40 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes on to child-2, child-2 passes on to child-4, child-4 passes on to child-7 and so on. After how many such changes (including child-1) will the ring be in the hands of child-1 again?

Correct Option (B)

The problem describes a sequence of ring passing among 40 children arranged in a circle. The passing rule follows an increasing increment pattern:

• Child-1 passes to Child-2 (an increment of 1 position).
• Child-2 passes to Child-4 (an increment of 2 positions).
• Child-4 passes to Child-7 (an increment of 3 positions).
• Child-7 passes to Child-11 (an increment of 4 positions).

This establishes a pattern where the increment in child number increases by one with each successive pass. Since the children are in a circle numbered 1 to 40, any child number exceeding 40 wraps around (e.g., Child-41 is equivalent to Child-1, Child-42 to Child-2, and so on, by subtracting 40).

Let's trace the sequence of children holding the ring and the number of changes:

1. Child-1 (Initial state, 0 changes)
2. Child-2 (1st change: 1 + 1 = 2)
3. Child-4 (2nd change: 2 + 2 = 4)
4. Child-7 (3rd change: 4 + 3 = 7)
5. Child-11 (4th change: 7 + 4 = 11)
6. Child-16 (5th change: 11 + 5 = 16)
7. Child-22 (6th change: 16 + 6 = 22)
8. Child-29 (7th change: 22 + 7 = 29)
9. Child-37 (8th change: 29 + 8 = 37)
10. Child-6 (9th change: 37 + 9 = 46; 46 - 40 = 6)
11. Child-16 (10th change: 6 + 10 = 16)
12. Child-27 (11th change: 16 + 11 = 27)
13. Child-39 (12th change: 27 + 12 = 39)
14. Child-12 (13th change: 39 + 13 = 52; 52 - 40 = 12)
15. Child-26 (14th change: 12 + 14 = 26)
16. Child-1 (15th change: 26 + 15 = 41; 41 - 40 = 1)

The ring returns to Child-1 after exactly 15 changes.

Incorrect Options:

Options (A) 14, (C) 16, and (D) 17 are incorrect. The detailed step-by-step calculation of the ring's position, following the described increment pattern and accounting for the circular arrangement of 40 children, demonstrates that the ring is precisely in the hands of Child-1 after the 15th change, and not at any of these other change counts.