Cycle length of passing a ring in a circle
Question
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?
Options
14
15
16
17
Explanation
Track the positions of the ring by calculating cumulative movements modulo 40[cite: 3989, 3990]. The jump step size increases by 1 each time :
The position after ⟨MATH⟩n⟨/MATH⟩ changes is given by the triangular number formula: P_n = 1 + n(n+1)/2. We need the ring to return to Child 1, meaning n(n+1)/2 must be a clean multiple of 40 (40k)[cite: 3989, 3992]. This means n(n+1) must be a multiple of 80.
Test consecutive values for n to find a pair of consecutive numbers whose product is a multiple of 80:
Let's verify the position: P₍15₎ = 1 + 15 × 16/2 = 1 + 120 = 121. 121 ± od{40} = 1, which confirms the ring lands exactly back in Child 1's hands after 15 changes.
Answer: (b).
Question details
Year
2023
Paper
CSAT
Question
Q54
Section
Logical & Analytical Reasoning
Sub-topic
Number Sequence / Patterns
Type
Sequence & pattern
Difficulty
Hard
Source hint
Pattern recognition
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