Problem 2 · 2009 Math Kangaroo
Medium
Counting & Probability
careful-countingspatial-reasoning
What is the minimum number of points that must be removed from the diagram shown so that no three of the remaining points lie on one line?
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Answer: C — 3
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Hint 1 of 2
List every straight line of three dots in the 3×3 array — rows, columns, and the two diagonals.
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Hint 2 of 2
Find the largest set of dots with no three in a line; the answer is 9 minus that.
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Approach: find the most dots you can keep, then subtract
- The 3×3 array of 9 dots has lines of three along 3 rows, 3 columns, and 2 diagonals.
- If you keep 7 or more dots, some full line of three survives, so 7 is too many to keep.
- You can keep 6 dots with no three in a line: remove the three dots of one diagonal, leaving two dots in every row and column.
- So the fewest removals is 9 − 6 = 3.
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