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A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and same perimeter?

Note: The questions have natural analogs in higher dimensions.

Further thought (in appreciation of the affirmative answers given below for polygons defined on the square lattice): Perhaps such examples be constructed even when one goes from square lattice to a parallelogram one.

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and same perimeter?

Note: The questions have natural analogs in higher dimensions.

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and same perimeter?

Note: The questions have natural analogs in higher dimensions.

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and same perimeter?

Note: The questions have natural analogs in higher dimensions.

Further thought (in appreciation of the affirmative answers given below for polygons defined on the square lattice): Perhaps such examples be constructed even when one goes from square lattice to a parallelogram one.

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and perimeter?

Note: The questions have natural analogs in higher dimensions.

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.

1. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?

2. If answer to 1 is yes, are there convex lattice polygons which can be cut into some integer number of convex lattice polygons which are not all congruent and all have same area and same perimeter?

Note: The questions have natural analogs in higher dimensions.