# 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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