Overview

A fill-a-pix puzzle consists of a partially filled grid of numbers, generally rectangular. The goal is to black out certain squares so that the total number of black squares adjoining the each numbered square (including itself) matches the number in that square. The solution is generally a picture.

Here is a sample unsolved fill-a-pix grid and the solution (a camel with a sun in the background):

Fig. 1: Unsolved grid		Fig. 2: Solved grid

Strategy Summary

Basic Strategy

Color all squares around 0s white.

Color all squares around 9s black; do the same for 6s on the border and 4s in a corner.

If neighboring squares have a difference of three (two on the border), color the opposite squares white and black.

Fill in white and blacks for other numbers, as possible.

Advanced Strategies

If two neighboring numbers have a difference of n, then the greater number must have n more black non-overlapping neighbors.

Look for cases where one set of clues results in "at most n" and another set results in "at least n" for the same set of squares.

Basic Strategy

Color all squares around 0s white.

No black squares can touch a 0. Most fill-a-pix cells have large areas of white, which allow us to quickly complete a section of the grid.

Fig. 3: Whiten all cells around 0

Note that a 0 in the middle of a grid creates nine white squares, a 0 on the border creates six, and a 0 in the corner creates four.

Color all squares around 9s black; do the same for 6s on the border and 4s in a corner.

There are no 6s on the borders, but there is a 9 and a 4. Most fill-a-pix puzzles are larger and will contain many more.

Fig. 4: Blacken all cells around 9, 6, and 4

If neighboring squares have a difference of three (two on the border), color the opposite squares white and black.

The key to solving fill-a-pix puzzles is in thinking about how the territories of each number overlap each other. For instance, if two numbers are next to each other, they share six squares. As a result, any difference between the two numbers must be because of the squares that don't overlap. In the sample puzzle, there are two cases of 3/6 neighbors and two of 5/8 neighbors (there can also be 0/3, 1/4, 2/5, 4/7, and 6/9 neighbors).

Fig. 5: Blacken/whiten opposite cells

When the numbers are on the border, there are four overlapping cells and two non-overlapping cells. In the sample puzzle, there are three cases of 2/4 neighbors on the border (there can also be 0/2, 1/3, 3/5, and 4/6 neighbors).

Fig. 6: Blacken/whiten opposite cells (border)

There are four other ways in which territories can overlap: By four squares (numbers are diagonal), by three squares (numbers are two squares apart), by two squares (numbers are a knight's move away), and by one square (numbers are a square apart diagonally). In the first case, the neighboring number possibilities for filling in the non-overlapping squares are 0/5, 1/6, 2/7, 3/8, and 4/9. In the second, they're 0/6, 1/7, 2/8, and 3/9. In the third, they're 0/7, 1/8, and 2/9. In the fourth, they're 0/8 and 1/9. These are less common combinations, but they're useful to keep in mind.

Fill in white and blacks for other numbers, as possible.

At this point, consider the 8 above the 9. It now neighbors one white square, meaning the other squares have to be black.

Fig. 7: Using the process of elimination

Likewise, look at the 4 in the eighth row. It currently has four black neighbors (including itself), so the rest of the neighbors must be white.

Fig. 8: Using the process of elimination

At this point, you can solve much or all of the puzzle by repeating the last step over and over. Applying it repeatedly to the sample puzzle yields:

Fig. 9: Using the process of elimination		Fig. 10: Using the process of elimination

Fig. 11: Using the process of elimination		Fig. 12: Using the process of elimination

Fig. 13: Using the process of elimination		Fig. 14: Using the process of elimination

Fig. 15: Using the process of elimination		Fig. 16: Simple strategies complete

There are only three squares that are still undetermined. We'll do that in the next section.

Advanced Strategies

If two neighboring numbers have a difference of n, then the greater number must have n more black non-overlapping neighbors.

This is a generalization of the rule above that if neighbors are different by 3, then the non-overlapping squares must all be black. To see how it works, look at the remaining section. The 2 in the fourth row is next to a 1; the difference is one.

Fig. 17: Advanced techniques

There are two white squares in the non-overlapping squares. Therefore, the third non-overlapping square must be black.

This allows us to complete the puzzle, noting that the 2 in the third row now touches two black neighbors:

Fig. 18: Finishing up		Fig. 19: Finishing up

Look for cases where one set of clues results in "at most n" and another set results in "at least n" for the same set of squares.

Consider the following situation:

3 and 5 have a difference of two, meaning that at least two of the a squares have to be black (that is, two or three of them have to be black). 7 and 8 have a difference of one, meaning that at most two of the a squares have to be black (that is, zero, one, or two of them have to be black). Thus, exactly two of the a squares have to be black. We don't know which two, but this also means that all three of the b squares have to white, while all three of the c squares have to black.

Here's another example:

Because 4 is two greater than 2, we know that either zero or one of the a squares are black and that either two or three of the b/b* squares are black. This means that either one or two of the b* squares must be black. However, because of the 1, we know that at most one of the b* squares are black. Therefore exactly one of the b* squares is black. From this it also follows that the b square is black, meaning two of the b/b* squares are black; the a squares are all white, as are all the squares around the 1 except the b* squares.

Conclusion

These strategies should be sufficient to solve most any fill-a-pix puzzles you choose to tackle. As you solve more puzzles, you will begin to see other patterns that will simplify solving. All puzzles will have a unique solution which can be entirely determined by logic. Have fun!

All content copyright Paul and/or Valerie Hartzer, all rights reserved, except where indicated. Contact Paul Hartzer with comments. This file was last modified on Friday, December 18, 2009.