Problem:
Sara makes a staircase out of toothpicks as shown:
This is a -step staircase and uses toothpicks. How many steps would be in a staircase that used toothpicks?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
In the staircase with steps, the number of vertical toothpicks is
There are an equal number of horizontal toothpicks, for a total of toothpicks. Solving with yields .
By inspection, the number of toothpicks for staircases consisting of , and steps are , and , respectively. The -step staircase is obtained from the -step staircase by adding horizontal toothpicks and vertical toothpicks. With this observation, the pattern can be continued so that , and are the numbers of toothpicks used to construct staircases consisting of through steps, respectively. Therefore toothpicks are needed for the -step staircase.
The problems on this page are the property of the MAA's American Mathematics Competitions