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Answer by Earlsdon(6294)   (Show Source): You can put this solution on YOUR website!
While the picture would be helpful, one can easily imagine how it would look!
Let's first look at the given example of 39 squares in a 6 by 4 jagged rectangle.
First, each individual square would be tilted at an angle of 45 degrees, so that one of its corners would be in the upper-most position.
Now, lay 6 such squares side by side so that each square touches a adjacent square at a corner.
Now, beneath that row of 6 squares, place 5 squares so that they fit in the triangular spaces between two adjcent squares of the first row of squares.
Continue this arrangement until you have 4 rows of squares in 6 columns.
The pattern is then is 1 row of 6 squares, then 1 row of 5 squares, then 1 row of 6 squares, in an alternating fashion.
You can see that the total number of squares is the sum of 4 rows of 6 squares plus 3 rows of 5 squares, or 4*6+3*5 = 24+15 = 39 squares.
Now, applying the same reasoning to the larger jagged rectangle, you can see that there would be alternating rows of 9 squares and 8 squares.
There would be 7 rows of 9 squares plus 6 rows of 8 squares for a total of 7*9 + 6*8 = 63 + 48 = 111 squares.
I suppose one could generalise this for a jagged rectangle of m by n squares.
Total number of squares = m*n+(m-1)(n-1) = 2mn-m-n+1

Never knew such a helpful website existed with such helpful tutors, who are... (answered by vleith)