The history of Balance

Most classic logic puzzles are decades old. The binary puzzle is a newcomer, barely older than the smartphone, and it took off because it is so easy to learn and so pleasant to finish in a few minutes.

A puzzle in two symbols

The idea is as old as binary numbers themselves: a grid of 0s and 1s, with rules strict enough that only one arrangement fits. As a published puzzle, though, it's recent. It's usually credited to two Belgian puzzle makers, Frank Coussement and Peter De Schepper, who began publishing it around 2009.

It spread quickly through puzzle magazines and books in the Low Countries, and then further afield under other names: in many collections simply as the binary puzzle or binary Sudoku, or by the names of the apps that carried it onto phones in the 2010s. Some versions use 0 and 1, some use two colours, and ours uses dots and rings.

Why it caught on

The rules fit in one breath, there's no arithmetic, and a small grid takes only a few minutes. Yet the logic is real: on larger grids, with the no-identical-lines rule switched on, a well-made puzzle can hold an experienced solver for a good while. That range — a warm-up on a 6×6, a proper workout on a 12×12 — is why it's the opening game of our daily edition.

Some numbers

Two symbols sounds like very little to work with. The rules cut it down much further:

• A row of 6 cells can be filled in 64 ways with two symbols, but only14 of them have three of each and no three alike in a row. For 8 cells it's 34, and for 12 cells 208.
• There are exactly 11,222 ways to fill a whole 6×6 grid by the basic rules. Add the rule that no two rows and no two columns may match, and only 4,140 remain.

Like most of the puzzles here, the general problem is hard for computers: solving binary puzzles of any size has been shown to be NP-complete, so no fast method is known that works for every grid. At the sizes people actually play, a program solves one in a blink — the craft is in making puzzles that are fair to a human.

Variations

• Extra signs between neighbouring cells, saying whether the two match or differ. They let a puzzle give fewer symbols away.
• No identical lines, or not. Some versions always use this rule; others leave it out on smaller grids. Ours keeps it for Hard.
• Bigger grids of 14×14 and beyond, for long sessions.