2. Given that a match has a length of 1 unit, it is possible to place 12 matches on a plane in various ways to

Question

form polygons with integral areas. For example, one could construct a square with an area of 9 sq. units and a cross with an area of 5 sq. units. Can you use all 12 matches to form the perimeter of a polygon with an area of exactly 4 sq. units? The entire length of each match must be used.

Answer 1

Yes.

Start by constructing a 1 by 4 rectangle. That uses 10 matches and has area 4.

Now let's modify it.

There are four horizontal matches across the top and four across the bottom. And one vertical match on each side. On the bottom, take out the second horizontal match from the left. Replace it with two matches that make a "v" shape. We now have an additional triangular area enclosed at the bottom of the rectangle. We have now used 11 matches and have an area greater than 4. One the top also take out the second horizontal match from the left. Replace it with two matches that make a "v" shape. This takes out a triangular area from the top. We have now used 12 matches and once again an area equal to 4.

The triangular shape added at the top took out the same area as the triangular shape added at the bottom. So the area is 4. The same as the rectangle made with 10 matches.