Question 15
Twelve balls are numbered by the numbers $1$, $2$, $3$, $\ldots$, $12$. Each ball is coloured either red or green, so that the following two conditions are satisfied:
a. If two balls marked by different numbers $a$ and $b$ are coloured red and $a+b \lt 13$, then the ball marked by the number $a+b$ is coloured red, too.
b. If two balls marked by different numbers $a$ and $b$ are coloured green and $a+b \lt 13$, then the ball marked by the number $a+b$ is also coloured green.
How many ways are there of colouring the balls?
a. If two balls marked by different numbers $a$ and $b$ are coloured red and $a+b \lt 13$, then the ball marked by the number $a+b$ is coloured red, too.
b. If two balls marked by different numbers $a$ and $b$ are coloured green and $a+b \lt 13$, then the ball marked by the number $a+b$ is also coloured green.
How many ways are there of colouring the balls?