How to solve a binary puzzle (dots and rings)
Binary puzzles are also known as Takuzu or Binairo. You fill a grid with two symbols, following a few simple rules. WitRally's daily puzzle uses dots and rings; on other sites you will see 0 and 1 or black and white. The method is the same.
The three rules
- Balance. Every row and every column has the same number of dots and rings. In a 6×6 grid that is three of each; in an 8×8 grid, four of each.
- No three. You may not put three of the same symbol next to each other in a row or column. Two in a row is fine.
- All different. No two rows are identical, and no two columns are identical.
Each WitRally puzzle has exactly one solution, and it can be reached by logic alone. You never need to guess.
Step-by-step techniques
1. Pairs. If two dots sit side by side, the squares on either side must be rings, otherwise you would have three dots. The same goes for two rings.
2. Gaps. If you see dot, empty, dot, the empty square must be a ring. Putting a dot there would make three in a row.
3. Counting. When a row already holds all the dots it is allowed, every empty square left in it must be a ring, and the other way round. For example, in a 6×6 row that reads dot, dot, ring, ring, empty, empty, each symbol has two so far, so the last two squares need one dot and one ring. The no-three rule then decides the order.
4. Look for what would break a row. If putting a ring in a square would force three rings together or give a row too many rings, it must be a dot. Try each option in your head for just one step.
5. Compare rows. This one matters late in the puzzle. If a row is complete and another row is nearly the same with only two squares empty, those two squares must differ from the complete row so that the rows are not identical. Do the same for columns.
6. Repeat. Each square you fill gives new information to the row and column that cross it. Work in loops across the whole grid.
If you get stuck
- Look for three in a row you may have missed, or a row with too many of one symbol. A mistake early on can block you later.
- In WitRally, a rule you break lights up: the squares with the problem get a red outline and the matching rule highlights. Use that to find the slip.
- Tap an empty square to place a dot, tap again for a ring, and again to clear it. Squares given at the start are locked.
- A hint is available but adds 15 seconds to your time, because the clock matters on the team leaderboard.
Practise with a real puzzle
A new puzzle appears every day. Play today's puzzle or start a team and see how you compare with your colleagues. For the idea behind the game, read a daily puzzle for your team chat.