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: Understanding the Concept:
The problem asks for the number of distinct ways to divide a 4x4 grid into two identical (similar) pieces, each with an area of 8 squares. Such a division is possible only if the dividing line has point symmetry about the center of the grid. The pieces formed are called octominoes. We are looking for the total number of such octominoes that can tile a 4x4 square with two copies of themselves, and then subtracting the one example given.
Step 2: Detailed Explanation:
It is a known result in recreational mathematics that there are exactly 6 distinct octominoes (8-square polyominoes) that can form a 4x4 square when two copies are put together. These are the octominoes that possess central symmetry. The question provides one of these 6 shapes in Figure Q and asks for the number of *other* ways. Therefore, the answer is \(6 - 1 = 5\).
Let's visualize the other 5 possible cuts/shapes:
The Straight Cut: A straight line cut vertically or horizontally through the middle. This divides the 4x4 grid into two 2x4 rectangles.
The 'L' Shape Cut: A Z-shaped cut that divides the grid into two L-shaped octominoes.
The 'S' Shape Cut: A different Z-shaped cut that divides the grid into two S-shaped (or Z-shaped) octominoes.
The 'T' Shape variant Cut: A cut that results in two pieces that resemble thick T-shapes.
The Jagged Cut 2: Another irregular, jagged cut with central symmetry, different from the one shown in Q.
Since the question asks for the number of ways *excluding* the one shown, we count these 5 other possibilities.
Step 3: Final Answer:
There are a total of 6 ways to make such a cut. Since we must exclude the one shown in the example, there are \(6 - 1 = 5\) other ways.
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Approach Solution -2
Since the shown cut in Figure Q is one specific way of splitting the 4x4 grid into two matching, connected halves, the question is really asking how many other such splits exist, so checking each candidate count against a direct recount of the possible symmetric cuts settles it.
3: this undercounts the possibilities, since it misses at least two of the distinct staircase-style cuts that also split the grid into two congruent, connected halves.
4: this also undercounts, missing one more valid cut once every distinct centrally symmetric dividing line through the 4x4 grid is listed out.
5: recounting every distinct way to draw a centrally symmetric line through the grid, from the plain straight cut to the different staircase and zigzag variants, gives six cuts in total; removing the one already shown in Figure Q leaves exactly five others.
6: this overcounts by including the cut shown in Figure Q itself, which the question explicitly says to exclude.
Once the cut from Figure Q is removed from the full list of six valid symmetric cuts, five distinct ways remain.
