Problem 22 · 2013 AMC 8
Medium
Counting & Probability
grid-of-segments
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
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Answer: E — 3932 toothpicks.
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Hint 1 of 2
Each toothpick lies either horizontally or vertically, so sort them into those two piles and count each pile separately — never try to count the whole grid at once.
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Hint 2 of 2
The classic fence-post catch: a row of cells 60 long needs 61 vertical lines (one extra at the far end), and the lines stack 33 high (one more than 32). Always "cells + 1" for the dividing lines.
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Approach: split into horizontal and vertical toothpicks
- Horizontal toothpicks form rows of length 60. The grid is 32 cells tall, so there are 32 + 1 = 33 such rows: 33 × 60 = 1980.
- Vertical toothpicks form columns of length 32. The grid is 60 cells wide, so there are 60 + 1 = 61 such columns: 61 × 32 = 1952.
- Add the two piles: 1980 + 1952 = 3932.
- Watch the trap: using 60 and 32 (instead of 61 and 33) drops the boundary lines and gives 1920 or 1952 — both are decoy choices. The "+1 fence-post" is the whole problem.
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