Rules, solutions and levels

What a binary puzzle is

A binary puzzle is a square grid, in Nuleen 6×6, 8×8 or 10×10, partly filled with 0s and 1s. The given digits are the clues. You fill in the rest so that three rules hold:

Balance: every row and every column has as many 0s as 1s.
No three in a row: never three equal digits next to each other, across or down.
Unique lines: no two rows are the same, and no two columns are the same.

A good puzzle has exactly one solution, and it can be reached by reasoning alone, without guessing.

One solution, up to sixteen look-alikes

A solved grid can be turned and flipped without breaking any rule:

The 8 symmetries of the square: the grid as it is, turned a quarter, half or three quarters, and mirrored left-right, top-bottom, or along either diagonal.
The binary switch: swapping every 0 for a 1 and every 1 for a 0.

Combined, these are 16 transformations. Applied to one solution they give up to 16 grids that look different but are really the same puzzle, with the same reasoning in the same places. Together those grids form the solution's symmetry family.

Not every family has 16 members. If a grid has symmetry of its own, some transformations give a grid already in the family:

If 2 of the 16 transformations leave a grid unchanged, its family has 8 members.
If 4 do, the family has 4 members.
If 8 do, the family has 2 members.

A family never has just 1 member: the binary switch always changes a grid.

Nuleen treats all members of a family as the same puzzle for scores and favorites. The transform buttons under the board let you look at any member.

How many solutions there are

The number of valid lines on its own is small. Lines with equal 0s and 1s and no three equal digits in a row:

Size	Valid lines
6×6	14
8×8	34
10×10	84

Complete solutions (all three rules, rows and columns), grouped by family size:

Size	Solutions	Families	16 members	8 members	4 members	2 members
6×6	4,140	321	203	107	7	4
8×8	4,111,116	259,400	254,541	4,764	79	16
10×10	48,183,195,384	3,011,899,016	3,011,001,482	895,438	1,988	108

The same, counted in solutions (number of families × members):

Size	In 16-member families	In 8-member	In 4-member	In 2-member
6×6	3,248	856	28	8
8×8	4,072,656	38,112	316	32
10×10	48,176,023,712	7,163,504	7,952	216

How the figures were obtained:

6×6 and 8×8: every solution was generated and its family size determined.
10×10: the total of 48,183,195,384 solutions is from the On-Line Encyclopedia of Integer Sequences (OEIS A253316). The 8-, 4- and 2-member families were counted directly, by building only the grids that some transformation leaves unchanged. The 16-member families are the remainder.

Rare puzzles. The families of 2 and 4 members are rare: 11 of 321 families for 6×6, 95 of 259,400 for 8×8, and 2,096 of about 3 billion for 10×10. Puzzle type Rare in Nuleen plays only these. Every grid in them is unchanged by a half turn.

How puzzles are made

Nuleen carries no puzzle collection: every puzzle is made on your iPhone when you start it.

Pick a solution.
- Random: a valid grid is built from scratch, with random choices.
- Rare: these grids are too scarce to find by chance (2,096 of about 3 billion families for 10×10), so they are constructed. This is done once, in the background after the first launch, and takes about 34 KB of storage. A rare puzzle then picks a family size (2 or 4, at even odds), a family, and one of the 16 transformations, all at random.
Remove clues. Clues are taken out one at a time, in random order. A removal is kept only if the puzzle still has exactly one solution and can still be solved with the techniques allowed for the chosen level. Digging stops at a floor of clues for the level.

Which clues remain differs every time, so one solution gives many different puzzles. For scale: a 10×10 puzzle with 25 clues could have its clues in about 2.4 × 10²³ sets of positions. Only a small fraction of those sets gives a unique solution, but that fraction is still far more puzzles than there are solutions.

Making puzzles yourself works the other way round: on the Enter puzzle screen you add clues until the solution is unique, then remove clues to make it harder. See Your own puzzles, scores and favorites.

Solving techniques

The techniques, from simplest to hardest. The examples use one row of a 6×6 puzzle; a dot is an empty cell. Everything works the same way for columns.

Local rules

The rules applied directly to what is in a line:

Two equal digits side by side: the cells on both sides get the other digit.
```
. 0 0 .   →   1 0 0 1
```
Two equal digits with one empty cell between them: the middle gets the other digit.
```
1 . 1   →   1 0 1
```
A line that already has all its 1s (or all its 0s): the empty cells get the other digit.
```
1 0 1 . 1 .   →   1 0 1 0 1 0      (three 1s already: the rest are 0)
```

Duplicate avoidance

A line with two empty cells left that need one 0 and one 1 can be completed in two ways. If one of them would copy a line that is already complete, the other one is right. (When the two cells need two equal digits, the local rules already fill them.)

complete row:   0 1 1 0 1 0
this row:       0 1 1 0 . .    needs one 0 and one 1:
                               0 1 1 0 1 0 would copy the complete row
                →  0 1 1 0 0 1

Line enumeration

List every way a line can still be completed so that it is balanced and has no three in a row. A cell that has the same digit in all of them can be filled in, even when no local rule applies.

this row:          0 . . . . 0     needs one more 0 and three 1s
the 0 in cell 2:   0 0 1 1 1 0     three 1s in a row: no
the 0 in cell 3:   0 1 0 1 1 0     valid
the 0 in cell 4:   0 1 1 0 1 0     valid
the 0 in cell 5:   0 1 1 1 0 0     three 1s in a row: no
both valid ways have a 1 in cells 2 and 5   →   0 1 . . 1 0

Line enumeration with duplicate avoidance

The same listing, also leaving out every completion that would copy a complete line, however many cells are still empty. It combines the two techniques above.

complete row:   0 1 1 0 1 0
this row:       . . 1 . . 0
valid ways:     0 1 1 0 1 0   copies the complete row: no
                1 0 1 0 1 0
                1 0 1 1 0 0
the two left agree on cells 1 and 2   →   1 0 1 . . 0

Without leaving out the copy, the three ways agree on no empty cell, so plain line enumeration finds nothing here.

Lookahead (trial and error)

Try a digit in a cell and follow what the simpler techniques then fill in. If that leads to a broken rule, the cell must hold the other digit. Unlike the techniques above, this can involve many lines at once.

The star levels

Level	Techniques that may be needed	Clue floor
★	Local rules only	30%
★★	Local rules and duplicate avoidance	25%
★★★	Up to line enumeration; the local rules run out at most 3 times	27%
★★★★	Up to line enumeration; the local rules run out at most 4 times	25%
★★★★★	Line enumeration with duplicate avoidance, needed at least once	20%

No level needs lookahead.

Puzzles from other sources. A puzzle entered on the Enter puzzle screen gets an estimate in outline stars (☆☆☆) for the level it most resembles, based on the hardest technique it needs. A puzzle that needs more than line enumeration with duplicate avoidance (lookahead) is marked ☆☆☆☆☆+.

Tips for learning

Start at the lowest level and work up:

★ teaches the local rules.
★★ may need duplicate avoidance.
★★★ and ★★★★ need line enumeration and duplicate avoidance, each applied on its own. They differ in the number of clues and in how often the local rules run out before the puzzle is solved.
★★★★★ needs line enumeration combined with duplicate avoidance.

While you play, the status bar shows when the local rules have run out, and the hint button shows the next step and the technique it uses. See Playing a puzzle.