# Symmetric axis-parallel integral hexagon

Geometry Level 4

A polygon is called a symmetric axis-parallel integral hexagon if:

• It has six sides (a hexagon). It is also not degenerate (no vertex has an angle of $$0^\circ$$ or $$180^\circ$$) and it doesn't self-intersect.
• Its side lengths are (positive) integers.
• Each of its sides is parallel to either the x-axis or the y-axis.
• It has a line of symmetry: upon reflection on this line, the reflection coincides exactly with the original.

Consider a symmetric axis-parallel integral hexagon $$H$$. Suppose its perimeter is $$P(H)$$ units of length and its area is $$A(H)$$ units of length squared (that is, $$P(H), A(H)$$ are its perimeter and area respectively without the dimension). Let $$f(H) = |P(H)-A(H)|$$. Determine the minimum value of $$f$$.

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