Grade 7 Math Puzzle 5: Build a count, then correct for overlap

A free grade 7 math puzzle that teaches Build a count, then correct for overlap. Try it first — the hint and the full solution open when you are ready.

Problem 5 · Build a count, then correct for overlap · 4 points

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A convex polygon has exactly 20 diagonals. Two vertices joined by a diagonal are colored blue; how many diagonals have no blue endpoint?

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Imagine adding a new corner between two neighboring vertices. What happens to the diagonal count?
Once you know how many vertices there are, count the diagonals touching each blue vertex. Is any diagonal in both counts?

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Answer: D — 11

A triangle has no diagonals. Adding a corner to make a k-vertex polygon creates k−3 diagonals from the new corner and turns the old side into a diagonal, adding k−2 in all. The totals for 4 through 8 vertices are therefore 2, 5, 9, 14, and 20, so this is an octagon. Each blue vertex touches 5 diagonals, but the diagonal joining them is counted twice. Thus 20−(5+5−1)=11 diagonals have no blue endpoint. The tempting answer 10 subtracts their shared diagonal twice.

Grade 7 puzzle 5: Build a count, then correct for overlap

Problem 5 · Build a count, then correct for overlap · 4 points