Advanced Binary Puzzle Techniques for Harder Grids

Advanced binary puzzle techniques start where pairs, gaps and counts stop working. At that point the grid is not stuck: you only need to look at a line as a whole, test a digit for a few steps, or compare lines with each other. This page works on one real 8 x 8 puzzle that stops moving after 39 cells have been filled by deduction, and shows three ways to move it on.

When the basic rules stop

The basic checks are: two equal digits side by side or around a gap, a line that has its full count of one digit, and a line that would copy another line. If you do not know them yet, read the binary puzzle rules and how to solve a binary puzzle first.

The puzzle

This is a medium LogikLand puzzle. We followed pairs, gaps and counts until none of them could add a digit. That happened after 39 cells had been found by deduction, with 8 cells left, all in columns 4 and 5 and rows 2 to 5. The solver that produced the puzzle finishes it with the duplicate-lines rule, which we come back to in technique 3.

The grid at that standstill, with an underscore for each of the 8 cells left, reads:

row 1   0 1 0 0 1 1 0 1
row 2   1 0 1 _ _ 1 0 0
row 3   1 1 0 _ _ 0 1 0
row 4   0 0 1 _ _ 0 1 1
row 5   0 1 0 _ _ 1 0 1
row 6   1 1 0 0 1 1 0 0
row 7   1 0 1 1 0 0 1 0
row 8   0 0 1 0 1 0 1 1

Technique 1: fit the missing digits

Take column 4. It reads, from the top, 0 _ _ _ _ 0 1 0. A column of 8 holds four 0s and four 1s. It already has three 0s and one 1, so the four empty cells need exactly one more 0 and three more 1s. Now add the no-three rule. Three 1s in four cells without a run of three means the 0 must be in the middle, in row 3 or row 4. So the outer cells, rows 2 and 5, are both 1.

Column 5 reads 1 _ _ _ _ 1 0 1. It has three 1s and one 0, so it needs one more 1 and three more 0s. Three 0s in four cells without three in a row means the 1 sits in the middle. Rows 2 and 5 of column 5 are 0. We checked both columns by listing every legal way to fill them: the outer cells are the same in every one.

A second line from the same family: a column such as 0 1 0 0 1 _ _ _ needs two more 1s and one 0 in its last three cells. The 0 cannot be last, because a 1 in the fifth cell followed by 1 1 would make three, so the last cell is a 1. Rules 1 and 2 applied one at a time do not see that; applied together they do.

Technique 2: assume and test

Choose a cell and try the digit you suspect is wrong. Take row 2, column 4 and try 0. Column 4 would then hold four 0s (rows 1, 2, 6 and 8), which is its full count. Every other empty cell of the column would be a 1, so rows 3, 4 and 5 would be 1 1 1, three in a row. That breaks the rule, so row 2, column 4 is not a 0: it is a 1, as technique 1 found.

We tried this on the same standstill, for every empty cell and both digits, with a program that applies the pair, gap and count rules after each assumption. In this puzzle it rules out the same four wrong digits as technique 1 (a 0 in rows 2 and 5 of column 4, a 1 in rows 2 and 5 of column 5), and no digit that belongs to the solution.

Technique 3: the last block and twin lines

After techniques 1 and 2, rows 2 and 5 are finished. Rows 3 and 4, columns 4 and 5, form a block of four cells where the digits can still go either of two ways: row 3 gets 0 then 1 and row 4 gets 1 then 0, or the opposite. Counts and no-three are happy with both. Only the third rule can choose.

Look at row 4: 0 0 1 _ _ 0 1 1. Row 8 is complete: 0 0 1 0 1 0 1 1. If row 4 took 0 then 1 in the empty cells, it would be a copy of row 8, which is forbidden. So row 4 gets 1 then 0, and row 3 gets 0 then 1.

Keeping track on 10 x 10 and 12 x 12

• Write the needs next to each line. A small "needs 1 zero, 3 ones" beside a row or column turns technique 1 into a glance.
• Look at columns as often as rows. In our standstill, the whole block was only visible from the columns.
• Save duplicates for last. The third rule only applies to a line with two empty cells, so it is wasted effort early.
• Limit assumptions. Two or three steps of consequences are enough. Longer ones are where errors come in.

Hard grids in a book

LogikLand has no free single-puzzle tool for binary puzzles. Create a free account and use the Binary Puzzle assistant to make a book with 10 x 10 and 12 x 12 grids. Our suggestion, not a rule: tell your readers in the "How to play" page that no guessing is needed. Every grid in the assistant can be solved with the basic rules, and this page's techniques are only shortcuts.

Questions