I noticed that Lionel added a new 3" Fastrack straight section in the 2023 catalog (page 219). As someone who's created plenty of track plans with Fastrack, including my own layout, I was curious to see how many new lengths were possible with the 3" section.
It turns out this is a version of the "Subset Sum" problem. After some research and writing some code to use an existing subsetsum Python library, I can now answer that question!
Note: I know that cutting Fastrack to make custom lengths is possible and has been done. There are posts on this forum about it. I'm not considering that here as I expect many of us (myself included) don't want to cut up our relatively expensive Fastrack sections. No need to comment and suggest that as an option
The following lengths that were previously impossible to create can now be created using the 3" straight section:
Additionally, some of the lengths that could already be created without the 3" section now use fewer sections when the 3" section is involved - one example is a 7 1/2" length, which would have required 4x 1 3/4" sections before, but can now be created with 1x 4 1/2" section and 1x 3" section. Here's an updated "Fastrack Formulas" table which includes the new 3" section.
PDFs for both of these images, as well as the full table which includes the lengths still not possible, are attached. Happy track planning!