Halves strategy: pairs, gaps, counting and unique lines

The five techniques behind every Halves puzzle, from the pairs and gaps that finish Easy grids to the counting and line comparisons that Hard ones need, with worked examples.

Halves has four rules and only two symbols, yet a 10×10 grid can hold you up for a good while. The good news is that every step in every puzzle comes from one of five techniques, and the difficulty levels introduce them in a fixed order. This guide goes through them level by level, using the same names as the hints, so that when a hint says counting or unique lines you know exactly what it means.

The rules, and what they really say

Every square holds a disc or a ring. On top of that:

No more than two of the same symbol may sit side by side in a row or column.
Every row and every column has as many discs as rings.
No two rows are identical, and no two columns are identical.

Each rule turns into a technique. The first rule gives you pairs and gaps, the second gives you counting, and the third gives you unique lines. Advanced counting combines the first two. Keep the line length in mind: an Easy grid is 6×6, so each line holds three of each symbol; Medium is 8×8, with four of each; Hard is 10×10, with five of each.

The difficulty label tells you which rule you will lean on hardest, not how many squares are empty. It is set by the hardest technique our step-by-step solver needed to finish the puzzle, and every published puzzle has exactly one solution that can be reached without guessing. The controls are on the How to play page.

What Easy puzzles need: pairs and gaps

Every Easy puzzle can be finished with just two techniques. Both come from the rule against three alike in a row, and both are about a single empty square next to symbols that are already there.

Pairs

When two identical symbols sit side by side, the squares at either end of them must hold the other symbol. A third disc next to two discs would make three in a row, so the only thing that can go there is a ring.

To spot pairs quickly, run your eye along each line looking for two of the same touching, then glance at both ends. A pair against the edge of the grid has only one open end, so it gives you one square instead of two.

Gaps

When two identical symbols have exactly one empty square between them, that square must hold the other symbol, for the same reason: filling it with the same symbol would join the three. Gaps are easy to overlook because the two symbols don’t touch. Train yourself to see the pattern disc, empty, disc as a single shape.

Square 2 sits in a gap between two discs, so it must be a ring. Squares 5 and 6 are a pair of rings, so square 4, next to them, must be a disc.

Follow the chain

Pairs and gaps feed each other. Every symbol you add sits in one row and one column, and it may have just completed a new pair or gap in either. The most efficient habit on an Easy grid is to check both lines through each new symbol straight away, rather than going back to scanning from the top. A single printed pair can unzip a whole corner of the grid this way.

The printed pair of discs in the top row forces rings at both ends, shown in blue. The new ring in column 4 now sits two squares above a printed ring, so the highlighted square between them must be a disc.

Easy puzzles never require counting or comparing lines, although nothing stops you using them. If you find yourself stuck on an Easy grid, there is a pair or gap somewhere that you have missed, often in a column.

What Medium puzzles need: counting

Medium grids are 8×8, with a smaller share of their squares printed than on Easy, and at some point the pairs and gaps run dry. The rule that takes over is the balance rule: each line holds exactly half of each symbol.

Counting is the simplest use of it. When a line already has all its discs (four, in an 8×8 grid), every empty square left in that line must be a ring, and the other way round. It doesn’t matter where the empty squares are or what surrounds them.

This row of eight already holds its four discs, so all three empty squares must be rings. No pair or gap applies to them; only counting gives them away.

The tip for spotting it is to count the symbol you have more of. A line with four discs and one ring is settled, while a line with three discs and two rings is not, even though both have three empty squares. Lines that are more than half full deserve a count every time you pass them.

Counting and pairs work well together. Filling a line by counting often creates new pairs and gaps across it, in the columns if you counted a row and in the rows if you counted a column. After each count, go back to the pair-and-gap routine before counting again.

What Hard puzzles need: advanced counting and unique lines

Hard grids are 10×10, with only about a fifth to a quarter of the squares printed. Every Hard puzzle needs at least one of two techniques that look at a whole line at once. Neither is difficult once you know what to look for, but both are easy to miss if you only ever scan for local patterns.

Advanced counting

This applies to a line that needs exactly one more of a symbol, say one more ring. That ring has to go in one of the empty squares, and every other empty square in the line will then be a disc. So try each empty square in turn: imagine the ring there and discs everywhere else. If that produces three discs in a row, the ring can’t go in that square, which means that square is a disc.

In practice you rarely need to try every square. Look for a stretch that would break the rule if it were filled entirely with discs: three empty squares in a row, or two empty squares next to a disc. The missing ring has to go somewhere in that stretch, so every empty square outside it is a disc.

This row of ten has four of its five rings, so it needs exactly one more. If that ring went in the last square, the three empty squares in the middle would all be discs, three in a row. So the last square must be a disc, and the missing ring is somewhere in the middle three.

Notice what the example doesn’t tell you: which of the middle three squares holds the ring. Advanced counting often settles only one square at a time, and that is enough. The new disc may complete a pair or gap in its column, and the chain starts again.

The tip is to watch for lines that are one short of half. Those are the lines where counting nearly works, and advanced counting is what you do instead.

Unique lines

The third rule says no two rows are the same and no two columns are the same. The hints use it in one precise situation: a line with exactly two empty squares that needs one disc and one ring. There are only two ways to fill it, and if one of those ways would make it an exact copy of a line that is already finished, it must be the other way.

Two rows from the same 10×10 grid, placed together for comparison. The lower row needs one disc and one ring. A disc in square 4 and a ring in square 9 would copy the finished upper row, so it must be the other way round: a ring in square 4 and a disc in square 9.

Both orders are fine as far as the lower row’s own rules go, which is why neither pairs nor counting can decide it. Only the comparison with another line does.

To find these, list the lines that are down to their last two squares with one of each symbol missing. For each, compare its filled squares with the finished lines running the same way. Only rows are compared with rows and columns with columns; a row may perfectly well read the same as a column. You can also turn the search around: whenever you finish a line, look for any line in the same direction that matches it in all but two places.

Common mistakes

Being stricter than the rules. Two alike side by side are allowed. The rule forbids three, so a pair is a clue, not a mistake.
Forgetting the columns. Most people scan rows naturally and columns only as an afterthought. Stalls on Easy grids are usually a missed pair or gap running down the page.
Using the wrong half. A line holds three of each symbol on Easy, four on Medium and five on Hard. Moving between sizes, it is easy to count to the wrong number.
Comparing with an unfinished line. Unique lines only works against a line that is complete. If the other line still has empty squares, it may end up different, and nothing follows.
Filling a square to see what happens. A guess can sit on the board for many moves before it causes trouble. If your board already contains a wrong entry, the hint tells you so rather than building on it, and Check shows which entries don’t match the solution.

A good solving routine

The Hint button always offers the simplest step available, trying pairs first, then gaps, counting, advanced counting and unique lines, in that order. The same order makes a good routine at the table:

Sweep every row, then every column, for pairs and gaps. Follow each new symbol into its row and column.
When that dries up, look for lines that already have half of one symbol, and fill the rest by counting.
Next, look for lines one short of half and try advanced counting on them.
Then check lines with two empty squares against finished lines in the same direction.
As soon as any of these gives you a symbol, go back to step 1. The easy techniques do most of the work, even on Hard grids.

Near the end of a puzzle, counting is usually the quickest tool: once most lines are well filled, a quick count often settles several squares at once. If you are new to logic puzzles in general, the getting started guide covers how to use hints without spoiling a puzzle. Otherwise, pick an Easy grid on the Halves page and practise spotting gaps until you see them without looking.