SOLUTION: For every real number k, let ⌊k⌋ represent the "floor function", i.e., ⌊k⌋ is the greatest integer less than or equal to k. in the coordinate plane, what is

The only way

⌊x⌋² + ⌊y⌋² = 1

can be true is for values of (x,y) such that 
either 

⌊x⌋² = 0 and ⌊y⌋² = 1 or 
⌊y⌋² = 0 and ⌊x⌋² = 1 or

⌊x⌋² = 0 when and only when  0 ≦ x < 1
⌊x⌋² = 1 when and only when -1 ≦ x ≦ 1
⌊y⌋² = 0 when and only when  0 ≦ y < 1
⌊y⌋² = 1 when and only when -1 ≦ y ≦ 1


⌊x⌋² = 0 and ⌊y⌋² = 1 when and only when 0 ≦ x < 1 and -1 ≦ y ≦ 1
⌊y⌋² = 0 and ⌊x⌋² = 1 when and only when 0 ≦ y < 1 and -1 ≦ x ≦ 1
 
Thus it is true at all points interior and on the square excpt at the
four corners



Its area is 2x2 or 4 square units.

Edwin