Shooting Zone

Let "the champion" be the living person with the smallest index, and let a "turn" end when the gun crosses the 100-1 line. After every turn, the champion changes if and only if the number of survivors is odd; in that case its index increases by a power of two given by the turn number. This gives that if $n$ people around a table play this kind of determistic russian roulette, the index of the last survivor is given by $2\cdot\left(n-2^{\left\lfloor\log_2 n \right\rfloor}\right)+1,$ that is a cyclic shift of the binary representation of $n$ , where the leading $1$ becomes the least significant bit.

This can be solved by computer science: using C/C++ complier: //Linked List struct Node { int _Val; Node* _pNext; };

void deleteNode(Node* & head, Node* _del) { Node* cur = _head; if (cur == _del) { Node* cur 2 = head; while (cur 2-> pNext != _head) cur 2 = cur 2-> pNext; //cur 2 in the tail Node* temp = cur-> pNext; delete cur; head = temp; cur 2-> pNext = _head; } else { while (cur-> pNext != head) { if (cur-> pNext == del) { Node* temp = cur-> pNext-> pNext; delete cur-> pNext; cur-> pNext = temp; } cur = cur-> pNext; } } }

int countEle(Node* head) { int c = 1; Node* cur = _head; if (cur-> pNext == head) return c; while (cur-> pNext != head) { c++; cur = cur-> pNext; } return c; }

void main() { Node* head = NULL; //Create circle linked list for (int i = 1; i <= 100; i++) { if (head == NULL) { head = new Node; head-> Val = i; head-> pNext = head; } else { Node* cur = head; while (cur-> pNext != head) cur = cur-> pNext; cur-> pNext = new Node; cur-> pNext-> Val = i; cur-> pNext-> pNext = head; } } //============================ Node* cur = head; while (countEle(head) != 1) { deleteNode(head, cur-> pNext); cur = cur-> pNext; } cout << cur-> Val << endl; }