Suppose that Mr. X will be giving away a golden homework pass to one student in each class. The recipient of this pass is excused from homework all year. He has developed a unique system for determining who will get the pass.

1. First, he wants all students, x, who are present today to stand in a circle
2. Second, he will decide on a random number and call it the "Eliminumber"
3. Third, he will randomly select a starting position and count up to his "Eliminumber" The student standing in the "Eliminumber" is eliminated. The count then starts over at the next student in the circle, until the "Eliminumber (which does not change) is reached again, eliminating the next student. This process will repeat until only one student remains

TASK: Determine the position in the initial circle (after Mr. X tells you a new "Eliminumber") that will win the golden homework pass if the activity was played again. If you complete this your task is you find your position in any sized circle for any given "Eliminumber".

To determine the position in the initial circle that will win the golden homework pass, we need to understand the pattern of elimination.

Let's take an example to illustrate the steps. Suppose there are 10 students in the initial circle (positions numbered 1 to 10) and Mr. X chooses an "Eliminumber" of 3.

1. In the first round, starting from position 1, students 1, 2, and 3 will be eliminated. The circle now becomes: 4, 5, 6, 7, 8, 9, 10.

2. In the second round, starting from position 4 (the next student after 3), students 4, 5, and 6 will be eliminated. The circle now becomes: 7, 8, 9, 10, 4.

3. In the third round, starting from position 7 (the next student after 6), students 7, 8, and 9 will be eliminated. The circle now becomes: 10, 4, 7.

4. In the fourth round, starting from position 10 (the next student after 9), students 10 and 4 will be eliminated. The circle now becomes: 7.

Therefore, position 7 is the position that wins the golden homework pass in this example.

To find your position in any sized circle for any given "Eliminumber," you can follow these steps:

1. Write down the positions of all the students in a clockwise direction starting from 1 in a circle.

2. Determine the "Eliminumber" given to you by Mr. X.

3. Starting from position 1, count the number of students according to the "Eliminumber." The student at the "Eliminumber" position gets eliminated.

4. Repeat step 3 from the next student in the circle, until there is only one student remaining.

The position of the last remaining student will be the position that wins the golden homework pass in the initial circle.

By following these steps, you can determine the winning position for any sized circle and any given "Eliminumber."