Problem 10 · 2009 AMC 8
Easy
Counting & Probability
interior-of-board
On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?
Show answer
Answer: D — 9/16.
Show hints
Hint 1 of 2
"Doesn't touch the outer edge" means: peel off the whole one-square-thick border ring. What's left in the middle is the safe region.
Still stuck? Show hint 2 →
Hint 2 of 2
Peeling one ring removes a row/column from EACH of the four sides — so an 8-wide board becomes 8 − 2 = 6 wide inside.
Show solution
Approach: subtract the border ring, then take the ratio
- Drop the outer ring: the inside is a 6×6 block (8 minus 1 row off the top and 1 off the bottom, same for the sides). That's 6 × 6 = 36 interior squares.
- Probability = interior ÷ total = 36 / 64 = 9/16.
- Why this transfers: "not on the edge" problems shrink an n×n grid to (n−2)×(n−2) — the −2 is one layer off opposite sides. Same idea for borders on rugs, frames, or seating around a table.
Another way — count the edge squares instead:
- Border of an 8×8: 4×8 − 4 corners counted twice = 32 − 4 = 28 edge squares.
- Interior = 64 − 28 = 36, so 36/64 = 9/16. (Subtracting the ring directly is faster.)
Mark:
· log in to save