Camp Koeller offers exactly three water activities: canoeing, swimming and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.
We start by creating a Venn diagram:

![](https://cdn.mathpix.com/cropped/2024_04_30_dfe99ade8da3ee5d11fdg-09.jpg?height=582&width=635&top_left_y=1975&top_left_x=802)

We want to determine the minimum possible value of $a+b+c+d+e+f+9$.

From the given information $a+b+d=15$ and $b+c+e=22$ and $d+e+f=12$.

Adding these equations, we obtain $a+2 b+c+2 e+f+2 d=49$ or $a+b+c+d+e+f+(b+d+e)=49$.

Thus, $a+b+c+d+e+f=49-(b+d+e)$ and so we want to minimize $49-(b+d+e)$, which means that we want to maximize $b+d+e$.

Since $a, b, c, d, e, f$ are non-negative integers, then $a+b+d=15$ means $b+d \leq 15$. Similarly, $b+e \leq 22$ and $d+e \leq 12$.

Adding these inequalities, we obtain $2 b+2 d+2 e \leq 49$ and so $b+d+e \leq \frac{49}{2}$.

Since $b, d, e$ are integers, then in fact $b+d+e \leq 24$.

It is possible to make $b+d+e=24$ by setting $a=1, c=0$ and $f=0$. This gives $b+d=14$ and $b+e=22$ and $d+e=12$, and so $b=12$ and $d=2$ and $e=10$.

We can see this in the completed Venn diagram here:

![](https://cdn.mathpix.com/cropped/2024_04_30_dfe99ade8da3ee5d11fdg-10.jpg?height=586&width=635&top_left_y=821&top_left_x=802)

Therefore, the maximum possible value of $b+d+e$ is 24 and so the minimum possible number of campers is $58-24=34$.

ANSWER: 34