How to Solve Picture Logic Puzzles: Fill-by-Number Grids

To solve a picture logic puzzle, read each row and column clue as a list of solid blocks, then fill only the cells that every possible arrangement of those blocks agrees on and mark the cells that none of them can reach. Work line by line, row then column, and each new mark makes the next line easier until the hidden picture appears.

Below you will find how the clues work, five moves that solve both grids here, one puzzle solved step by step and a bigger grid to try on your own, answer included.

What is a picture logic puzzle?

A picture logic puzzle, also called a fill-by-number grid, is a blank grid with numbers beside every row and above every column. The numbers tell you which cells to fill. When every cell is either filled or left empty, the filled cells draw a simple picture.

Each clue is a list of block lengths, in order. A row clue of 3 1 means the row holds a block of three filled cells, then at least one empty cell, then a single filled cell. A clue of 0 means the whole line is empty.

Nothing is said about where the blocks start. That is what you work out, and a well-made puzzle allows exactly one finished picture.

How do you read the clues?

Three facts hide in every clue, and the method uses all of them:

• Order. Blocks appear in the same order as the numbers, from left to right in a row and from top to bottom in a column.
• Gaps. Two blocks in the same line are always separated by at least one empty cell.
• Totals. The numbers in a clue add up to the number of filled cells in that line.

It helps to use two marks. Shade a cell when you know it is filled. Put a small dot in a cell when you know it is empty. Never leave a known empty cell blank, because dots are what make the later moves possible.

Which solving moves should you learn first?

Five moves solve both puzzles in this guide from start to finish, and they are the base for larger grids too. The table compares them.

Exact fit is the easiest. A clue of 3 1 in a line of five cells needs three, one gap and one, which is five cells. There is only one way to place it.

Overlap is the move you will use most in this guide. Slide a block as far left as it can go, then as far right. Cells covered in both positions must be shaded. A block of four in a line of five always covers the middle three cells.

In general, a block of length k in a free stretch of n cells fixes its middle cells, and there are twice k minus n of them (two times k, then take away n), when that count is above zero.

Finished line is the tidy-up move. Once the shaded cells match the clue exactly, every other cell in that line gets a dot.

Edge push uses the border. If the first cell of a line is shaded, the first block must start there, so shade as many cells as that block is long and put a dot just after it. If the shaded cell sits a little way in, the block still cannot start before the edge. A first block of three that covers the second cell, for example, must also cover the third cell, so shade it.

Out of reach works from the other side. If every block in a line is pinned near one end, cells at the far end can hold nothing, so they get dots.

What is the step-by-step method?

1. Start with the big numbers. Look for lines where the clue nearly fills the line, and apply exact fit and overlap.
2. Switch direction after every pass. A cell you shade in a row is also a cell in a column, so it gives that column a new fact.
3. Dot the finished lines at once. Each dot shrinks the free space in a crossing line and can turn a weak clue into an exact fit.
4. Treat dots as walls. A stretch of cells between two dots is a smaller line. Ask which blocks can fit in it and apply overlap inside it alone.
5. Check every clue at the end. Read each row and each column against its clue once the picture looks complete.

Neither puzzle in this guide needs a guess. If you feel stuck, look again at every line that gained a mark since your last full pass. The fact you need may have arrived a moment ago.

How do the moves crack a five by five house picture?

Here is an original puzzle. The column clues run across the top and the row clues down the left side. Row 1 is the top row and column 1 is on the left.

A computer check of every possible filling confirms that these clues allow exactly one picture. Try it yourself before reading on.

Step 1: exact fit in row 3. The clue is 5 in a line of five cells, so shade the whole row.

Step 2: overlap in columns 2 and 4. Each clue is a single block of four in a column of five. Whether the block starts in row 1 or row 2, it covers rows 2, 3 and 4. Shade those three cells in both columns.

Step 3: exact fit in column 3. The clue 3 1 needs three cells, a gap and one cell, which is exactly five. So column 3 is shaded in rows 1, 2 and 3, has a dot in row 4 and is shaded in row 5.

Step 4: finished lines in columns 1 and 5. Each clue is 1, and row 3 already gave each column its one shaded cell. Put dots in every other cell of both columns.

