This problem involves deducing the positions of four people (P, Q, R, S) on a staircase with six steps, based on given relational conditions. We need to find S's step position if P is on the first step.

Staircase Problem: Understanding the Setup

We have a staircase with six distinct steps, numbered from 1 (ground floor) to 6 (first floor). There are four people: P, Q, R, and S. An important rule is that no two people can occupy the same step, ensuring each person has a unique position.

Analyzing the Conditions for People's Positions

Let's carefully examine each condition provided, which describes the relative positions of the people on the staircase:

• Condition I: P is two steps below R.
  This implies that R is two steps higher than P.
  We can represent this as: Position of R = Position of P + 2, or mathematically, \(R - P = 2\).
• Condition II: Q is on the next step to S.
  This means Q is directly adjacent to S, either one step above or one step below S.
  This can be written as: Position of Q = Position of S + 1, OR Position of Q = Position of S - 1.
  Mathematically, this relationship is \(|Q - S| = 1\).
• Condition III: R is two steps below S.
  This indicates that S is two steps higher than R.
  We can express this as: Position of S = Position of R + 2, or mathematically, \(S - R = 2\).

Step-by-Step Solution to Find S's Step

We are given a starting piece of information: P is on the first step. We will use this fact along with the established conditions to systematically determine the position of S on the staircase.

1. Identify the Initial Given Position:
   • The problem states that P is on the first step.
   • So, the Position of P = 1.
2. Use Condition I to Determine R's Position:
   • Condition I tells us: P is two steps below R. This means R is 2 steps higher than P.
   • Using the formula: Position of R = Position of P + 2.
   • Substitute the known position of P (which is 1): Position of R = \(1 + 2 = 3\).
   • Therefore, R is on the third step.
3. Use Condition III to Determine S's Position:
   • Condition III states: R is two steps below S. This means S is 2 steps higher than R.
   • Using the formula: Position of S = Position of R + 2.
   • Substitute the position of R (which we found to be 3): Position of S = \(3 + 2 = 5\).
   • Therefore, S is on the fifth step.
4. (Optional) Verify with Condition II for Q's Position and Consistency:
   • Condition II states: Q is on the next step to S.
   • Since S is on the fifth step (Position of S = 5), Q can be either on step 4 (5 - 1) or step 6 (5 + 1).
   • Let's check the current known positions: P=1, R=3, S=5.
   • If Q is on step 4, the positions are P=1, Q=4, R=3, S=5. All are distinct and within the 6 steps, so this is a valid arrangement.
   • If Q is on step 6, the positions are P=1, R=3, S=5, Q=6. All are distinct and within the 6 steps, so this is also a valid arrangement.
   • Since the question specifically asks only for S's position, and we have uniquely determined it, the exact position of Q is not required for the final answer, but this verification confirms the overall consistency of our deductions about the staircase setup.

Let's summarize the confirmed positions of the people based on our analysis:

Conclusion on S's Step Position

Based on our logical deduction, starting with P on the first step and applying the given conditions, we unequivocally determined that S is located on the fifth step of the staircase.