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?

$9$

$18$

$20$

$22$

$24$

Solution:

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.

2015 AMC 10A Problem 3

Solution:

Problem 3 in Other Years