How to solve Binary Diagonals
Binary Diagonals is a binary puzzle — zeros and ones, the same number of each in every row and every column, never three alike one after another — with one thing added: the two long diagonals, corner to corner, are lines as well. Each holds as many zeros as ones, and three alike may not follow one another along it.
Two lines more does not sound like much, and it changes the whole board: a diagonal crosses every row and every column, so a mark it settles is a mark handed to both. Most boards here cannot be finished by somebody who forgets the diagonals are there. This page walks through the shapes to look for, simplest first, each on a small board you can step through at your own pace. The boards are drawn the way the screen draws them — a zero is a hollow square, a one a filled square; on a printed sheet the same two are the digits 0 and 1.
How to play
Fill every empty cell with a hollow square or a filled one, so that each row, each column and each of the two long diagonals — the lines drawn corner to corner — holds as many of one as of the other.
Never three alike one after another — not in a row, not in a column, and not along either of those two diagonals.
Tap a cell for a filled square, tap again for a hollow one, once more to clear it. The squares the board came with stay put: tap one and they all give a shake, to show which they are.
Shown in red: three of the same kind one after another, and squares a row, column or diagonal has too many of.
A few squares are given at the start and those stay put; the rest of the cells are empty, and every one of them is yours to fill. Only the two long diagonals count — the short ones beside them answer to nothing. The side of the board is even, so the two never share a cell; they pass each other in the four cells at the centre.
Every board has exactly one answer, and it can be reached without guessing. If you know Binary you know its third rule, that no two rows and no two columns read the same: it holds here too, and no board needs it.
Pairs, gaps and counting
Three moves do most of the work in any row or column, and all three come straight from the rules: two alike side by side close the cells either side of them, two alike with one cell between them close that cell, and a line that already has all of one kind takes the other everywhere else.
A pair on a diagonal
Along a marked diagonal, the cell “next door” is the one that touches corner to corner. Two alike in neighbouring cells of a diagonal are a pair like any other, and they close the diagonal’s next cell on either side.
A gap on a diagonal
Two alike with one diagonal cell between them leave that cell no choice, exactly as in a row. And whatever a diagonal settles lands in a row and a column too, where it often starts the next move.
Counting a diagonal
A diagonal is as long as a row, and it is held to the same balance: on an 8 × 8, four hollow squares and four filled. Once one kind has its four, every other cell of the diagonal is the other.
Reading a whole diagonal
When no pair, gap or count is left, take one line and ask what it can still be. It works on any row or column, and the larger boards ask for it a few times each; the diagonals are the lines people forget to read.
No timer. Leave and come back — it’s saved.
Play today’s Binary Diagonals