Depth · one concept from first principles · step 2 of 4
The shapes below are made by connecting dots and line segments in a regular pattern. Find how many line segments the th shape has.
| Position | 1st | 2nd | 3rd |
|---|---|---|---|
| Number of dots | |||
| Number of line segments |
The segment counts are 6, 12, 18, which are 6x1, 6x2, 6x3. So the nth shape has 6 times n line segments.
Starting at 6 and adding 6 each step gives the multiples of 6, so the position number times 6 is the count.
4.OA.C.5Solve An Easier Related ProblemThe nth shape has 6 times n line segments.
The first shape already has 6 segments and every next shape adds another 6, so the nth shape is 6 counted n times over.
The nth shape's segments are n equal parts of 6 joined with no gaps or overlaps, so the total is simply those n sixes added together.
Adding 6 to itself n times is the same as n equal groups of 6, and that count equals 6 times n.
Making n equal groups of 6 and adding them up is exactly what n times 6 means.
n groups of 6 counts the same dots as 6 groups of n, so n times 6 is the same number as 6 times n.
Substitute n = 100 into 6 times n.
The 100th shape is just 100 groups of 6 segments.
4.OA.C.5Look For A PatternThis only needs Grade 4 pattern sense: every step adds 6, so the 100th shape is 100 sixes!
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The same shape shows up later on the AMC, as Casework by Position / Vertex Type, Lattice / Grid Pattern from Small Cases, Overlap / Complement Counting, Small-Domain Constrained Search. Why the shapes repeat →