Step 5: exact fit in what is left of row 5. The clue is 3, and columns 1 and 5 now hold dots. The only free stretch left is columns 2 to 4, exactly three cells long, so shade all three. Columns 2 and 4 now have their block of four in rows 2 to 5, so each is a finished line and row 1 gets a dot in both.

Step 6: finish the rows. Row 1 needs one cell and column 3 already gives it. Row 2 needs three and has them in columns 2 to 4. Row 4 needs 1 1 and has shaded cells in columns 2 and 4 with a dot between them. Every line is complete.

Here is the finished grid, with a filled square for each shaded cell and a dot for each empty one.

The picture is a small house: a pointed roof, two walls and a doorway in the middle of the fourth row.

What does mathematics say about pictures made of squares?

A finished grid of this kind is a table of zeros and ones, with one for a filled cell and zero for an empty one. Mathematicians call such a table a binary matrix.

A paper published on 24 July 2009 took a binary matrix and asked how to write it as a sum of as few solid rectangles of ones as possible ^[1]. Its author pictured the matrix as an array of unit squares, so that the ones form a polygon with only horizontal and vertical sides ^[1].

For a polygon like that, a known theorem gives the smallest number of rectangles from four counts: its corners, its separate pieces, its holes and the largest set of non-crossing chords ^[1]. A chord is a horizontal or vertical segment inside the shape that joins two corners pointing inwards ^[1]. In that formula, the number of holes is added to the total, while the number of separate pieces is taken away ^[1].

The main result of the paper concerns a relaxed version of the problem in which each rectangle may be used with a fractional weight between zero and one ^[1]. When the matrix holds only zeros and ones, the best relaxed total equals the smallest number of whole rectangles ^[1].

To see holes matter in our own house picture, we checked by computer, as this guide's own check and not the paper's, every way of cutting it into rectangles. The house has one hole, the doorway. Fill the doorway in and the house can be cut into fewer rectangles than before.

Why are the blocks like tiles?

Every block in a clue is a straight strip of cells, the simplest shape you can build from squares joined edge to edge. Shapes of this kind are called polyominoes.

A paper published in 2003 looked at which rectangles can be filled exactly with copies of given shapes made from unit cubes, with polyominoes as the flat case ^[2]. Its main theorem says that whatever set of pieces you start with, a finite list of prime bricks decides everything: a box can be built from the pieces exactly when it can be built from those bricks ^[2].

The same paper points out that deciding whether a given collection of shapes can fill any rectangle at all is algorithmically undecidable ^[2]. It also lists rectangles filled by single small polyominoes, and it names the six-square Y-hexomino as the smallest case still open ^[2].

A picture logic grid is far simpler. Its strips only run along rows and columns, and the clues pin down their lengths. That is why careful line-by-line work is the natural way in, and it solves both puzzles in this guide from start to finish without a single guess.

Which mistakes should you avoid?

• Leaving known empty cells blank. Without dots you cannot see the small stretches that make overlap work.
• Forgetting the gap. Two blocks in one line never touch. A clue of 2 2 needs at least five cells, not four.
• Mixing up the order. In a clue of 1 3, the single cell comes first. Do not slide the three past it.
• Working only in rows. After a row pass, switch to the columns, since new marks in the rows can unlock them.
• Shading on a hunch. One wrong cell spreads through every crossing line. Only shade what a clue forces.

How do you get moving again when a grid stalls?

• Count the free stretches in each line between dots and edges, and check which blocks can fit in each one.
• Look for a block that can only fit in one stretch. Overlap inside that stretch can give a cell.
• Revisit every line that crosses a cell you marked during your last pass.
• If a large grid still resists, choose one cell, try shading it in your head and follow the result. If it breaks a clue, that cell gets a dot.

Can you solve this practice puzzle?

This seven by seven grid hides an everyday object. A clue of 0 means the row is empty.

A hint if you need one: the empty second row splits several columns, and three of them then have only one possible arrangement.

Answer: the picture is a mug with steam rising above it and a handle on the right. A computer check confirms it is the only solution.

Column 2, with clue 1 5, shows how the hint works. The single cell cannot sit in the empty second row, and the block of five needs the five rows below it, so the single cell goes in row 1 and the five fill rows 3 to 7. Columns 3 and 4 settle just as quickly, because their blocks cannot use the empty second row either.

Want another puzzle where a hidden shape comes out of a grid of squares? Try this mosaic and look for the letter tucked inside it.