Tessellate S.T.E.M.S. (2019) - Computer Science - School - Set 4 - Subjective Problem 1

There are $10$ houses in a row which are being painted red, blue and green.

The mayor of the town however passed a law.

If a house is painted red or blue, the house to its right has to be green.

How many ways are there of painting the $10$ houses?

This problem is a part of Tessellate S.T.E.M.S (2019)

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Comments

Let's first try smaller cases and write $a_n$ as the number of ways to paint $n$ houses according to the rules.

1 house

Obviously, there are $a_1 = 3$ ways.

2 houses

If the left house is green, then there are $a_1 = 3$ ways.
If the left house is red or blue, then there is only one way each.

$a_2 = a_1 + 2 \cdot 1 = 5$

3 houses

If the leftmost house is green, then there are $a_2 = 5$ ways.
If the leftmost house is red or blue, then the next house has to be green and the third house has 3 possibilities each.

$a_3 = a_2 + 2 \cdot 3 = 11$

4 houses

Leftmost house green: $a_3 = 11$ Leftmost house red/blue: next house green, $a_2 = 5$ ways for remaining houses.

$a_4 = a_3 + 2 \cdot a_2 = 21$

$n$ houses

Leftmost house green: $a_{n-1}$ ways for the remaining houses.
Leftmost house red/blue: next house green and $a_{n-2}$ ways each for the remaining houses.

So, in general:

$a_n = \begin{cases} 3, &\ \text{for } n=1 \\ 5, &\ \text{for } n=2 \\ a_{n-1} + 2 a_{n-2}, &\ \text{for } n>2 \end{cases}$

With this formula, we can calculate $a_{10} = \boxed{1365}$ .

Henry U - 2 years, 5 months ago

Yup...........same approach as mine!!! :)

Aaghaz Mahajan - 2 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$