Welcome to Open Problem #5! This is a problem I've picked, but I'd like Open Problem #6 to come from our suggestion thread which has been quiet lately. Please make some suggestions if you can!
If you're new here, you should start by reading some general information about this group.
Using copies of the polynomio above, you can use repeated copies to form a rectangle (with rotations and reflections allowed).
Using just copies of the polyomino below, is it possible to form them into a rectangle?
The answer incidentally is thought to be "no", but nobody has proven it yet.
This problem of the week from a month ago gives something a flavor of the type of proof needed.
Typically, you figure out some value ("pointiness = 0" or the like) that all rectangles share. Then you find that adding a polynomio increases or decreases that value by some set amount, such that the target value is never reached.
This problem hasn't gotten a lot of attention, so it's a possible things are that simple. You may decide a different strategy altogether, though. Good luck!
Quick updates: Open Problem #2 has the paper done, I'm still looking into the possibility of publishing in a journal.
I will be soon contacting the Online Encyclopedia of Integer Sequences about Open Problem #4, which while not finding any "new" sequences found out information about existing ones and made connections between previously unrelated ones.