2015 AMC 10A Problem 3

Question

Ann made a

3

-step staircase using

18

toothpicks as shown in the figure. How many toothpicks does she need to add to complete a

5

-step staircase?

9

18

20

22

24

Look at how many new toothpicks the next stair step adds

The number of new toothpicks increases by

2

with each added step

Let us try to find a pattern between the number of toothpicks needed for the staircases.

For a

1

-step staircase, we would only need

4

toothpicks (just a square).

For a

2

-step staircase, we would need

10

toothpicks according to the diagram.

Similarly, we would need

18

toothpicks for a

3

-step staircase.

A

2

-step staircase needs

10 - 4 = 6

more toothpicks than a

1

-step staircase. A

3

-step staircase needs

18 - 10 = 8

more toothpicks than a

2

-step staircase.

Following this pattern, we can see that a

4

-step staircase will need

18 + 10 = 28

toothpicks, and a

5

-step staircase will need

28 + 12 = 40

toothpicks.

This means that Ann would need to add

40 - 18 = 22

more toothpicks.

Thus, D is the correct answer.

Problem 3 in other years