The History of the Binary Puzzle: Binairo, Takuzu and Beacon Balance

Binairo in a Belgian newspaper, 2009

The firmest date in this history comes from the publisher. PeterFrank, a Belgian media content company, says it published its first puzzle of this kind in the summer of 2009, in what it calls Belgium’s largest newspaper. The company named the puzzle Binairo. PeterFrank was founded in 2000 as a partnership between the companies of Peter De Schepper and Frank Coussement, and the puzzle is usually credited to the two of them.

The rules were already the ones you play today. Fill the grid with zeros and ones. Every row and column gets as many of one as of the other. No more than two of the same digit may sit next to or under each other. No two rows and no two columns may be the same. PeterFrank also promised a single solution that could be reached by logic alone.

By the company’s own account the paper later dropped the puzzle, and readers complained until it came back. PeterFrank insists that Binairo is not a Sudoku derivative. The two share simple rules and a habit-forming quality, and not much else.

A second inventor, and a thin trail

Belgium is not the whole story. The puzzle archive kept by Angela and Otto Janko credits the Italian puzzle author Adolfo Zanellati with devising the same kind of puzzle under the name Tohu wa Vohu, a Hebrew phrase from the Book of Genesis. The puzzle publisher Conceptis repeats that credit and describes Binairo as an independent invention from about the same time. Simon Tatham’s puzzle collection also names Zanellati as the inventor.

None of these pages gives a date for Zanellati’s puzzle, so we can’t say which came first. Some websites mention still earlier versions, but we found no dated publication before 2009 that we could open and check. Tatham’s version, called Unruly, has only two rules: balance in every line and no triples. The rule against repeated lines comes from the Binairo side of the family.

One puzzle, many names

The puzzle spread quickly through Dutch and French puzzle pages, and it changed names as it went. Conceptis lists Dutch magazines printing it as bineros and Binaire puzzles, and notes that the French paper Le Figaro and the publisher Ludojeux call it Takuzu. Other names on record include Binary Puzzle, Binary Sudoku, Binoxxo, Zernero and Unruly. PeterFrank marks Binairo as a registered trademark, and other names belong to their publishers too.

The websites followed. BinaryPuzzle.com, which offers daily grids of zeros and ones from 6×6 to 14×14, carries a copyright notice that starts in 2011. Conceptis publishes the puzzle with Xs and Os as Tic-Tac-Logic and presents it as tic-tac-toe turned around for one player, where the aim is to avoid three in a row. Its pages don’t say when it began. In April 2018 the puzzle maker Jim Bumgardner added it to his KrazyDad site as Binox, after many requests from readers.

The mathematics: a hard problem in general

In July 2012 Marzio De Biasi wrote a paper titled Binary Puzzle is NP-complete. It shows that deciding whether a partly filled n×n grid has any valid solution is NP-complete. The proof translates a known hard problem about logical formulas, planar 3CNF satisfiability, into grids of zeros and ones. De Biasi also showed that the problem stays hard when only the no-triples rule is kept.

That result is about grids of unlimited size. It says nothing bad about a 6×6 daily puzzle, which a person solves in minutes with a few simple patterns. It does explain why the puzzle interests computer scientists. In 2017 Putranto Utomo and Rusydi Makarim compared three ways of solving it by machine: backtracking search, a satisfiability solver and polynomial equations over the binary field.

The puzzle has turned up in cryptography as well. A 2016 paper by Xavier Bultel and three colleagues describes how to convince someone that you know the solution to a Takuzu grid, by a physical procedure, without revealing any part of it.

Counting the grids

The On-Line Encyclopedia of Integer Sequences keeps a tally of finished grids. A line of six cells with three of each symbol and no triple can be written in exactly fourteen ways. Dave Langers pointed out in 2016 that an existing sequence, about paths that never take three steps in the same direction, counts these lines for every grid size.

Whole grids are counted in a sequence that Brian Kell added in December 2014. There are 72 finished 4×4 grids, 4,140 finished 6×6 grids and 4,111,116 at 8×8. The count for 12×12 was only added in 2026. Every Beacon Balance solution is one of those 4,140 grids. The fixed beacons and links of a daily puzzle are chosen so that exactly one of them fits.

Links between cells

For most of its life the puzzle gave you one kind of clue: cells already filled in. Daily versions have added a second kind, a small sign on the border between two neighboring cells. An equals sign says the two cells match. A cross, or in some versions a not-equals sign, says they differ.

The best-known example is the game known as Tango, which LinkedIn announced on October 7, 2024 as its fourth daily game. LinkedIn’s help page gives two rules: an equal number of each symbol in every row and column, and no more than two alike next to each other. Puzzles of the same type published by Grandmaster Puzzles add that cells separated by = hold the same symbol and cells separated by × hold different ones.

Thomas Snyder of Grandmaster Puzzles, who worked with LinkedIn on the design, wrote in January 2025 that binary puzzles were among the ideas the team considered. He credits the edge clues with making the puzzle deeper and easier to approach, and he says the rule against identical rows and columns was deliberately left out. We found no earlier dated source for these signs in a binary puzzle, so we can’t say where they first appeared.

Zeros and ones, Xs and Os, suns and moons

Nothing in the rules depends on the digits. The puzzle needs two symbols that are easy to tell apart, and publishers have chosen freely. Binairo and most printed versions use 0 and 1. Tic-Tac-Logic and Binox use X and O. Simon Tatham’s Unruly colors squares black or white, and the web game 0h h1 by Martin Kool of Q42 uses two colors of tile.

Suns and moons are a recent choice. Reports on LinkedIn’s game in October 2024 describe boxes that flip between a sun and a moon when tapped. The theme suits the puzzle, since day and night are a pair of opposites that everybody reads at a glance. Beacon Balance uses suns and moons too, lit as beacons on a signal grid.

How LarkSpark plays it

Beacon Balance is a 6×6 grid, the smallest size in common use, with one puzzle a day that is the same for everybody. It keeps all three classic rules: three suns and three moons in every row and column, never three alike in a line, and the unique-line rule that no two rows and no two columns may match. On top of those it adds = and ≠ links between some neighboring cells.

Fixed beacons carry a diamond mark and can’t be changed. You tap any other tile to cycle through sun, moon and blank, and conflicts show as you work. A hint places one beacon and adds fifteen seconds to your time. The rules page has the details, and the solving tips cover pairs, gaps, counting and links in the order you’ll need them.

The history of Beacon Balance