and column has exactly one 1 and one 9, and every box does too. Mark their possible positions early; one placed 1 or 9 often fixes a whole sandwich.

Common mistakes

• Counting the 1 or the 9 as part of the sum.
• Assuming the 1 comes before the 9. Either order is allowed.
• Treating a missing clue as 0. It simply means no information.

How the ranks scale difficulty

Every puzzle type on Hoot’s Puzzles comes in four ranks, named after the stages of a mathematical idea. Lemma is the warm-up: about a dozen more digits are given than logic strictly needs, and singles plus the variant’s own rule will carry you through. Proposition needs a real idea or two, typically pointing and claiming. Theorem is near-minimal, so almost every clue is load-bearing and you may need pairs. Conjecture gives the fewest digits and sparser rule clues: expect to think hard.

Lemma and Proposition puzzles show every one of the eighteen sandwich clues. Theorem hides two of them and Conjecture hides five, so you have to work out the missing rows and columns from the rest.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Keep exploring

• Killer SudokuCages with sums instead of given digits. Partition arithmetic and the rule of 45.
• Little Killer SudokuOutside clues add up the diagonals their arrows point along.
• Arrow SudokuThe circled digit equals the sum along its arrow. Bound circles and arrows.
• Solving techniquesSingles, pairs, cage arithmetic, line logic and Latin squares, all in one guide.
• All puzzle guidesEvery puzzle type on Hoot’s Puzzles, with rules and tips.