Binary puzzle techniques: pairs, sandwiches, signs and counting
A binary puzzle asks you to fill a grid with just two symbols. In Sun & Moon, every cell gets a sun or a moon. Each row and column holds equal numbers of each, three and three on a 6 by 6 board and four and four on 8 by 8. No more than two of the same symbol may sit side by side, across or down. Cells joined by = match, and cells joined by × are opposite. The family is also known as Takuzu or Binairo. One rule that some binary puzzles use is missing here: two rows may be identical, so never reason that a row can't copy another.
How to read the examples
Each example is a row of six cells, written left to right, with ☀︎ for a sun, ☾ for a moon and _ for an empty cell. Cells are numbered 1 to 6 from the left, and signs are described in words. Everything works the same way down columns.
One row, start to finish
Here is how the basic rules chain together. Take the row ☀︎ ☀︎ _ ☾ _ _.
- Cells 1 and 2 are a pair of suns, so cell 3 must be a moon: ☀︎ ☀︎ ☾ ☾ _ _.
- Now cells 3 and 4 are a pair of moons, so cell 5 must be a sun: ☀︎ ☀︎ ☾ ☾ ☀︎ _.
- The row has its three suns, so cell 6 is a moon: ☀︎ ☀︎ ☾ ☾ ☀︎ ☾.
Two rules did all the work: pairs need guards, and a line stops at its quota. The sections below add the rest, one at a time.
Pairs need guards
Two of the same side by side need the opposite symbol at both ends, or there would be three in a row. _ ☀︎ ☀︎ _ _ _ becomes ☾ ☀︎ ☀︎ ☾ _ _. Pairs appear constantly as a board fills up, so check both ends of every new pair, across and down.
Fill the sandwich
Two of the same with one gap between them need the opposite in the middle. ☀︎ _ ☀︎ _ _ _ becomes ☀︎ ☾ ☀︎ _ _ _. Sandwiches hide well on a busy board, so look for them along every line that has a gap.
Finish a line at its quota
Once a line holds all its suns, every empty cell left in it is a moon, and the other way round. ☾ ☀︎ _ ☀︎ _ ☀︎ already has three suns, so it becomes ☾ ☀︎ ☾ ☀︎ ☾ ☀︎. Count before you hunt for patterns: a line two cells short of full is often settled by counting alone.
Follow the signs
When one end of a sign is filled, the other end follows: the same symbol across =, the opposite across ×. Signs help even before either end is filled. A × pair always holds one sun and one moon. An = pair is a pair in disguise, so it needs guards like any other pair.
Take _ ☀︎ _ _ _ ☾ with an = between cells 3 and 4. If that pair were two suns, cells 2 to 4 would be three suns in a row, so it is two moons: _ ☀︎ ☾ ☾ _ ☾. The moon pair needs a sun at cell 5, and the row now has its three moons, so cell 1 is a sun: ☀︎ ☀︎ ☾ ☾ ☀︎ ☾.
Count the × pairs
Because a × pair holds exactly one of each, you can count it before you know which way round it goes. Take ☀︎ _ _ _ _ _ with a × between cells 2 and 3 and another between cells 5 and 6. Each pair adds one sun, so together with cell 1 the row already has its three suns. Cell 4 must be a moon: ☀︎ _ _ ☾ _ _.
The same counting works with = pairs. In a row of six that already holds two moons, an = pair can't be two moons, because that would make four, so it must be two suns.
Find the only safe place
When a line needs just one more of a symbol, try it in each empty cell and see which choices avoid three in a row. ☀︎ _ ☾ _ _ ☀︎ needs one more sun and two more moons. A sun in cell 2 would leave cells 3 to 5 as three moons. A sun in cell 5 would leave cells 2 to 4 as three moons. Only cell 4 works: ☀︎ ☾ ☾ ☀︎ ☾ ☀︎.
Long gaps between two equal symbols are good places to try this. In ☾ _ _ _ _ ☾ the middle four cells hold three suns and one moon. If the moon sat in cell 2 or cell 5, the other three middle cells would be three suns side by side. So the moon is in cell 3 or 4, and cells 2 and 5 are suns: ☾ ☀︎ _ _ ☀︎ ☾.
It helps to know how few rows are possible at all. Only 14 different rows of six obey the balance and no-three rules, and only 34 rows of eight. A row of six with three cells filled never has more than three ways to finish, and often just one.
A short what-if
On the hardest boards the simple steps eventually run dry. Then pick a cell, suppose a symbol, and follow only the simple rules for a step or two. If a rule breaks, the cell must hold the other symbol.
Here is the idea on a single row, ☀︎ _ _ ☾ _ ☀︎. Suppose cell 2 is a sun. Cells 1 and 2 become a sun pair, so cell 3 is a moon, and the row now has its three suns, so cell 5 is a moon as well. That makes cells 3, 4 and 5 three moons in a row, which breaks a rule. So cell 2 is a moon, the sandwich of moons around cell 3 makes it a sun, and the quota fills cell 5: ☀︎ ☾ ☀︎ ☾ ☾ ☀︎. On a real board the chain usually runs from a row into a column and back. Choose a cell at the end of a sign or beside a pair, where a wrong guess breaks something quickly.
A routine for a whole board
Work in passes, and after every new symbol glance along its row and its column, because each one can create a pair or a sandwich:
- Fill the far end of every sign whose other end is known.
- Guard every pair and fill every sandwich, across and down.
- Complete any line that has reached its quota.
- Count × and = pairs in lines that are nearly balanced.
- Find the safe places in lines that need one more of a symbol.
- Only then, try a what-if.
A broken rule shows in red at once, so a slip never hides for long, and each puzzle has three free hints that give their reason in one sentence. Levels 1 to 300 are 6 by 6 and levels 301 to 400 are 8 by 8. The daily puzzle is a 6 by 6 board, easiest on Monday and hardest on Saturday. If you like this kind of reasoning, the same habits solve queens puzzles, and our history of binary puzzles explains where these grids came from.