Problem 6 · 1998 AJHSME
Medium
Geometry & Measurement
rearrangement
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Answer: B — 6 square units.
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Hint 1 of 2
A slanted edge looks scary but it only TRADES area: whatever triangle it cuts off one side, it adds an equal triangle on the other. So nothing is really lost.
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Hint 2 of 2
Mentally straighten the slanted cuts into a tidy rectangle, then just count its width times height.
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Approach: slide the slanted pieces to make a clean rectangle
- Each slanted edge cuts a little triangle off one place and pastes an identical triangle somewhere else — the area doesn't change, it just relocates. Picture sliding those triangles straight, and the wiggly shape becomes a flat 2 × 3 rectangle.
- Its area is 2 × 3 = 6 square units.
- Why this transfers: a diagonal cut between two dots one step apart always splits its unit square exactly in half, so cut-off and pasted-on pieces match. Reshaping an awkward region into a rectangle (or pair of rectangles) is the go-to move for dot-grid areas.
Another way — box-and-subtract (boxing method):
- Box the whole figure in the smallest grid rectangle that contains it, then subtract the empty corner triangles the slanted edges leave outside the shape.
- Each slanted edge carves off a half-unit triangle and adds one back, so the subtractions and additions cancel and you land back on 6 square units. Boxing-and-subtracting is the reliable fallback whenever a grid shape has slanted sides.
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