How to solve Paint by Numbers puzzles

Overview

A paint by numbers puzzle consists of a blank grid, generally rectangular, with numbers to the left and the top. The goal is to black out certain squares. The numbers to the left and the top represent the number of contiguous squares that are black in the respective rows and columns. For instance, "1 4" to the left would indicate that that row contains some number of white squares (possibly none), then one black square, then at least one white square, then four black squares, then some number of white squares (possibly none). Note that the groups are listed from left to right; that is, "1 4" to the left could not indicate a row with four black squares to the right of one black square. The solution is generally a picture.

Here is a sample unsolved paint by numbers grid and the solution (a camel with a sun in the background):


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

In color paint by numbers, different colored blocks don't need to have white spaces in between them. For instance, "42" to the left would indicate that that row contains zero or more white squares, then four red squares, then zero or more white squares, then two green squares, then zero or more white squares.

Strategy Summary

Place some black squares based on overlapping block possibilities.
Fill in squares black and white based on clues in the other direction.
Fill in white squares that can't be reached.
Fill in white squares too small for blocks.
Find advanced overlaps.
Repeat until complete.

Strategy

Overview

The key to solving paint by numbers puzzles is visualizing the blocks of squares and deciding which cells must be black or white based on where those blocks can be positioned. The most trivial row or column is the case where the number of black squares plus the number of gaps is equal to the width/height of the grid. For instance, imagine a grid is 15 squares wide. If the number to the left is 15, that means all fifteen of the squares have to be black. If instead the left says "10 4," then there must be ten black squares, one white square, and four black squares. For most rows and columns, though, the number of black squares plus the number of gaps is less than the width/height of the grid, and you have to use the process of elimination to solve the puzzle.

Step one: Place some black squares based on overlapping block possibilities.

Consider the eighth row. There is "8" to the left of the grid. If we start the block as far left as possible, the block would be positioned as in the blue bar shown; if we start the block as far right as possible, the block would be positioned as in the red bar shown. This means that the six squares that are contained in each bar have to be black.

Fig. 3: Overlapping bars

Now consider the seventh row. There is "2 6" to the left of the grid. There are three possible ways to lay out the blocks: The red bars, the blue bars, or the blue bar for the 2 and the red bar for the 6. We can thus color the overlapping squares (purple in the picture) as black.

Fig. 4: Overlapping bars

Finally, consider the sixth row, which is marked "2 4." There are a variety of ways of placing those two blocks, shown by the red, green, and blue bars. However, in all cases, there is one square always in a bar, which is therefore black.

Fig. 5: Overlapping bars

The general formula for determining how many fully overlapped squares a block contains is os = bs + tbs + nb - 1 - gs, where os = overlapped squares, bs = block size, tbs = the total of all the block sizes, nb = number of blocks, and gs = grid size. For instance, for the 4 block in the sixth row, bs = 4, tbs = 4 + 2 = 6, nb = 2, and gs = 10, so os = 4 + 6 + 2 - 1 - 10 = 1. For the 6 block in the sixth row, bs = 6, tbs = 6 + 2 = 8, nb = 2, and gs = 10, so os = 6 + 8 + 2 - 1 - 10 = 5. If os <= 0 for a particular block, there are no black squares that can be determined using this method.

Step two: Fill in squares black and white based on clues in the other direction.

At this point, consider the fifth column. There is a "5" at the top of the column. If we place the block as low as possible, then the purple square has to be black. We could move the block higher, but no higher than the two black squares already marker. Therefore, the purple square has to be black regardless. In the same way, we can color the purple square in the eighth column.

Fig. 6: Close to the edge

Using similar reasoning, consider the seventh column. There are three black squares in a group, with two squares below them. The clue says "1 4 1." The only way to satisfy the "4 1" portion of the clue is shown below.

Fig. 7: Only possible scenario

Step three: Fill in white squares that can't be reached.

Now consider the second column. The black square that is already placed has to be included in the 3 block. Therefore, none of the orange squares can be part of the 3 block, and must be white.

Fig. 8: Unreachable squares

We can fill in some white squares in other columns and rows as well, now.


Fig. 9: Unreachable squares		Fig. 10: Unreachable squares

Step four: Fill in white squares too small for blocks.

Consider row two. There is a block of two squares somewhere in that row, but there are two cases of single squares surrounded by white. Therefore, since the block wouldn't fit there, these squares must be white.

Fig. 11: Not enough room

There are some cases in several columns that can be similarly ruled out.

Fig. 12: Not enough room

Step five: Find advanced overlaps.

In column 4, we have to have a 1 block below a 2 block. We currently have one square marked black. It could either be part of the 2 block (as shown with the red bars) or part of the 1 block (as shown by the blue bars). In either case, we know that the orange square can't be part of either block, and therefore must be white. Since there are no other ways to place the blocks so that the clue is satisfied, we can color the orange square white.

Fig. 13: Advanced overlaps

Step six: Repeat until complete.

Now consider the second column. The number of possible black squares has been reduced, and so now we know the purple square must be black. Likewise, we can complete the fifth column.

Fig. 14

This allows us to complete some more squares based on the row clues.

Fig. 15

Continuing back and forth, columns then rows, allows us to finish this puzzle:


Fig. 16		Fig. 17


Fig. 18		Fig. 19

Conclusion

In some cases, puzzles will require more sophisticated applications of these strategies, through trial and error, but these strategies are usually sufficient to solve easier paint by numbers puzzles. As you solve more puzzles, you will begin to see patterns that will simplify solving. All puzzles will have a unique solution which can be entirely determined by logic. Have fun!