Let the number of boys initially in the hall be $x$, and the number of girls initially in the hall be $2x$ (since there were twice as many girls as boys).

We know the following:

1. After 105 girls leave the hall, 21 girls remain. So, the number of girls initially was: $2x - 105 = 21$ Solving for $x$: $2x = 126 \quad \Rightarrow \quad x = 63$ Thus, the number of boys is $x = 63$, and the number of girls initially was $2x = 126$.

2. After 105 girls leave, 21 girls remain, and the problem tells us that three times as many boys as girls remain. The number of boys still in the hall is $63$, and the number of girls is $21$. This satisfies the condition that the number of boys is three times the number of girls: $63 = 3 \times 21$

Finally, the total number of children initially in the hall was: $x + 2x = 63 + 126 = 189$

Thus, there were 189 children in the hall at first.

Would you like further details or clarification? Here are five related questions you might want to explore:

1. What if 120 girls had left the hall instead of 105? How many children were initially present?
2. If the number of boys had been doubled, how would the total number of children change?
3. What if there were four times as many girls as boys initially? How would that affect the solution?
4. How many children would be left if 30 boys left the hall after the 105 girls left?
5. What is the ratio of boys to girls if 50 boys left the hall after the 105 girls left?

Tip: Always set up relationships between variables clearly to simplify solving word problems!