If you were told to draw a rectangle along the lines of a sheet of graph paper such that its area is 40 squares, you could choose rectangles measuring 8x5, 10x4, 20x2 or 40x1.
For two of these, 8x5 or 10x4, you would find that you could draw a diagonal across the rectangle that would pass through exactly 12 squares.
What is the smallest number of squares that could be the area of three different rectangles whose diagonals pass through the same number of squares? How many squares does this diagonal pass through?
A------------------B
C------------------D
E-------------------F
G--------------------H
There are two squares above. Points A, B, D and C form a square. Another square is found on the opposite side of E, F, H and G. These two squares are placed in such a way that the edge of square, GH, is parallel to the edge of another square, CD. Not only that, the edge of a square, EF, is parallel to the edge of another square, AB.
From the diagram above, 4 rectangles are formed and there are, GHCD, ACGE, ABFE and BDHF.
The number of diagonal lines can be found from G to D and from C to H for the rectangle, CDHF; diagonal lines from A to G and from C to E for the rectangle, ACGE; diagonal lines from A to F and from B to E for the rectangle, ABFE; and diagonal lines from B to H and from D to F.
From the above diagonal lines, the diagonal lines do not go through any squares.
Thus, none of the squares that the diagonal lines should pass through in this solution.
Edited on April 28, 2005, 1:20 pm
Edited on April 28, 2005, 1:24 pm
Solution - zero square
