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Using grid would probably help you avoid this problem too. With grid, you should be able to make one column or row diagonal, though I have not yet tried using grid with hexagonal boards. My CSS code for hexagonal boards uses absolute positioning, as grid was not yet part of CSS when I wrote the code. Anyway, absolute positioning is another alternative to using tables.

I am not familiar with this technique, but a problem could be that it is not just about positioning the images, but also about translating mouse clicks to cell coordinates. (Which must also work for empty cells, for entering non-captures.) In the table version I have attached event handlers to the cells for this.

The I.D. represents the board through a HTML table,

Game Courier used to use HTML tables for hexagonal boards, but I gave up on this in favor of using either CSS with a board image or drawing the board image. This reminds me that I was working on getting CSS to work with a generated board image so that you could view each position of the game without reloading the page, but I haven't completed that. I think I had two different approaches and wasn't sure which one to go with. I either wanted to change how I generated the board so that the image would be perfectly symmetrical, or I wanted to modify the code to work with the asymmetrical boards I was already generating. I lost sight of what I was doing when I got caught up with other projects.

in theory it should be possible to create a table with a masonry-like tiling, shifting each subsequent rank by half a cell.

I imagine this would be easier to do with CSS grid, because, as far as I know, table columns remain vertical.

I have tried this, though, and it works to some extent. But for reasons that I do not grasp yet it also changes the height of the ranks in a way that I could not control.

Using grid would probably help you avoid this problem too. With grid, you should be able to make one column or row diagonal, though I have not yet tried using grid with hexagonal boards. My CSS code for hexagonal boards uses absolute positioning, as grid was not yet part of CSS when I wrote the code. Anyway, absolute positioning is another alternative to using tables.

Is it possible to make an Interactive Diagram use a hexagonal board?

The current standard scripts do not support that. The I.D. represents the board through a HTML table, and in theory it should be possible to create a table with a masonry-like tiling, shifting each subsequent rank by half a cell. This could be done by giving the first cell on each rank colspan 1, 2, 3, 4, ..., and all other cells colspan="2".

I have tried this, though, and it works to some extent. But for reasons that I do not grasp yet it also changes the height of the ranks in a way that I could not control.

Once the pieces are displayed on the hexagonal grid defined by the table, suppressing cell borders and coloring would allow display of a custom hexagonal board image as background.

Is it possible to make an Interactive Diagram use a hexagonal board?

Since it's the featured variant for this month.