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Problem 328

Combinatorics Difficulty 2.5 Find the answer CEMC Pascal · Canada · 2020

A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. The first four figures are as shown:

How many tiles are used to form the figure in the sequence with perimeter 91 cm?

$6$
$11$
$13$
$15$
$23$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The first figure consists of one tile with perimeter $\text{3 7 cm = 21 cm}$ .

Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is “covered up” and two new side lengths of a tile are added to the perimeter for a net increase of one side length (or 7 cm).

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Since the first figure has perimeter 21 cm and we are looking for the figure with perimeter 91 cm, then the perimeter must increase by $\text{91 cm - 21 cm = 70 cm}$ .

Since the perimeter increases by 7 cm when each tile is added, then $\text{70 cm 7 cm/tile = 10}$ tiles need to be added to reach a perimeter of 91 cm.

In total, this figure will have $1 + 10 =11$ tiles.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.