Problem 25 · 1993 AJHSME
Stretch
Geometry & Measurement
coveringtilting
A checkerboard consists of one-inch squares. A square card, 1.5 inches on a side, is placed on the board so that it covers part or all of the area of each of n squares. The maximum possible value of n is
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Answer: E — 12 or more.
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Hint 1 of 2
The word 'maximum' is the signal: don't settle for the obvious lined-up placement. The squares you touch are exactly the squares the card's edges cross into — so you want the card to straddle as many grid lines as possible.
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Hint 2 of 2
Tilt the card off the grid. Rotating it makes its corners poke past grid lines they'd otherwise stay short of, and its long diagonal (about 2.1 in) now spans more rows and columns than the 1.5-in sides did.
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Approach: tilt to cross the most grid lines
- Count what the card overlaps. Lined up square-with-the-grid, a 1.5-inch card fits inside a 2×2-to-3×3 footprint — at most 9 squares (a 3×3 block when it straddles lines both ways).
- Now tilt it. A rotated card sticks its sharp corners across extra grid lines, and its diagonal (≈2.12 in) reaches farther than 1.5 in did — so the card can overlap pieces of more than 9 squares. A good tilt covers parts of 12 or more squares.
- Why this transfers: 'cover the most squares' really means 'cross the most grid lines,' because each crossing slices the card into another square. Aligning with the grid wastes that — tilting maximizes crossings. Whenever a 'maximum overlap' problem lets you rotate, suspect the answer comes from tilting off the axes.
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