You can merge cells in Excel. “Duh!” you say. So how does Excel ensure that the cells you want to merge don’t overlap any existing merged cells? How do you check *programmatically *that the cells you want to merge won’t overlap any existing merged cells?

Well, that was the problem I faced while writing my spreadsheet library software (check out SpreadsheetLight here!). The inkling of an idea was formed when I visited StackOverflow. Can’t remember what I was there for, but the answer page wasn’t about Excel or spreadsheets. It was about collision detection.

Yes, the type of collision detection used in games so you don’t walk through walls and stuff.

And thus I learnt a new mathematical theorem: the separating axis theorem (or hyperplane separation theorem). From what I understand, given two N-dimensional thingies, if you can find one (N-1)-dimensional hyperplane that separates those two thingies, then those two thingies are separate (or not overlapping).

Yes, “thingies” is a technical term.

For example, given two 3D objects, if you can find one 2D plane that separates them, then the 3D objects are not overlapping. Another example is when you’re given two 2D objects, and if you can find one line (which is 1D) that separates them, then the 2D objects are not overlapping.

There are some conditions to be fulfilled, such as the objects being convex and disjoint and whatever. (yes I’m a mathematician…) I’ll leave it to you to do your own reading.

But for our purposes, how do we check if that rectangle of Excel cells we want to merge won’t overlap with any existing merged cells? We have these conditions that make our lives easier:

- The merged cells are rectangular
- The 4 end points (top-left, top-right, bottom-left, bottom-right) of a merged cell are in whole numbers
- The (merged) cells map strictly to a 2D rectangular grid

Since the merged cells are rectangular, they’re also convex (I’m not going to explain why, just trust me). Since the 4 end points are in whole numbers, line boundaries can be easily calculated (it’s easy to check if something is <7 or >=8 for example). And since they map strictly to a 2D rectangular grid, the separating axis is a line. And even better, you can always find a subset of solutions of those separating axes that are either horizontal or vertical.

Ok, diagrams are in order.

So in our case, as long as you can find *one* horizontal line or *one* vertical line that separates the 2 merged cells we’re checking, then the 2 merged cells are separate.

Let’s try an example. Excel uses cell references that are in the A1 format, which means row 3 column 4 becomes “D3”. The column index uses alphabets and is in base-26. Read more here.

We’re going to simplify that. Let’s say our existing set of merged cells only has one entry. This merged cell has the top-left corner at row 1 column 1, and the bottom-right corner at row 3 column 4. Suppose we want to check if this merge cell overlaps: top-left corner at row 7 column 8, bottom-right corner at row 10 column 10.

The horizontal line of 5 fits the bill, or the line y=5 if you want to be mathematical about it (but y goes from negative to positive downwards instead of the usual Cartesian y). Or y=6. Or even y=7 (note that the line can “touch” one of the merged cells, but not *both*. This is where the “whole number” condition comes in).

The vertical lines x=5, x=6 or x=8 also fit the bill.

Thus, our 2 merged cells don’t overlap.

So what’s the programmatic way to check? You’d be surprised at the simplicity. To make it easier, the variables with “StartRowIndex” mean the top row of the merged cell, “EndRowIndex” mean the bottom row of the merged cell. And “StartColumnIndex” mean the left-most column of the merged cell, and “EndColumnIndex” mean the right-most column of the merged cell.

if (!(iEndRowIndex < mc.StartRowIndex || iStartRowIndex > mc.EndRowIndex || iEndColumnIndex < mc.StartColumnIndex || iStartColumnIndex > mc.EndColumnIndex)) { // there's an overlap! }

So the merge cell we want to check has top-left corner at (iStartRowIndex, iStartColumnIndex) and bottom-right corner at (iEndRowIndex, iEndColumnIndex).

The variable “mc” refers to a custom class I use to represent merged cells. Obviously, you’d run that condition in a loop through all your existing merged cells. If you can’t find an overlap after running through that loop, then the merge cell you’re checking is good to go.

Let’s run through the individual conditions.

- (iEndRowIndex < mc.StartRowIndex) means our merged cell is completely above the existing merged cell
- (iStartRowIndex > mc.EndRowIndex) means our merged cell is completely below the existing merged cell
- (iEndColumnIndex < mc.StartColumnIndex) means our merged cell is completely to the left of the existing merged cell
- (iStartColumnIndex > mc.EndColumnIndex) means our merged cell is completely to the right of the existing merged cell

The first 2 conditions check for the existence of a horizontal separating axis. The next 2 conditions check for the existence of a vertical separating axis.

Note the negation of the entire boolean condition. Those 4 conditions check for existence of solutions. Negation means checking for overlaps.

Note also that we use strictly greater than or strictly less than checks. If the merge cells share a row or column, then they overlap, right?

Did you know my first draft of writing the code had me checking 6 different conditions? Each condition had its own if statements. This one just had one. I’m so lucky to have learnt the separating axis theorem. I was checking if a point was above, within or below the merge cell, then I had to check if the point was to the left, within or to the right of the merge cell, and … urgh, it was horrible…