In the 3-part grid of a square with side \(a+b+c\), how many rectangular cross regions are there in total?

Correct answer: 6

The direct answer is C, 6. Divide each side of the square into three labelled lengths, \(a\), \(b\), and \(c\). This creates a 3-by-3 grid. The three diagonal cells are squares with areas \(a^2\), \(b^2\), and \(c^2\). Every off-diagonal cell is a rectangle: the pairs are \(ab\) and \(ba\), \(ac\) and \(ca\), and \(bc\) and \(cb\). Thus there are six rectangular cross regions in all. A counts only the three pair types and ignores their two orientations. B misses two regions, while D counts all nine cells and incorrectly includes the three diagonal squares. The area identity represented is \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Memory cue: in an n-by-n grid, the off-diagonal count is \(n(n-1)\); for n=3 it is 6.