Answer for "How can this be true"?

1. Let's calculate the area of the pieces involved:
area of RED triangle = (8 * 3) / 2 = 12 units
area of GREEN traingle = (5 * 2) / 2 = 5 units
area of the BRAUN piece = 7 units
area of the GREEN piece = 8 units
2. in CASE 1 we are "putting" all the pieces on the GRID.
The TOTAL area of the pieces = 12 + 5 + 7 + 8 = 32 units
3. in CASE 2 we have all the pieces PLUS one square,
meaning we have VIZIBLE an area of 32 + 1 = 33 units
4. now let's calculate the area of the TRINAGLE that is VISIBLE on the GRID:
one leg of the right-angled triangle is 13 the othe one is other 5.
So the area is (13 * 5) / 2 = 32,5 units
CONCLUSION we are covering in case 1 the area of 32,5 units with pieces of 32 units;
and in case 2 the same area of 32,5 units with 33 units. The 0.5 unit difference can be visualized if we take a piece of paper, draw the grid and put the pieces on it.