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Strategies for Popular Number Puzzles

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
Guaranteed a single solution! Now with Print option




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CommentsTalk

... by: FHessel

Monday 22-Sep-2025
Hi Andrew!

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.
Thinkist replies:Tuesday 23-Sep-2025
@FHessel, cool math (I saw your other comment btw), and thanks for noticing, though I was mostly noting that in Andrew's Binairo puzzles that the "must have unique content in rows or columns" is rarely, if ever, invoked. In solving another Binairo puzzle elsewhere though, I noticed a "unique rectangle" type of situation that only this rule could fix. It's something like

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.
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... by: Denk-mal

Sunday 24-Aug-2025
I really love that new puzzles. It is also a very good base for education on solving (game) problems and programming those solution.
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.
Andrew Stuart writes:
Good idea. New "check" options in the settings - see the "cog" icon
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... by: Thinkist

Friday 1-Aug-2025
It's now been over 20 (well, 240) Binairo puzzles (such a strange name too, with no immediately intuitive way to say it...), and while I'm loving the flow state of solving these daily (reeeeeaaally taking me back to Islands of Insight...), I've noticed that the extra rules aren't typically needed, especially in the regular Binairo, even at Level 3.

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...).
Andrew Stuart writes:
I’m glad you’re enjoying them

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!
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... by: nono

Wednesday 16-Jul-2025
We don't play online. We print the grids and fill them with zeros and 1s. How do I do this please ?
Andrew Stuart writes:
I will add a print option shortly.
nono replies:Monday 28-Jul-2025
But the black circles are not printed. Is it perhaps possible to replace the circles with 0 for white and 1 for black colors ? Many thanks.
Andrew Stuart writes:
Another option is to tick "Print Background Images" on the printer options at the preview just before printing
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... by: Thinkist

Wednesday 16-Jul-2025
First I guess. Also: woo! Cool to see this puzzle on here!

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.
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Article created on 14-July-2025. Views: 17078
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