This question presents a logical reasoning puzzle centered around the placement of individuals on a staircase. We are given specific conditions about the relative positions of four people (P, Q, R, S) on a six-step staircase. The core rule is that no two people can occupy the same step. Our task is to determine the correct arrangement of people based on these initial conditions and then apply an additional condition involving a fifth person (T) to identify which of the given pairs of person-step assignments is incorrect.

Staircase Problem Setup

We have a staircase with six steps, numbered 1 through 6, starting from the ground floor. There are four people, P, Q, R, and S. A fundamental rule is that no two people can be on the same step. We are provided with three statements describing their positions:

• Statement I: P is two steps below R.
• Statement II: Q is on the next step to S.
• Statement III: R is two steps below S.

People's Positions Deduction from Statements

Let's use the given statements to systematically deduce the possible positions of P, R, and S first, as their relationships are directly linked.

• From Statement I, "P is two steps below R," we can express this relationship mathematically:
  $ \text{R} - \text{P} = 2 $
  This implies that if P is on a certain step, R will be exactly two steps higher.
• From Statement III, "R is two steps below S," we can write:
  $ \text{S} - \text{R} = 2 $
  This implies that if R is on a certain step, S will be exactly two steps higher.

Now, by combining these two relationships, we can find the relationship between P and S. Since $ \text{R} = \text{P} + 2 $ and $ \text{S} = \text{R} + 2 $, we can substitute the value of R into the second equation:

$ \text{S} = (\text{P} + 2) + 2 $

$ \text{S} = \text{P} + 4 $

This means P, R, and S are positioned such that R is two steps above P, and S is two steps above R (or four steps above P). They maintain a consistent two-step interval between each other. Given that the staircase has 6 steps (1 to 6), let's list the possible absolute step numbers for P, R, and S:

• Possibility 1: If P is on step 1, then R would be on step $1 + 2 = 3$, and S would be on step $1 + 4 = 5$. This arrangement (P=1, R=3, S=5) is valid as all positions are within the 1-6 steps.
• Possibility 2: If P is on step 2, then R would be on step $2 + 2 = 4$, and S would be on step $2 + 4 = 6$. This arrangement (P=2, R=4, S=6) is also valid as all positions are within the 1-6 steps.
• Possibility 3: If P is on step 3, then R would be on step $3 + 2 = 5$, and S would be on step $3 + 4 = 7$. This arrangement is not valid because S would be on step 7, and the staircase only has 6 steps.

Thus, we have two primary possible sets of positions for P, R, and S:

Next, let's incorporate Statement II: "Q is on the next step to S." This means Q and S are adjacent steps. Therefore, Q can be either $ \text{S} - 1 $ or $ \text{S} + 1 $.

• For Arrangement A (where S is on step 5):
  • If Q is on step $ \text{S} + 1 = 5 + 1 = 6 $. This is a valid step. In this case, the people are P=1, R=3, S=5, Q=6. All positions are distinct.
  • If Q is on step $ \text{S} - 1 = 5 - 1 = 4 $. This is a valid step. In this case, the people are P=1, R=3, S=5, Q=4. All positions are distinct.
• For Arrangement B (where S is on step 6):
  • If Q is on step $ \text{S} + 1 = 6 + 1 = 7 $. This is not valid because there are only 6 steps on the staircase.
  • If Q is on step $ \text{S} - 1 = 6 - 1 = 5 $. This is a valid step. In this case, the people are P=2, R=4, S=6, Q=5. All positions are distinct.

So, considering all initial statements, we have three possible complete arrangements for P, Q, R, and S:

T's Position and Arrangement Validity

The question provides an additional condition: "If T was on the third step". We must now check which of the above three arrangements remains valid when T is placed on step 3, remembering the rule that no two people can be on the same step.

• Checking Scenario A-1 with T on step 3: In this arrangement, P is on step 1, Q on step 6, R on step 3, and S on step 5. If T is also placed on step 3, this creates a conflict because both R and T would be on the same step (step 3), violating the rule "No two people can be on the same step". Therefore, this arrangement is invalid under the given condition.
• Checking Scenario A-2 with T on step 3: In this arrangement, P is on step 1, Q on step 4, R on step 3, and S on step 5. Similarly, if T is placed on step 3, it causes a conflict with R being on the same step. Thus, this arrangement is also invalid.
• Checking Scenario B-1 with T on step 3: In this arrangement, P is on step 2, Q on step 5, R on step 4, and S on step 6. If T is placed on step 3, all five people (P=2, Q=5, R=4, S=6, T=3) are on distinct steps. There is no conflict, making this the only valid arrangement under the given condition.

Arrangement Summary

Based on all the provided conditions, including T's position, the final valid arrangement of the people on the staircase is:

Incorrect Pair Identification

Now, we will evaluate each of the given options against our derived final valid arrangement to determine which pair is incorrect.

• Option 1: Q-fifth step

  Our derived arrangement shows that Q is on step 5. This matches the option.

  This pair is Correct.

• Option 2: R-fourth step

  Our derived arrangement shows that R is on step 4. This matches the option.

  This pair is Correct.

• Option 3: Q-fourth step

  Our derived arrangement shows that Q is on step 5, not step 4. This does not match the option.

  This pair is Incorrect.

• Option 4: S-sixth step

  Our derived arrangement shows that S is on step 6. This matches the option.

  This pair is Correct.

The question asks for the incorrect pair. Based on our detailed analysis, the pair "Q-fourth step" is incorrect because Q is actually on the fifth step.