Given a set of ‘n’ rays in a plane, we mean by ‘a reversal’ the operation of reversing precisely one ray and obtaining a new set of ‘n’ rays. Starting from 2009 rays and performing one million reversals, is it possible to reverse all the rays?

To tackle the problem of reversing rays, we first need to clarify what we mean by a "ray" and the operation of "reversing" one of them. A ray can be thought of as a line that starts at a point and extends infinitely in one direction. When we reverse a ray, we essentially flip its direction, making it point in the opposite way. Now, let’s delve into whether it’s feasible to reverse all 2009 rays after performing one million reversals.

Understanding the Reversal Operation

When we reverse a ray, we change its direction, but the total number of rays remains constant. Each reversal affects only one ray at a time. Therefore, if we start with 2009 rays, we will still have 2009 rays after any number of reversals.

Analyzing the Reversal Count

Now, let’s consider the implications of performing one million reversals. Each reversal can be seen as toggling the direction of a ray. If we reverse a ray an even number of times, it will end up in its original direction. Conversely, if we reverse it an odd number of times, it will end up in the opposite direction. This toggling effect is crucial in determining whether we can reverse all rays.

Calculating the Total Reversals

• Each ray can be reversed multiple times.
• With one million reversals, we can think of this as having a large pool of toggles.
• To reverse all 2009 rays, each ray must be toggled an odd number of times.

Since one million is an even number, if we were to distribute these reversals evenly among the rays, we would find that some rays would be toggled an even number of times, while others could be toggled an odd number of times. However, the total number of reversals must still add up to one million.

Feasibility of Reversing All Rays

To reverse all rays, we need to ensure that each ray is toggled an odd number of times. Given that we have an even total of reversals (one million), it is impossible to achieve this for all rays simultaneously. If we were to reverse each ray an odd number of times, the total number of reversals would also need to be odd, which contradicts our starting point of one million reversals.

Conclusion on the Problem

In summary, with 2009 rays and one million reversals, it is not possible to reverse all the rays. The even nature of the total number of reversals prevents us from achieving the odd toggling required for each ray to point in the opposite direction. Thus, at least one ray will remain in its original direction after all reversals are completed.