Is there a way to know when a noun should end with or without an "e"
I see a word like "the table" (bordet) or "the picture" (bildet) and I'm confused as to why one has an "e" at the end when saying "et bilde" whereas the other, "et bord", does not.
Maybe I'm looking for a formula that isn't there?
There isn't really a 'formula'. You just have to remember that it is 'et bilde' and not 'et bild' and 'et bord' and not 'et borde', etc. 'Bilde' and 'bord' are the base forms, or dictionary forms if you will, so it shouldn't be too hard to remember those forms.
The neuters ending in an unstressed e are 'weak' neuter nouns, whereas those that end in any other sound are 'strong'. The strong neuter nouns do not change their form in the plural, but the weak ones add an r. So there's something to remember.
Hence: Et bord - mange bord, Et bilde - mange bilder
wiktionary suggests that both "bord" and "border" are acceptable plurals in bokmål.
Yeeees... But wiktionary is not the most reliable source.
You'll never hear anyone say 'border', although it might be an acceptable alternative form. It sounds like a typical error a child would make.
That really surprised me too, but it means that it actually is an allowed alternative form.
I honestly think Språkrådet (the norwegian language council) tend to go all to far in accepting alternative forms that virtually no-one uses, not in speech nor in writing. 'Border' still sounds like toddler speech, and if a Norwegian learner were to say it, it would just sound broken.
In other words, don't ever stress learning those alternative forms. The basic rule for forming these plurals is crystal clear, and native norwegians are not even likely to know about such bizarre forms as 'border' as plural for '(et) bord'. 'En bord' is another story though...
Ah, I see. That must be a bit confusing, yes.
I don't do the lessons myself, so I don't know. I just help out a bit in the forum, since I'm a native speaker :)
The online dictionary that we tend to use here, is this: https://ordbok.uib.no
If you look up a word, and then hit the blue letter behind it (for example the 'n' that shows that 'bilde' is a neuter word), you'll see all the forms listed, with alternative declensions as well.