Question

Вдоль береговой линий острова имеющегго форму круга, расположены 2016 маяков. Нерадивый чиновник изображая бурную деятельность каждый день наудачу меняет состояние трех маяков, либо подряд расположенных, либо идущих через один ( т.е. в последовательности ABABA он меняет состояние маяков А) Чиновник будет уволен, если в какой-то момент все маяки погаснут. Стоит ли ему опасаться за свое место, если он припоминает, что в какой-то момент не горел только один маяк. ( Поменять состояние маяка, значит включить его, если он выключен, и наоборот)

Answer 1

Cстояние маяка будет хорошее нет не стоит ему опасаться за свое место !

Answer 2

The Problem:

The problem states that there are 2016 lighthouses located along the coastline of an island that has a circular shape. Every day, a careless official randomly changes the state of three lighthouses, either in a consecutive sequence or in a pattern where the changes alternate between two lighthouses (e.g., ABABA). The official will be fired if, at any point, all the lighthouses are turned off. The question is whether the official should be concerned about losing their job if they remember that at some point, only one lighthouse was not turned off.

Analysis:

To solve this problem, we need to determine if it is possible for the official to turn off all the lighthouses while leaving only one lighthouse on at some point. Let's consider the different scenarios and analyze the possibilities.

Scenario 1: Consecutive Sequence

In this scenario, the official randomly changes the state of three lighthouses in a consecutive sequence. Let's assume the lighthouses are labeled A, B, and C. The official can change the state of the lighthouses in the following ways:

1. ABC 2. BCA 3. CAB

In each of these cases, the official changes the state of three lighthouses, but there will always be at least one lighthouse that remains in the same state. Therefore, it is not possible for the official to turn off all the lighthouses while leaving only one lighthouse on in this scenario.

Scenario 2: Alternating Sequence

In this scenario, the official randomly changes the state of three lighthouses in a pattern where the changes alternate between two lighthouses. Let's assume the lighthouses are labeled A and B. The official can change the state of the lighthouses in the following ways:

1. ABABA 2. BABAB

In each of these cases, the official changes the state of three lighthouses, but there will always be at least one lighthouse that remains in the same state. Therefore, it is not possible for the official to turn off all the lighthouses while leaving only one lighthouse on in this scenario as well.

Conclusion:

Based on the analysis of the two scenarios, we can conclude that it is not possible for the official to turn off all the lighthouses while leaving only one lighthouse on, regardless of whether the changes are made in a consecutive sequence or an alternating sequence. Therefore, the official does not need to be concerned about losing their job if they remember that at some point, only one lighthouse was not turned off.

Please note that the analysis provided above is based on the information given in the problem statement. If there are any additional constraints or details that need to be considered, please let me know.