Question

Есть три деревянных столбика разной высоты. От любого столбика можно отпилить разность между высотами других двух столбиков. Можно ли, повторив эту операцию несколько раз, получить три столбика одинаковой высоты?

Answer 1

Ответ нет можно судя по логике для примера это не может продолжаться бесконечно

Answer 2

Problem Analysis

We have three wooden posts of different heights, and we can cut off the difference in height between any two posts. The question is whether it is possible to repeat this operation multiple times and eventually end up with three posts of the same height.

Solution

No, it is not possible to obtain three posts of the same height by repeating this operation multiple times. Let's consider the initial heights of the three posts as A, B, and C, where A > B > C.

After the first operation, we can cut off the difference between A and B, resulting in two posts of heights B and C. Let's call this new configuration A1.

After the second operation, we can cut off the difference between B and C, resulting in two posts of heights B-C and C. Let's call this new configuration A2.

At this point, we have two posts of different heights, and no matter how many times we repeat the operation, we will always end up with two posts of different heights. Therefore, it is not possible to obtain three posts of the same height by repeating this operation.

Example

Let's consider an example to illustrate this:

Initial heights: A = 10, B = 7, C = 4

After the first operation, we cut off the difference between A and B (10 - 7 = 3), resulting in two posts of heights 7 and 4.

After the second operation, we cut off the difference between B and C (7 - 4 = 3), resulting in two posts of heights 3 and 4.

No matter how many times we repeat the operation, we will always end up with two posts of different heights.

Conclusion

In conclusion, it is not possible to obtain three wooden posts of the same height by repeating the operation of cutting off the difference in height between any two posts.