Problem 22 · 2024 Math Kangaroo
Stretch
Spatial & Visual Reasoning
symmetry
Tarek wants to colour two more cells of the 4×4-square black so that the pattern of white and black cells has exactly one axis of symmetry. In how many ways can he do that?
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Answer: E — 6
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Hint 1 of 3
A 4×4 grid can be symmetric about four possible lines: vertical, horizontal, and the two diagonals.
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Hint 2 of 3
For each candidate axis, reflect the two black cells already there and see what the two new cells would have to be.
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Hint 3 of 3
Be careful to keep exactly one axis — a placement that accidentally creates a second axis does not count.
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Approach: make the picture symmetric about each axis in turn and count the valid two-cell completions
- Reflecting the two given black cells across the vertical axis forces one extra colouring that is symmetric only about that vertical line.
- The horizontal axis similarly forces one valid colouring.
- The two diagonal axes are more flexible and give the remaining colourings, for a total of four diagonal-symmetric ways.
- Adding them up, 1 vertical + 1 horizontal + 4 diagonal = 6 ways.
Mark:
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