A hundred people participated in a flash mob. First, they formed a circle. MJ marked as number one had a torch. He pointed the to

Question

A hundred people participated in a flash mob. First, they formed a circle. MJ marked as number one had a torch. He pointed the torch at the person next to him, Ravi, and passed the torch to the person marked as third. Meanwhile, Ravi walked away from the circle. The passing of the torch and the walking away from the circle continued till only one dancer remained. What’s the number of the last dancer?

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2 years 2021-07-15T14:20:18+00:00 2 Answers 0 views 0

1. Given : A hundred people participated in a flash mob. First, they formed a circle. MJ marked as number one had a torch. He pointed the torch at the person next to him, Ravi, and passed the torch to the person marked as third. Meanwhile, Ravi walked away from the circle. The passing of the torch and the walking away from the circle continued till only one dancer remained.

To Find :   number of the last dancer ​

a) 88

b) 78

c) 85

d) 73

Solution:

cycle 1

walk away  2  4    6   8  10  _____________  98  100

remaining  1   3  5  7    9  _________ 97  99

Cycle 2 :

walk away     3   7    11   15  _________ 95  , 99

remaining   1    5   9   _______  93 , 97

Cycle  3

walk away       5   13    21   _______  93

Remaining   1     9   _____  89  ,  97

Cycle 4

walk away    1        17     33     49    65   81   97

remaining       9     25   41        57  73    89

Cycle 5

walk away              25       57      89

Remaining     9         41        73

Cycle 6

walk away              41

Remaining       9           73

Cycle  7

Walk away           9

Remaining 73

Hence dancer number 73 is remaining

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2. Solution :

Case 1) :-

• MJ(1) given pointed the torch to Ravi(2) and passed the torch to the person marked as third.
• Ravi(2) walked away .
• Now, (3) pointed the torch at the person next to him(4) and give torch to 5th . (4)th walked away .
• Similarly, 5th pointed the torch to 6th and he walked away and give torch to 7th .
• This process continue and in the end 99 pointed the torch to 100 and he walked away .

Conclusion :-

• All people with even positions walked away .

So,

Left People are :- 1, 3, 5 , 7, 9, 11 , ____________ 99 .

Case 2) :-

• 1 pointed the torch at the person next to him, who is 3 and he walked away and gave torch to 5th .
• 5th pointed the torch to 7th and he walked away and gave torch to 9th .
• Similarly, in the end , 97 pointed the torch to 99 and he walked away .

So,

Left People are :- 1, 5, 9 , 13, 17, 21 , ____________ 97 .

Case 3) :-

• 1 pointed the torch at the person next to him, who is 5 and he walked away and gave torch to 9th .
• 9th pointed the torch to 13th and he walked away and gave torch to 17th .
• Similarly, in the end , 89 pointed the torch to 93 and he walked away and 97 torch was in the hand of 97 .

So,

Left People are :- 1, 9, 17 , 25, _______ 89 ,97 .

Case 4) :-

• Now, since they were in circle, 97 pointed the torch at the person next to him, who is 1 and he walked away and gave torch to 9th .
• 9th pointed the torch to 17th and he walked away and gave torch to 25th .
• Similarly, in the end , 89 pointed the torch to 97 and he walked away .

So,

Left People are :- 9, 25, 41, 57, 73, 89 .

Case 5) :-

• 9 pointed the torch at the person next to him, who is 25 and he walked away and gave torch to 41 .
• 41 pointed the torch to 57th and he walked away and gave torch to 73th .
• Similarly, in the end , 73 pointed the torch to 89 and he walked away .

So,

Left People are :- 9, 41, 73 .

Case 6) :-

• 9 pointed the torch at the person next to him, who is 41 and he walked away and gave torch to 73th .

therefore, In the end left people are 9 and 73 . now, the torch is in the hand of 73 . Since they are in circle,

• Next person next to 73 is 9 .
• Now, 73 pointed the torch at 9 and he also walked away .