2018 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 8 Easy

Sara makes a staircase out of toothpicks as shown:

This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks? (2018 AMC 10B Problem, Question#8)

  • A.

    10

  • B.

    11

  • C.

    12

  • D.

    24

  • E.

    30

Answer:C

A staircase with n steps contains 4+6+8+\cdots+2n+2 toothpicks. This can be rewritten as (n+1)(n+2)-2.

So, (n+1)(n+2)-2=180,

So, (n+1)(n+2)=182,

Inspection could tell us that 13*14=182 , so the answer is 13-1=\rm (C)~12.

Layer 1:4 steps,

Layer 1, 2:10  steps,

Layer 1, 2, 3:18 steps,

Layer 1, 2, 3, 4:28 steps,

From inspection, we can see that with each increase in layer the difference in toothpicks between the current layer and the previous increases by 2 . Using this pattern:

4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180,

From this we see that the solution is \rm (C)~12.

We can find a function that gives us the number of toothpicks for every layer. Using finite difference, we know that the degree must be 2 and the leading coefficient is 1. The function is f(n)=n^2+3n where n is the layer and f(n) is the number of toothpicks. We have to solve for n when n^2+3n=180\Rightarrow n^2+3n-180=0. Factor to get (n-12)(n+15). The roots are 12 and -15. Clearly -15 is impossible so the answer is \rm (C)~12.

Notice that the number of toothpicks can be found by adding all the horizontal and all the vertical toothpicks. We can see that for the case of 3 steps, there are 2(3+3+2+1)=18 toothpicks.Thus, the equation is 2S+2(1+2+3\cdots+S)=180 with S being the number of steps. Solving, we get S=12, or \rm (C)~12.

If you are trying to look for a pattern, you can see that the first column is made of 4 toothpicks. The second one is made from 2 squares: 3 toothpicks for the first square and 4 for the second. The third one is made up of 3 squares: 3 toothpicks for the first and second one, and 4 for the third one. The pattern continues like that. So for the first one, you have 0 "3 toothpick squares" and 1 "4 toothpick squares". The second is 1 to 1. The third is 2:1. And the amount of three toothpick squares increase by one every column.The list is as follow for the number of toothpicks used\cdots 4, 4+3, 4+6, 4+9, and so on. 4, 7, 10, 13, 16, 19, \cdots

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