Puzzle 81 | 100 people in a circle with gun puzzle

100 people standing in a circle in an order 1 to 100. No. 1 has a sword. He kills the next person (i.e. No. 2) and gives the sword to the next (i.e. No. 3). All people do the same until only 1 survives. Which number survives at the last?
There are 100 people starting from 1 to 100.

 
 
 
 

Solution:  73rd person will survive at last

Explanation: 
Here, we can define an array with 100 elements with values from 1 to 100.

  • No.1 has a sword. He kills next person (i.e. no. 2) and gives sword to next to next (i.e no. 3).
    We have taken array element as a person. 1st person kills the next. So, starting from 1, we’ll remove next element i.e. 2.
  • Then first person gives sword to next to next i.e. 3. That person will also kill next person and this continues. Means, in array, we need to start with 1 and remove the every other (alternate) element till 100. (all the even numbers will be removed and we’ll be left with odd numbers only in array).

Round 1: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99
Round 2: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97
Round 3: 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97
Round 4: 9, 25, 41, 57, 73, 89
Round 5: 9, 41, 73
Round 6: 9, 73
Round 7: 73

To avoid the manual calculation done above, here the general algorithm :
Step 1 : For a given value of N, find the “Power of 2” immediately greater than N. Let’s call it P
Step 2 : Subtract N from (P-1). Lets call it M, i.e, M = (P-1)- N
Step 3 : Multiply M by 2. i.e M*2
Step 4 : Subtract M*2 from P-1. Let’s call it ans, i.e, ans = (P-1) – (M*2)
So, the person with number “ans” will survive till last.

This puzzle is contributed by Jyoti. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



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