Test of Pentominoes

Question

These are pentominoes, with letter codes:

Create 4 yes/no questions which uniquely classify each pentomino.

Examples of such questions are:

• Does it have rotational symmetry?
• Does it have reflection symmetry?
• Is it the net of an open box?
• Does it have point symmetry?

The idea is to create a set of questions where no pentomino has the same answers as another.

Source: https://www.mathsisfun.com/pentomino_challenge.html

Answer 1

Here my first go:

1. Does it fit (perpendicularly) inside a rectangle smaller than 9 squares of area?
2. Is there a square with more than 2 neighbours?
3. Can you rotate the mirror image of the piece back to the original?
4. Find the longest straight run that isn't a dead end at both ends. Is it exactly three squares long?

Answer 2

Not sure if this kind of reasoning is allowed, but:

this seems trivial if we can simply refer to the letter labels themselves. e.g. four questions could be:

Q1: Is the letter in TUVXYZ?

Q2: Is the letter in FILTUV?

Q3: Is the letter in LMNUVX?

Q4: Is the letter in MFVY?

Each pentomino will have a unique set of answers to those four questions.

Answer 3

Late answer:

                                       U:      P:      N:
                                       ┌┬┐     ┌┬┐     ┌┐
                                       ├┼┘     ├┼┤     ├┼┐
                                       ├┼┐     ├┼┘     └┼┤
                                       └┴┘     └┘       ├┤
                                                        └┘


     square bbox:                      False   False   False
     cut from ∞ "+":                   False   False   False
     simultaneously sort x and y:      False   False   True
     connect □ centres w/o leaving:    False   True    False




                                       Y:      L:      I:
                                       ┌┐      ┌┬┐     ┌┐
                                       ├┼┐     ├┼┘     ├┤
                                       ├┼┘     ├┤      ├┤
                                       ├┤      ├┤      ├┤
                                       └┘      └┘      ├┤
                                                       └┘
     square bbox:                      False   False   False
     cut from ∞ "+":                   True    True    True
     simultaneously sort x and y:      False   True    True
     connect □ centres w/o leaving:    False   False   True




                                       F:      Z:      W:
                                       ┌┬┐     ┌┬┐     ┌┬┐
                                       └┼┼┐    └┼┤     └┼┼┐
                                        ├┼┘     ├┼┐     └┼┤
                                        └┘      └┴┘      └┘




     square bbox:                      True    True    True
     cut from ∞ "+":                   False   False   False
     simultaneously sort x and y:      False   True    True
     connect □ centres w/o leaving:    False   False   True




                                       T:      X:      V:
                                       ┌┬┬┐     ┌┐     ┌┬┬┐
                                       └┼┼┘    ┌┼┼┐    ├┼┴┘
                                        ├┤     └┼┼┘    ├┤
                                        └┘      └┘     └┘




     square bbox:                      True    True    True
     cut from ∞ "+":                   True    True    True
     simultaneously sort x and y:      False   False   True
     connect □ centres w/o leaving:    False   True    False