Staircase People Arrangement Solution

This problem involves understanding the relative positions of four people (P, Q, R, S) on a six-step staircase. We need to use the given conditions and a specific starting point to determine their exact or possible positions and then evaluate the truthfulness of the provided statements.

Understanding the Staircase Constraints

• The staircase has six steps, numbered 1 to 6 (from ground floor to first floor).
• There are four people: P, Q, R, and S.
• A crucial rule is that no two people can be on the same step, meaning each person occupies a unique step.

Analyzing the Relational Statements

Let's translate the given statements into simple mathematical relationships, where a person's name represents their step number.

1. P's Relationship with R: "P is two steps below R."
   This means R is two steps higher than P.
   Mathematically, this can be written as: \(R = P + 2\)
2. Q's Relationship with S: "Q is on the next step to S."
   This implies that Q is immediately adjacent to S, either one step above or one step below S.
   Mathematically, this means: \(Q = S - 1\) or \(Q = S + 1\)
3. R's Relationship with S: "R is two steps below S."
   This means S is two steps higher than R.
   Mathematically, this can be written as: \(S = R + 2\)

Applying P's Position to Deduce Others

The question provides a specific condition: P is on the first step. Let's use this information to deduce the positions of the other people step-by-step.

• Step 1: Determine R's position.
  We are given that P is on the first step, so \(P = 1\).
  From statement I, we know \(R = P + 2\).
  Substituting the value of P: \(R = 1 + 2 = 3\).
  So, R is on the third step.
• Step 2: Determine S's position.
  From statement III, we know \(S = R + 2\).
  We just found that R is on step 3, so substituting R's value: \(S = 3 + 2 = 5\).
  So, S is on the fifth step.
• Step 3: Determine Q's possible positions.
  From statement II, Q is on the next step to S. We know S is on step 5.
  This means Q can be either one step below S or one step above S.
  Possible positions for Q:
  • \(Q = S - 1 = 5 - 1 = 4\). So, Q could be on the fourth step.
  • \(Q = S + 1 = 5 + 1 = 6\). So, Q could be on the sixth step.

So far, we have the following fixed and possible positions:

All these positions (1, 3, 5, and either 4 or 6) are unique and within the 1 to 6 range of the staircase.

Evaluating Each Option's Truthfulness

Now let's check each given option based on our deductions:

1. "Q is on the second step"
   Based on our analysis, Q can be on the 4th or 6th step, but not necessarily on the 2nd step. Therefore, this statement is not true.
2. "R is on the fourth step"
   We deduced that R is on the 3rd step. Therefore, this statement is not true.
3. "S is on a higher step than R"
   We found S is on the 5th step and R is on the 3rd step. Since 5 is greater than 3, S is indeed on a higher step than R. This statement is true.
4. "Step four is empty"
   Q can be on either the 4th or 6th step. If Q is on the 4th step, then step four is not empty. Since it's not definitively empty, we cannot conclude that this statement is always true.

Conclusion on People's Positions

Based on the logical deductions from the given statements and the condition that P is on the first step, only one statement consistently holds true.

The statement "S is on a higher step than R" is the only true statement among the given options because S is on step 5 and R is on step 3.