Binary Puzzle Rules

Three rules, and every one of them earns its place. Here is what each forbids, shown on a single line of six cells rather than buried in a full grid.

Rule one: never three in a row

No three identical symbols may sit next to each other in any row or any column. Two is fine and is in fact necessary — a line of alternating symbols is legal but rare — and three is not, anywhere, in either direction.

This is the rule that does most of the early work, because it acts locally. Two cells you can see at once are enough to force a third, which is why the first technique anybody learns comes straight out of this rule and nothing else. It is also the rule that makes a binary puzzle feel unlike a Sudoku: constraints propagate along lines rather than jumping between boxes.

It cuts far more than it looks like it does. There are 3,432 ways to put seven of each symbol into a line of fourteen, and only 518 of them survive this rule — about one in seven. Every one of those three thousand eliminations is available to you, for free, before you have counted anything.

Rule two: half and half, in every line

Every row and every column contains exactly as many of one symbol as of the other. In a 6×6 that is three each, in a 14×14 seven each. This is why the grid is always an even size: an odd line cannot be split in half.

The rule is global where the first one is local. It says nothing about any particular cell and everything about the line as a whole, which is why counting feels like a different kind of move — you stop looking at neighbours and start taking inventory. The moment a line has its full quota of one symbol, every remaining cell in it is decided at once, sometimes five or six cells from a single observation.

It is also the rule most often broken by accident, because nothing looks wrong at the time. Three cells filled with the same symbol at opposite ends of a long row do not catch the eye the way three adjacent ones do. The error highlighter here will show it if you turn it on, and it is off by default, because a great many players would rather find their own mistakes.

Rule three: no two lines alike

No two rows may be identical, and no two columns may be identical. A row may of course match a column — they are different families and the rule does not compare across them.

This is the rule beginners are told about and then never use, and that is a fair description of how it behaves: it is nearly useless early and decisive late. It cannot say anything until at least one line is completely full, because two half-written rows that agree so far are not yet twins and may never become them. Once a line is finished, though, it forbids that exact arrangement to every other line of its kind for the rest of the puzzle.

On this site it is what separates expert from hard. An expert grid contains at least one deduction that is available only because some other line already used the arrangement being considered — and nothing else in the puzzle will get you past it.

Four things the rules do not say

They do not say the two symbols alternate. Pairs are not merely allowed, they are unavoidable: a line of six with three of each and no triples cannot be strictly alternating more than twice over.

They do not say anything about diagonals. A diagonal of six identical symbols is perfectly legal and turns up regularly. Players arriving from other logic puzzles look for a diagonal rule for about a week.

They do not balance the grid as a whole beyond what the lines already force, and they say nothing about any block or region. There are no boxes in this puzzle; the only structures are rows and columns.

And they do not permit guessing. A published puzzle here is checked to be finishable by reasoning alone, so if nothing will place, something has been missed rather than something being required. That is a promise about the generator, and it is the reason the generator exists at all.

Common questions

Everything else on the site

• Daily puzzleA new grid every day
• Binary puzzleUnlimited grids
• How to solveA 6×6, worked
• TechniquesThe whole ladder
• Doubles and sandwichesTechnique — easy
• Counting a lineTechnique — medium
• Line analysisTechnique — hard
• Twin eliminationTechnique — expert
• The assumptionTechnique — refused
• Printable puzzlesFor paper
• Daily archiveEvery past day