Figure P shows a square tile with a 4×4 grid marked on it. One of the ways the tile can be cut along the grid lines into two similar and contiguous pieces is shown in figure Q. The different configurations of this cut, shown in figure R, are counted as the same cut. In how many ways, excluding the cut shown in Q, can the tile be cut into two similar and contiguous pieces along the grid lines?
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Problems involving dividing a shape into two identical pieces often rely on the principle of central (point) symmetry. The dividing line must look the same if you rotate it 180 degrees around the center of the shape.
Step 1: Set up the condition for a valid cut.
Since the two pieces must be identical in shape, the cutting line has to pass through the exact centre of the 4x4 grid and be symmetric under a half-turn about that centre point.
Step 2: Trace paths from one edge to the opposite edge.
Every valid cut is a path drawn along grid lines from one outer edge of the square to the point directly opposite it through the centre, moving only up, down, left or right one grid step at a time.
Step 3: Count the distinct symmetric paths.
Listing these paths out, from the plain straight line to the different step patterns that stay symmetric about the centre, gives a total of six distinct valid cuts across the whole grid.
Step 4: Remove the one already shown.
Figure Q shows one of these six paths, so the number of other ways to make such a cut is
\[ 6 - 1 = 5 \]
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Approach Solution -2
A more visual way to settle this is to simply draw out and name every distinct shape the two matching pieces can take, rather than counting cuts as abstract paths.
Straight cut: a single straight line across the middle, either horizontal or vertical, splitting the grid into two 2x4 rectangles; both directions count as the same shape once rotated, so this is one distinct case.
Simple staircase cut: a jagged line that steps once through the centre, producing two interlocking L-shaped pieces.
Double staircase cut: a line with two steps, producing a different pair of interlocking pieces from the simple staircase.
Offset zigzag cut: a line that steps in the opposite direction partway through, giving yet another distinct interlocking shape.
Pinwheel-style cut: a more elaborate symmetric line that spirals slightly around the centre before reaching the opposite edge.
Together with the cut already shown in Figure Q, these five distinct shapes make up the full set of six possible symmetric cuts, so excluding the one shown leaves exactly five other ways.
