Explanation:

- Let's define variables: Let $C$ be the number of cricket players and $F$ be the number of football players. $B$ is the number of students playing both sports.

- We know all students play either or both sports: $C+F-B=100$.

- The school states $C=2F$, meaning twice as many students play cricket as football.

- It's given that students playing only cricket are three times those playing only football. So, $C-B=3(F-B)$.

- Simplifying:

- Replace $C$ with $2F$: $2F-B=3(F-B)$.

- This simplifies to $2F-B=3F-3B$, which leads to $F=2B$.

- Substituting back:

- $C+F-B=100$ becomes $2F+F-B=100$ and $F=2B$.

- So, $3B+2B-B=100$ leads to $4B=100$, giving $B=20$.

Thus:

- The number of students playing both sports is 20.