Logical Analysis of Ranks and Marks

To solve this problem, we need to analyze how the relationship between the marks given by the two judges affects the final ranking of the participants.

1. The Relationship Between Marks

The problem states that if the first judge gives marks 'x', the second judge gives marks 'y', according to the formula:

y = 1 + x

This means that for every participant, Judge 2's score is always exactly one point higher than Judge 1's score.

2. The Effect on Ranking

A participant's rank is determined by their score relative to the scores of all other participants. The person with the highest score gets rank 1, the second highest gets rank 2, and so on.

Let's consider two participants, P1 and P2.

• Judge 1 gives them marks x₁ and x₂.
• Judge 2 gives them marks y₁ = x₁ + 1 and y₂ = x₂ + 1.

If Judge 1 ranks P1 higher than P2, it means x₁ > x₂. If we add 1 to both sides of this inequality, the relationship remains the same: x₁ + 1 > x₂ + 1. This means y₁ > y₂, so Judge 2 will also rank P1 higher than P2.

Since adding a constant value to every participant's score does not change their relative order, the final ranking given by both judges will be identical.

Conclusion

Based on a logical analysis of how ranking works, the ranks given by both judges for any participant will be the same. This would make option D the correct answer.