Symmetrical Patterns

Kathleen owns a beautiful rug store. She likes to group the rugs into 4 mutually exclusive categories.

• imperfect
• horizontally symmetric
• vertically symmetric
• perfect

An imperfect rug is one that is neither horizontally nor vertically symmetric. Here is an example of an imperfect rug:

[
  ["a", "a", "a", "a"],
  ["a", "a", "a", "a"],
  ["a", "a", "b", "b"]
]

The following is an horizontally symmetric rug. You could "fold" the rug across a hypothetical x-axis, and both sides would be identical. A horizontally symmetric rug is not vertically symmetric (otherwise this rug would be classified as perfect ).

[
  ["c", "a", "a", "a"],
  ["b", "b", "b", "b"],
  ["c", "a", "a", "a"]
]

The following is a vertically symmetric rug. You could "fold" the rug across a hypothetical y-axis, and both sides would be identical. A vertically symmetric is not horizontally symmetric (otherwise this rug would be classified as perfect ).

[
  ["a", "b", "a"],
  ["b", "b", "b"],
  ["a", "b", "a"],
  ["a", "b", "a"]
]

Finally, a perfect rug is one that is both vertically and horizontally symmetric. That is, folded either length-wise or width-wise will yield two identical pieces.

[
  ["a", "b", "b", "a"],
  ["b", "b", "b", "b"],
  ["a", "b", "b", "a"]
]

Given a rug of m x n dimension, determine whether it is imperfect, horizontally symmetric, vertically symmetric or perfect. Rugs are represented using a two-dimensional list.

Examples

classify_rug([
  ["a", "a"],
  ["a", "a"]
]) ➞ "perfect"

classify_rug([
  ["a", "a", "b"],
  ["a", "a", "a"],
  ["b", "a", "a"]
]) ➞ "imperfect"

classify_rug([
  ["b", "a"],
  ["b", "a"]
]) ➞ "horizontally symmetric"

classify_rug([
  ["a", "a"],
  ["b", "b"]
]) ➞ "vertically symmetric"

Notes

You can consider a 1 x n rug as being trivially horizontally symmetric, an n x 1 rug as being trivially vertically symmetric, and a 1 x 1 rug as being trivially perfect.