2006 AMC 10A Problem 7

Cutting and moving pieces never changes total area, so the square has area $8 \cdot 18 = 144$ and side $12$.

Now read the figure. The cut goes down $4$ (half the height), across, and down $4$ again. The left hexagon has a short top edge of length $y$ and a long bottom edge running from $D$ to the foot of the cut, of length $18 - y$; the right hexagon is the same piece turned around, with a long top edge $18 - y$ and short bottom edge $y$.

To build the square, slide the right hexagon up by $4$ and left so that the two steps interlock. The stacked heights $8 + 4 = 12$ match the square, and the two long edges become the top and bottom sides of the square. Hence
$$18 - y = 12 \quad\Longrightarrow\quad y = 6.$$

Consistency check: the middle horizontal segment of the cut has length $18 - 2y = 6 = y$, so the two short edges fit exactly into the notches, as a square with no gaps requires.

The answer is $\boxed{\textbf{(A)}\ 6}$.

Dissection problems almost always start with the invariant: area. That fixes the target shape's dimensions, after which one labeled edge of a piece matched against a side of the target gives an equation. Verify with a second edge to be sure the pieces really interlock.

The horizontal $18$ is made of three pieces that must be interchangeable for the hexagons to nest: the two labeled $y$ segments and the middle segment of the cut. Interlocking forces the middle segment to equal $y$ too, so $3y = 18$ and $y = 6$.

• Eyeballing $y$ from the drawing, or assuming $y$ is half of $18$ or of $12$, instead of first finding the square's side from the area $144$.
• Forgetting the pieces have height $8$ with a step of $4$, and trying to make the square $8$ tall.