Question

В двух кучках по 100 камушков. На первом шаге перекладываем 5 камушков из первой кучки во вторую. На втором шаге перекладываем 6 камушков из второй куч- ки в первую. Затем снова перекладываем 5 камушков из первой кучки во вторую, потом перекладываем 6 камушков из второй кучки в первую. И так далее. Через сколько шагов обе кучки снова будут содержать по 100 камушков?

Answer 1

Через 11 шагов.  
95-105
101-99
96-104
102-98
97-103
103-97
98-102
104-96
99-101
105-95
100-100

Answer 2

Problem Analysis

We have two piles of stones, each initially containing 100 stones. On each step, we transfer 5 stones from the first pile to the second pile, and then transfer 6 stones from the second pile to the first pile. We need to determine how many steps it will take for both piles to again contain 100 stones each.

Solution

Let's analyze the number of stones in each pile after each step:

- Step 1: Transfer 5 stones from the first pile to the second pile. - First pile: 100 - 5 = 95 stones - Second pile: 100 + 5 = 105 stones

- Step 2: Transfer 6 stones from the second pile to the first pile. - First pile: 95 + 6 = 101 stones - Second pile: 105 - 6 = 99 stones

- Step 3: Transfer 5 stones from the first pile to the second pile. - First pile: 101 - 5 = 96 stones - Second pile: 99 + 5 = 104 stones

- Step 4: Transfer 6 stones from the second pile to the first pile. - First pile: 96 + 6 = 102 stones - Second pile: 104 - 6 = 98 stones

By analyzing the pattern, we can see that after each step, the total number of stones in both piles decreases by 1. This is because we transfer 5 stones from the first pile to the second pile and then transfer 6 stones from the second pile to the first pile. Therefore, after every two steps, the total number of stones decreases by 2.

To find the number of steps required for both piles to contain 100 stones each, we need to calculate how many times the total number of stones can decrease by 2 until it reaches 0.

Let's calculate the number of steps:

- Initial total number of stones: 100 + 100 = 200 - Desired total number of stones: 100 + 100 = 200

The number of steps required is equal to the difference between the initial and desired total number of stones divided by 2:

Number of steps = (Initial total number of stones - Desired total number of stones) / 2

Number of steps = (200 - 200) / 2 = 0 / 2 = 0

Therefore, it will take 0 steps for both piles to again contain 100 stones each.

Answer

After analyzing the given scenario, it is determined that both piles will contain 100 stones each again after 0 steps.

Please let me know if you need any further clarification.