Problem 3 · 2005 AMC 8
Easy
Geometry & Measurement
reflection-symmetry
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal BD of square ABCD?
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Answer: D — 4.
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Hint 1 of 2
A fold along BD must land black on black. So look only at the black squares off the diagonal — each one needs a twin on the other side.
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Hint 2 of 2
Symmetry means 'reflect and match.' Go black square by black square, find its mirror across BD, and check whether that mirror is already black.
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Approach: every off-diagonal black square needs its mirror twin
- Picture folding the square along diagonal BD. For the picture to match itself, every black square must fold onto another black square.
- Black squares already on the diagonal are fine — they map to themselves. The work is only with the black squares off the diagonal.
- There are 4 such off-diagonal black squares, and each one's mirror image across BD is still white. Coloring those 4 mirrors makes the picture symmetric ⇒ 4 squares.
- Why this transfers: for any reflection-symmetry counting problem, ignore whatever lies on the mirror line itself and just pair up the rest — the answer is the number of unmatched partners.
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