Daily Binairo
Rules
Click on cells and fill the board with white or black circles. To choose the correct color follow these rules:
- No more than two of the same colour next to each other
- Each row and column must have the same number of coloured circles
- In the non-Plus version no two rows or columns must be the same
- In 'Plus' a '=' symbol between two cells means these two cells are the same
- In 'Plus' a 'X' symbol between two cells means these two cells must be different
Comments
... by: FHessel
today's Binairo (non-Plus #352/3 star) is the first puzzle, that I have seen (maybe the first ever?), that is a bit harder to start:
First you apply the two basic rules (max 2 next to each other, same number in rows/columns).
Usually at this point (or earlier) the "diagonal-rule" is applicable (see my comment of 09-14-2025 below Binairo-Plus), and consequently it is easy to solve.
Today is different: the "diagonal-rule" cannot be used, since there is still the possibility of a continuous diagonal-up formation. Another means of deciding the next step is necessary.
That requires a second look:
Rows 2 and 3 and col 3 exclude the possibility of 5 black.
Cols 1 and 5 exclude the possibility of 3 black.
From there on it is a known field again.
That qualifies the puzzle imho for an additional star.
... by: FHessel
May I make a suggestion?
As I noted below the 8x8 Binairo Plus, the 8x8 binairos have an important disadvantage: there is almost no chance, that an overall number of 3 or 5 blacks will occur, and if it does, then it can be recognized qute fast. Consequently the number of black cells is not really a degree of freedom, as it should be based on the concept of the puzzle.
Using 9x9 instead would change the whole picture.
In numbers...
8x8:
2 blacks - 1 possible pattern - 0 valid boards
3 blacks - 16 possible patterns - 16 valid boards
4 blacks - 34 possible patterns - 4.111.116 valid boards
5 blacks - 16 possible patterns - 16 valid boards
6 blacks - 1 possible pattern - 0 valid boards
9x9:
3 blacks - 10 possible patterns - 0 valid boards
4 blacks - 45 possible patterns - 5.116.926 valid boards
5 blacks - 45 possible patterns - 5.116.926 valid boards
6 blacks - 10 possible patterns - 0 valid boards
Using 9x9 would not only more than double the possible number of puzzles, it would also introduce a real choice between 4 blacks vs 5 blacks, each having equal probability.
This could open the way for harder puzzles which not only @Thinkist hopes for.
WB
BW
or
BW
WB
except when swapped, there are (1) still no more than 2-in-a-row for W or B, and (2) still equal numbers of W and B in a given row or column.
Also, one of standard Binairo's rules actually requires the grid size to be even, so 9×9 wouldn't work, though 10×10 (12×12, etc.) would.
Really all this is why I (and others, actually) kinda suggested something like a "weekly diabolical" or the like.
In addition, the "X" and/or "=" symbols are redundant very often too.
@Andrew, could you revise the grading mechanism accordingly?
... by: Denk-mal
Thanks for that.
The only thing I would like to see is that the check would only shows IF something is wrong and not showing WHAT is incorrect on the field.
... by: Thinkist
And so, my request: some harder puzzles that use these (Weekly "Extreme", perhaps?). Also, no Hint button for these either.
Also, a fun combinatorics question for anyone reading this: how many possible Binairo solutions (i.e. solved grids) are there for a given 2n by 2n grid (NB this does not even begin to touch on the number of possible puzzles...).
I will have a look at the code again, maybe there is something I missed.
I do know that the regular Binario should have the rule about no rows or columns being identical but that this is not a rule in Binario+. By that I mean the generator enforces the no-duplicate rows/columns (I think!) but it is not a necessary rule to solve the puzzle. It’s very niche so I don’t think people will be up in arms either way
Not sure how to turbo charge the creator to make extremes. I don’t think the spectrum is wide enough or has a long tail like Sudoku. Or maybe I can change the scoring to do that.
Is a good combinatorics question!
- [Del]
... by: nono
... by: Thinkist
This reminds me of the many like puzzles in this game called Islands of Insight (https://store.steampowered.com/app/2071500/Islands_of_Insight/), where many different rules can be applied to make many different kinds of black-and-white puzzles in this game's Logic Grids.
For example, Binairo can generally be coded in using a couple of "Don't make this pattern" rules and some Darts.
Anyway, back to the webpages. For those using a mouse: left-click to add black, right-click to add white. Both options cycle between black, white, and empty upon subsequent clicks. I mention this because these precise details aren't exactly in the Rules. Also, in Islands of Insight, one could simply left-click to add black, and instantly right-click to change it to white without the need to erase a cell (or vice versa), and it would be nice to have that behavior here as well.