A of the region S can be obtained by summing up the areas of all the squares contained completely in the region S . This inner approximation is expected to yield an estimate lower or A . This outer approximation is expected to yield an estimate higher or A . A middle approximation might be obtained by adding to the inner approximation, the areas of those square which are completely in or more than half-in the region S . Other in-between approximations are possible.
Intermediate approximations yield an middle area estimate between the upper and lower estimates.
From a computational perspective, more than half-in but not completely in is not easy to define. To give a mathematical algorithm, the toss of a coin might be sufficient, or a judgment could be made on how many of the four triangles formed by the diagonals are included completely in the region S . One or more of the above approximations may be familiar to you from your elementary school days. The latter would cause the lower estimate to increase, the upper estimate to decrease while the middle estimate together with the area A presumably approximate, remaining in between. Such halving results three sequences of numbers or quantities.