Don't Fence Me In
Farmer Chuck purchased a plot of land surrounded by a fence. The former owner had marked off nine squares of equal size to subdivide the land, as shown. Chuck wants to divide the land into two plots of equal area. To divide the property, he wishes to build a single straight fence beginning at the far left corner (P). Is such a fence possible? If so, where should it be? Justify your answer with a clear rationale and explanation.
Note: A colleague (J.M.) gave this to two classes of prospective elementary teachers (42 total). Ten of them viewed the problem as "hard," while eighteen left the problem unanswered.
Hint: Can you rearrange the squares to make the problem simpler?
Solution Commentary: First, move the top square to fill the "corner gap" on the left, and assume each square is 1 x 1 with area of 1.
Two solutions seem feasible. First, the line AP is close, but the lower right hand square's area needs to be accounted for. So, lower this line to a intersect side AC at point B just below point A. Then, you want the area of the triangular slice APB to equal 1/2. So, solve [(1/2)(4)(2)]  [(1/2)(4)(BC)] = 1/2, or BC has length 1 3/4.
An alternate solution exists, but without actually determining the length of BC. Draw a vertical line through the midpoint M of the lower righthand box, and find it's intersection with the DA extended (as if cutting the lower righthand square into two slabs and moving one part to fit upper righthand corner). Then, the 2 x 4 1/2 rectangle can be cut into two equal pieces, by the diagonal...showing "exactly" where the farmer's line should be (i.e. point B)...but not knowing the length BC.
