One 15×15 Illustration Logic puzzle, solved from start to finish

The puzzle in this article

In this article, a square you have decided not to fill is called a "mark" (on paper, a square where you write a × or a dot). In the diagrams the navy × is a filled square (the game's embroidery look), and the small dot is a mark. Rows are counted from the top, columns from the left.

1Start with "0" rows and big numbers

The bottom row, row 15, is "0". Nothing in it is filled, so mark every square.

Next, go down the rows from the top. "3 2 8" in rows 4 and 5 and "2 3 8" in row 6: the numbers add up to 3 + 2 + 8 = 13, and the 2 gap squares bring it to 15, exactly the row's length. The runs have only one place to go, so fill them all.

The other rows use the overlap method. A row's length minus the sum of its numbers and the gaps between them is what we call its "slack". Any number bigger than the slack gives you squares to fill, as many as the difference. In row 2, "12" has slack 15 − 12 = 3, so you can fill the middle 9 squares. In row 3, "1 1 8" gives 5 of the 8's squares; in row 7, "2 1 9" has slack 1 and gives 1 of the 2's squares and 8 of the 9's; in rows 8 and 9, "2 10" gives 8 of the 10's squares; and in row 10, "1 10" gives 7 of the 10's squares. Row 1 and rows 11-14, with their small numbers, cannot be filled yet.

Techniques used: "0" rows and columns Exact-fit rows and columns The overlap method

2The overlap method on columns

Columns work the same way. In columns 8, 12 and 13, "13" has slack 2 and gives 11 squares; in column 14, "12" has slack 3 and gives 9 squares. Many of these are already filled from the rows; the new ones are in the red frames in the diagram.

In column 2 from the left, "1 9" has slack 4 and gives 5 squares of the 9. In column 6, "2 3 7" has slack 1 and gives 6 squares of the 7 (rows 9-14). In column 15, "8" has slack 7, which gives only the middle row, row 8.

Techniques used: The overlap method

3Mark the finished rows and columns

Rows 4-6 have all their runs filled. Mark the squares left between them.

"9" in columns 9, 10 and 11 is finished too, with the 9 squares in rows 2-10 filled. Mark row 1 and rows 11-14. Marks are clues for the rows they cross.

Techniques used: Finished rows and columns

4Work from the edges

When a row has a filled square at its edge, the run at that end starts there. In row 8, "2 10" has column 15 at the right edge filled, so the 10 takes columns 6-15. Mark column 5 next to it. In row 7, "2 1 9", the 2 ends with the 2 squares at the left edge, so mark column 3.

It works a little way in from the edge too. In column 13, "13" is filled down to row 13, and row 15 is a mark. Even pushed as far down as it goes, the run of 13 covers rows 2-14, so row 2 is always filled.

In the same way, the left-hand column's "7" contains row 4, so even pushed to the top it reaches row 7. In column 15, "8" contains row 4, so it reaches row 8; in column 14, "12" has its bottom end at row 14, so you can fill from row 3. In column 6, "2 3 7" runs 7 squares up from row 14, so mark row 7 next to it.

Techniques used: Work from the edges

5Settle the left-hand column

From here on, the basics alone will not get you far. The 3 squares on the right of rows 11 and 12 are as long as the biggest number in those rows, 3. They cannot grow, so mark column 15, just to their right.

In row 2, "12" has columns 4-13 filled and cannot reach column 1. Mark it, and the left-hand column's "7" gets a mark at row 2. "7" contains rows 4-7, so it cannot reach below row 11 either.

In the left-hand column, row 1 is a single square shut in between the end of the column and the mark at row 2. A 7 will not fit, so mark it. Now "7" cannot start above row 3, so it reaches at least as far as row 9. Fill rows 8 and 9.

In column 15, at the right edge, "8" contains rows 4-8, so it cannot reach below row 12 either. The mark puts its bottom end at row 10, so you can fill as far as row 3. Row 3 now has its 8 squares on the right, so mark column 7. In row 11, the left "1" is settled at column 2, so mark column 3.

Techniques used: Both sides of the longest run Squares out of reach Gaps too short Work from the edges

6Columns split by marks

In column 5 from the left, "2 4" has row 2 and rows 4-6 filled. Fill row 3 between them and the 5 squares in rows 2-6 join up, longer than the biggest number, 4. Mark row 3.

In column 3, "5 2" has been split by marks into 5 squares above (rows 1-5) and 3 below (rows 12-14). The 5 only fits above, and there are exactly 5 squares, so fill them all.

Then in row 1, "4" contains column 3, so working from the left edge you can fill columns 4 and 5 too, and everything from column 7 rightward is marks. In column 4, "2 1" is finished with rows 1 and 2 and row 6, so the rest are all marks.

Techniques used: Too long if joined The overlap method inside a gap Work from the edges Finished rows and columns Squares out of reach

7The marks in row 1 settle the right-hand columns

With marks lined up on the right of row 1, "13" in columns 8, 12 and 13 is settled at rows 2-14. Fill row 14. In row 2, "12" cannot reach column 15, so mark it. Then in column 15, "8" has marks on the top 2 squares and the bottom 5, which settles it at rows 3-10. In column 14, "12" can be filled as far as row 13, thanks to the mark at row 1.

As more squares fill, rows 9, 10 and 3 get finished one after another. Each time, mark the rest and go back over the columns they cross. The left-hand column's "7" has rows 4-9 filled and a mark at row 10, so you can fill as far as row 3.

Techniques used: Work from the edges Finished rows and columns Squares out of reach

8Settle the bottom rows

In row 14, "3 3": columns 2 and 3 are a gap of 2 squares between the marks in columns 1 and 4. A 3 will not fit, so mark them. The left "3" contains the filled columns 6 and 8, so it is columns 6-8. Fill column 7, between them.

In column 7, "1 8": after the 1 at the top, rows 7-10 and row 14 are filled. The run of 8 is settled at rows 7-14, so fill rows 11-13, between them.

That finishes row 11, so mark column 5. In rows 12-14, columns 6-8 now form a run of 3. That is as long as the biggest number, 3, so mark column 5, just to its left.

Techniques used: Gaps too short Work from the edges Both sides of the longest run

9The last few squares

Only a few squares are left. In column 2, "1 9" is split by the mark at row 3 into 2 squares above and 10 below, and the 9 only fits below. Use the overlap method inside those 10 squares and you can fill as far as row 12.

In row 12, "2" has column 1 marked and column 2 filled, so it is columns 2 and 3. In column 2, "9" now has its 9 squares in rows 4-12, so mark row 13. In row 13, the only square left for "1" is column 3, so fill it. In column 5, "2 4": the 4 only fits in the gap at rows 4-7, which is exactly 4 squares, so you can fill row 7 too.

In row 14, the "3" on the right starts next to the mark in column 11, so it is columns 12-14. The square next to a mark works just like the edge of the row.

Last, in column 14, "12" is finished with rows 3-14, so column 14 of row 2 is a mark. In row 2, "12" ends at column 13, so you can fill as far as column 2. That settles "1" in column 2, so column 2 of row 1 is a mark. In row 1, "4" has columns 1 and 2 marked, so it is columns 3-6. Every square is now settled.

Techniques used: The overlap method inside a gap Work from the edges Next to a mark Finished rows and columns

The finished picture: Elephant

With every square settled, the elephant appears. The first 3 steps work in the same order on any puzzle. From step 4 on, the trick is to go back over the rows and columns through each mark as you make it.

When you get stuck on another puzzle, look in the table of 7 techniques for when you get stuck for the one that fits what you see on the board.

Questions