These are the techniques our solver uses, in order of difficulty. The hint button uses the same ideas and explains each step in the same words.
1. Sign chains
Signs pass information along. If A = B and B × C, then A and C are opposite, even before you know any of them. When a chain reaches a known symbol, fill the whole chain at once.
2. Pairs and gaps
- A pair of the same symbol forces the opposite symbol at both ends.
- A gap between two of the same symbol must be the opposite.
Check every row and column for these after each move. They're behind most deductions in any grid.
3. Signs and the no-three rule together
An = sign means a pair, even while both cells are empty. So the cells on either side of an = pair must both be the opposite of whatever the pair turns out to be. If one of those neighbours is known, the pair is known.
4. Counting a line
A 6×6 line holds three of each symbol. Once one symbol reaches three, fill the rest with the other. Near the end, count what's left: if a row needs one more sun and two more moons, and one of the empty cells would make three moons in a row, it must be the sun.
5. What fits this row?
The technique for hard grids: take one row or column and list the ways it could still be completed. There are only 14 legal ways to fill a 6-cell line with three of each and no triples, and the cells and signs you already have usually rule out most of them. Any cell that's the same in every remaining option is solved.
A quick version: if a row needs two suns and two moons in four empty cells, and two of those cells are joined by ×, then that pair takes one of each, so the other two cells also hold one of each. Combine that with a pair or gap and the row often falls.
6. Expert grids (8×8)
The same ideas work on 8×8, with four of each symbol per line. Lines have more freedom, so look harder for sign chains and for rows that are nearly full. Start where the information is densest.