Inspired by this question series, which was inspired by this question. They give rise to beautiful pictures (at least in the eye of the
beholder mathematician) and some nice generalizable solutions.
Together, they make eight questions of the type 'take a pentomino and some rectangle-shaped tiles, and try to fit them in a (bigger) rectangular box'. There are twelve pentominos; so four of them are left, and it's easy to see that it doesn't really make sense to ask the question for I, L and P because they trivially tile a 1x5 or 2x5 rectangle:
It turns out that the Y pentomino (see above) also tiles a rectangle without help of other rectangles, but the solution is a bit trickier and a nice puzzle, certainly doable by hand (that's where the no-computers tag is for). So the actual puzzle here is:
What is the smallest rectangle which can be tiled by Y pentominos?
Rotations and reflections of the pentomino are allowed, so you actually have 8 forms to work with.
One observation/hint (which, as expected, wasn't necessary, but might help future readers who want to solve this puzzle by themselves):
The total area of the rectangle must be divisible by 5. Therefore, at least one of the sides must be divisible by 5 as well.