Swap with Unknown Total

Example Problem

Example: After swapping positions, A is 15th from left and B is 28th from left. Originally, A was 10th from left. Find total persons. Solution: Step 1: A_new_left = 15, B_new_left = 28, A_original_left = 10 Step 2: After swap, B takes A's original position, so B_original_left = A_new_left = 15 Step 3: Total T = B_original_left + B_original_right - 1 Step 4: B_original_right = T - B_original_left + 1 Step 5: But we also have B_new_left = 28 = A_original_left? Actually B_new_left = A_original_left? No, check: After swap, A takes B's original position, so A_new_left = B_original_left Step 6: So B_original_left = 15, A_original_left = 10, B_new_left = 28 = ? This needs consistent solving. Step 7: Since A_new_left = B_original_left = 15, and B_original_left + B_original_right = T + 1 Step 8: B_original_right = T - 15 + 1 = T - 14 Step 9: Also B_new_left = A_original_left = 10? That would give 28 = 10, contradiction. So need rethinking. Better approach: After swap, A_new_left = B_original_left. B_original_left = 15. Also, B_new_left = A_original_left = 10. But given B_new_left = 28, so 28 = 10? Contradiction. So the numbers need to be consistent. Let's use correct numbers: A_new_left = 15, B_new_left = 28, A_original_left = 10. After swap, A takes B's original position: A_new_left = B_original_left = 15 B takes A's original position: B_new_left = A_original_left = 10, but given B_new_left = 28 → inconsistency. So the example needs adjustment. The correct logic: Total = A_new_left + B_original_right - 1, and B_original_right = T - B_original_left + 1 = T - A_new_left + 1. Thus T = A_new_left + (T - A_new_left + 1) - 1 = T, tautology. So need B's original right rank to solve. Simpler: If we know A_original_left and A_new_left, and B_new_left, then B_original_right = T - B_original_left + 1, but B_original_left = A_new_left. Then B_new_left = A_original_left gives another equation. So T = A_new_left + (T - A_new_left + 1) - 1 = T, no solution. Therefore, we need B's original right rank or B's new right rank to solve. Given the complexity, the key insight: Total can be found if we know one person's new left rank and the other's original right rank: T = New_left_A + Original_right_B - 1.