How to play Twos

Fill the grid with two symbols: never three alike in a line, an even split in every row and column, and the = and × signs obeyed.

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Twos is a pure deduction puzzle. There is no luck in it, nothing hidden, and no guessing — every grid is proved to have exactly one answer before it is dealt. Everything below follows from three rules.

The three rules of Binairo

No three in a line. Reading across a row or down a column, you may never see the same symbol three times consecutively. Two together is perfectly normal and in fact most answers are full of pairs; it is the third that is banned.

An even split. Every row and every column contains exactly half of each symbol — three and three on a six-wide grid, four and four on an eight, five and five on a ten. This is what turns a nearly-full line into a decided one.

The signs. Some neighbouring pairs carry a mark between them. An equals sign means those two squares hold the same symbol. A cross means they hold different symbols. A sign is not a hint you may ignore; it is a rule, and a grid that breaks one is wrong even if everything else looks tidy.

Why two symbols makes this easier than it sounds

Every square has exactly two possible values. That sounds trivial and it changes the character of the whole puzzle: ruling one option out settles the square outright. There is no pencil-marking a list of candidates and whittling it down, because the list is only ever two long.

The consequence is that almost every deduction here is local and short. You look at two squares side by side that match, and you know their neighbours on both sides. You look at a gap between two matching squares, and you know what fills it. You count a row, find it already has all the circles it is allowed, and the rest of the row is squares. None of that requires holding a chain of reasoning in your head, which is why the puzzle works so well in short sittings.

The three deductions that do nearly all the work

A pair forces its neighbours. If two adjacent squares carry the same symbol, the squares on either end of that pair must carry the other one, or you would have three in a row.

A gap between two alike is forced. If two squares of the same symbol sit with exactly one square between them, that square must be the other symbol, for the same reason.

A line that is half done is finished. Once a row or column holds every one of a symbol it is allowed, every remaining square in it is the other symbol. This is the deduction people forget to look for, and on the bigger grids it is usually what breaks the board open.

What the difficulty labels mean

Each grid is rated by the hardest technique it genuinely forces. "Forces" is tested the hard way: the technique is switched off and the grid is re-solved to see whether it still falls. A technique that merely happens to fire on the way does not raise the rating.

Easy grids are settled by the three deductions above and the signs. Nothing more is needed.

Medium grids force a look ahead. At some point neither the pairs nor the counts give you anything, and the way through is to ask what a square would do to its own line: put a circle here, and the row can no longer be completed at all under the rules. That rules the circle out. It is not quite a proof by contradiction — you are only looking at one line, one step ahead — but it is the first technique that asks you to think about a square you cannot yet fill.

Hard and Expert grids force a genuine assumption. Suppose a square is a circle, follow the consequences, and find something that breaks — a line that can no longer come out even, or three in a row you cannot avoid. That rules the circle out and you carry on. Expert grids are not harder than Hard ones in kind; they are the same demand on a larger grid, which is the honest way to describe them.

The guarantee

Every grid starts as a complete, legal answer. Clues are then taken away one at a time in random order, and after each removal the grid is re-checked to confirm exactly one answer still fits — anything whose removal would allow a second answer is put straight back. What you receive is a grid that has been proved unique and proved reachable by the techniques above.

So if you are stuck, something is deducible. There is no point in any grid where two answers remain and you have to choose.

Hints, mistakes and the daily grid

Mistakes show immediately. Three in a row, a line with too many of one symbol, or a broken sign all turn red the moment they happen, and the Errors counter tells you how many squares are involved. Nothing is ever silently wrong.

A hint reads the board as you have actually left it and follows the line you are on. If you have filled a square that cannot be part of the answer, the hint says so and marks it rather than building advice on top of the mistake. Otherwise it names the next forced square and explains which rule forces it.

One grid a day, the same for everybody. The week ramps: Monday and Tuesday are easy, Wednesday and Thursday medium, Friday and Saturday hard, and Sunday is an expert grid. Solving extends your streak, solves finished without a hint are counted separately, and the share text carries your time without giving anything away. You can also send someone the exact grid you just played rather than a fresh one.

Controls

Tap a square to place a circle, tap it again for a square, and once more to clear it. Squares that arrived filled in are locked and will not change.

Keyboard

Arrow keys move the cursor around the grid.
Space or Enter cycles the square under the cursor.
Z undoes, H asks for a hint, R clears your working.

Strategy tips

Look for pairs before anything else. Two alike side by side immediately settle the squares at both ends, and on a fresh grid there are usually several waiting.
Then look for gaps. Two alike with a single square between them force that square, and this pattern is easy to miss because your eye reads the gap as empty rather than as a clue.
Work the signs early. A = or × next to a square that is already filled settles its partner for free, and settling that partner often starts a chain.
Count every line that looks nearly full. The moment a row holds every circle it is allowed, the rest of it is squares — this is the single most under-used deduction in the game.
Remember the columns. Almost everyone works across the rows and forgets that each column is under exactly the same two constraints, and stalled grids are very often unstuck by counting a column.
When nothing is forced, pick a square and ask what its own line would do. If one of the two symbols leaves that row or column with no legal way to finish, the answer is the other one.
On the hardest grids, assume and chase. Choose a square, suppose a symbol, and follow the consequences until either the grid breaks — in which case you have your answer — or you run out of consequences, in which case try elsewhere.
Do not guess to get unstuck. Every grid here is proved to have one answer reachable by reasoning, so a guess is never necessary and a wrong one costs more time than the deduction would have.