Problem 13 · 1993 AJHSME
Hard
Geometry & Measurement
area-subtraction
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Answer: D — 36.
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Hint 1 of 2
Counting the scattered white background squares directly would be a nightmare. Flip it: white = (whole sign) − (black letters). The black HELP is far easier to count.
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Hint 2 of 2
The strokes are exactly 1 unit wide, so each letter is just a count of unit squares. Tally H, E, L, P one letter at a time — and don't double-count squares where strokes meet.
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Approach: complementary counting — total minus the letters
- The white area is hard to count directly (it's the leftover background), so subtract instead. The whole sign is 5 × 15 = 75 unit squares.
- Now count the black letters (1-unit strokes), letter by letter, being careful at the junctions: the four block letters H, E, L, P cover 39 squares all together.
- White = 75 − 39 = 36.
- Why this transfers: when the shape you want is the messy 'everything else,' count the tidy part and subtract from the whole. This 'complementary counting' turns an ugly region into one clean subtraction — you'll reuse it constantly in area and counting problems.
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