Question

In a school drill, a number of children are asked to stand in a circle. They are evenly spaced and the 6th child is diametrically opposite the 16th child. How many children are made to stand in the circle?

• A. 16
• B. 20
• C. 22
• D. None of these

Hint:

On an evenly spaced circle of \(N\) people, two positions are diametrically opposite exactly when they differ by \(N/2\).

Answer 1

The Correct Option is B

A neat way to check this without jumping straight to the halfway rule is to count around the circle from child 6 to child 16 in both directions and use the fact that being directly opposite means both paths around the circle are the same length.

1. 16 children: going forward from position 6 to position 16 covers $16-6=10$ steps, and the remaining way around would only be $16-10=6$ steps. Since $10 \ne 6$, the two children are not opposite each other, so 16 does not work.
2. 20 children: going forward from position 6 to position 16 still covers 10 steps. The remaining way around the circle covers $20-10=10$ steps as well. Both paths are equal at 10 steps each, exactly what "diametrically opposite" requires, so 20 works.
3. 22 children: the forward gap is still 10 steps, but the remaining way around covers $22-10=12$ steps. Since $10 \ne 12$, the two children are not opposite, so 22 does not work.
4. None of these: since 20 already satisfies the condition exactly, this option does not apply.

Only when there are 20 children in total does the short way around from child 6 to child 16 (10 steps) match the long way around (also 10 steps), which is the actual meaning of being diametrically opposite.

Let's summarize:

• Two evenly spaced points are diametrically opposite only when both arcs between them are equal in length.
• The gap between position 6 and position 16 is fixed at 10 steps going one way.
• For the other way around to also be 10 steps, the circle must have $10+10=20$ children in total.

So the circle has 20 children, option (B).