From the edges

A filled cell next to the edge pins down the first group of the line: you know where it starts, and often the whole group and the X that closes it. After overlap, it is the technique that moves an ursograma forward the most. Every cell it fills can help pin down other groups.

How to play introduced the simplest case. Here is the rest: a filled cell near the edge, how to tell which group it belongs to, and what to do after closing a group.

A cell on the edge

The first group of the line is the one closest to the left edge. If the first cell is filled, it belongs to that group, and the group starts there: it has nowhere to go back to. Fill the whole group, as many cells as the first number says. Mark the next cell with an X, because between two groups there is always at least one empty cell.

3 2 | ■······· → ■■■×····

3 2 with the 1st cell filled: the 3 starts there and takes the first three cells. The fourth stays empty, and the 2 is left with the remaining four cells.

The other side works the same way, with the last number: a filled cell next to the right edge is the end of the last group.

2 3 | ·······■ → ····×■■■

2 3 with the last cell filled: the 3 ends there and takes the last three cells. The cell before them stays empty.

In columns, the edges work the same way: the top cell pins down the first group, and the bottom one the last.

Near the edge

A filled cell a few steps from the edge can also pin down the first group, but it leaves the group some room to slide. The group has to cover the filled cell and cannot start before the edge.

To find out what you can already fill, imagine the group in its two extreme positions: as far left as possible and as far right as possible, while still covering the filled cell. The cells filled in both positions must also be filled in the solution.

3 2 | ·■········ (the line) 3 2 | 111·22···· (pushed left) 3 2 | ·111····22 (pushed right) 3 2 | ·■■······· (certain so far; · undecided)

3 2 in ten cells, with the second cell filled: the 3 can start at the first or the second cell. In both positions the third cell is filled, so you can fill it now.

You can work it out in your head. If the group starts at the edge, it ends at the cell given by its size: a 4 ends at the fourth cell. If it starts further along, it also ends further along. So when the third cell is filled and belongs to the 4, the fourth must be filled too.

4 1 | ··■······· (the line) 4 1 | 1111·2···· (pushed left) 4 1 | ··1111···2 (pushed right) 4 1 | ··■■······ (certain so far; · undecided)

4 1: with the third cell filled, the 4 can start at the first, second or third cell. In any of these positions the fourth cell is filled. That is what you can add to the line.

Which group the cell belongs to

All of this only holds if the filled cell belongs to the first group. Next to the edge, that is always the case; further in, it may belong to another group. If you don't know which group it belongs to, this technique can't be used yet.

The second group cannot start before there is room for the first group and a gap. If the first group has 4 cells, the second never starts before the sixth cell: first come the four cells of the group, and then there has to be at least one empty cell. So a filled cell up to the fifth place necessarily belongs to the first group.

2 2 | ···■······ → ···■······

2 2: with the fourth cell filled, there are two possibilities: it may belong to the first 2, starting at the third or fourth cell, or to the second 2, starting at the fourth after the first takes the first two cells. Since both possibilities remain open, there is no new cell to fill yet. You need more information from elsewhere on the line or from a column that crosses it.

An X is an edge too

An X at the end of the line plays the same role as the edge: the first group cannot start before the next cell. Count from there, as if the line started at that point.

3 2 | ×·■······· (the line) 3 2 | ×111·22··· (pushed left) 3 2 | ×·111···22 (pushed right) 3 2 | ×·■■······ (certain so far; · undecided)

With an X in the first cell and the third cell filled, the 3 can only start at the second or the third cell. In both positions the fourth cell is filled, so you can fill it now.

After closing a group

When the first group is complete and closed with an X, the usable line starts after that X, and the next number becomes the first. Repeat the reasoning, one edge at a time.

2 3 | ■■×·■···· (the line) 2 3 | 11×222··· (pushed left) 2 3 | 11×·222·· (pushed right) 2 3 | ■■×·■■·×× (certain so far; · undecided)

The 2 is already complete and closed. The 3 is now the first group of the part still to solve. Since the fifth cell is filled, the 3 has to cover it. From the new edge it can start at the fourth or the fifth cell; in both positions the sixth is filled. As the 3 is the last group and ends at the seventh cell at most, the last two stay empty: no group reaches them.

In colour

In colour, the group at the edge is pinned down in the same way, but the next cell only stays empty if the next group has the same colour. If it has a different colour, the two groups can sit side by side.

3 2 | ■······· → ■■■·····

A red group of 3 at the edge takes the first three cells. The fourth doesn't get an X, because it could be the start of the blue group. All you know is that it doesn't belong to the red group just filled.

When to use it

When a cell next to the edge gets filled: look at the whole row or column straight away. It may pin down the first or the last group.
At both ends of every row and column: check what the first and the last numbers allow.
When a group is complete: the X that closes it creates a new edge for the next group.
When a filled cell is near the edge: first make sure it belongs to the group you are looking at; then use the extreme positions to find out what is already certain.

Together with overlap, it is one of the combinations that moves an ursograma forward the most at the start. Overlap on the first row fills cells at the top ends of the columns; those columns pin down their groups, which in turn fill cells in the rows they cross. So one deduction opens the way to the next.