Question

В новогоднюю ночь на подоконнике стояли в ряд (слева на право) фикус, ирис и кактус. каждое утро Маша, вытирая пыль, меняет местами цветок справа и цветок в центре. Днём Таня,поливая цветы, меняет местами тот, что в центре, с тем, что с лева. В каком порядке будут стоять цветы через 365 дней в следующую новогоднюю ночь?

Answer 1

Problem Analysis

In this problem, we are given that there are three plants on a windowsill: a ficus, an iris, and a cactus. Every morning, Masha swaps the rightmost plant with the plant in the center while dusting. During the day, Tanya swaps the center plant with the leftmost plant while watering. We need to determine the order of the plants after 365 days, on the next New Year's Eve.

Solution

To solve this problem, let's analyze the movements of the plants over time.

Initially, the order of the plants from left to right is: ficus, iris, cactus.

After Masha swaps the rightmost plant with the plant in the center, the order becomes: ficus, cactus, iris.

After Tanya swaps the center plant with the leftmost plant, the order becomes: cactus, ficus, iris.

Let's observe the pattern of the plant movements:

1. After each morning, the rightmost plant becomes the center plant, and the center plant becomes the rightmost plant. 2. After each day, the center plant becomes the leftmost plant, and the leftmost plant becomes the center plant.

Since there are 365 days in a year, we need to determine the final order of the plants after 365 days.

Let's simulate the movements of the plants:

1. After the first morning, the order becomes: ficus, cactus, iris. 2. After the first day, the order becomes: cactus, ficus, iris. 3. After the second morning, the order becomes: cactus, iris, ficus. 4. After the second day, the order becomes: iris, cactus, ficus. 5. After the third morning, the order becomes: iris, ficus, cactus. 6. After the third day, the order becomes: ficus, iris, cactus.

We can observe that the plants return to their initial order after every three days. Since 365 is not divisible by 3, we need to determine the remainder when 365 is divided by 3.

365 divided by 3 gives a quotient of 121 and a remainder of 2. This means that after 365 days, the plants will be in the same order as they were after 2 days.

After 2 days, the order of the plants is: iris, cactus, ficus.

Therefore, after 365 days, on the next New Year's Eve, the order of the plants will be: iris, cactus, ficus.

Answer

After 365 days, on the next New Year's Eve, the order of the plants on the windowsill will be: iris, cactus, ficus.