Staircase Logic Puzzle Solution

This problem involves determining the exact positions of five people (P, Q, R, S, and T) on a six-step staircase based on a set of relative positioning clues. The goal is to identify which step remains vacant after all positions are determined.

Understanding the Staircase and Constraints

• There are six steps, numbered 1 (representing the first step from the ground floor) to 6 (representing the sixth step leading to the first floor).
• Five people are involved in the arrangement: P, Q, R, S, and T.
• A crucial rule is that no two people can be on the same step. This means each person must occupy a unique step.

Analyzing the Initial Clues (P, Q, R, S)

Let's use the letters P, Q, R, and S to represent the specific step number each person occupies. We will break down each clue to understand their relative positions:

1. Clue I: P is two steps below R.
   • This implies that the step number of R is exactly 2 more than the step number of P.
   • In mathematical terms: $\text{R} = \text{P} + 2$, or equivalently, $\text{R} - \text{P} = 2$.
2. Clue II: Q is on the next step to S.
   • This indicates that Q and S are on adjacent steps. This means there is only one step difference between them.
   • Mathematically: $|\text{Q} - \text{S}| = 1$. This implies Q could be on the step immediately above S ($\text{Q} = \text{S} + 1$) or immediately below S ($\text{Q} = \text{S} - 1$).
3. Clue III: R is two steps below S.
   • This means that the step number of S is exactly 2 more than the step number of R.
   • In mathematical terms: $\text{S} = \text{R} + 2$, or equivalently, $\text{S} - \text{R} = 2$.

Deducing Relative Positions of P, R, and S

By combining Clue I ($\text{R} = \text{P} + 2$) and Clue III ($\text{S} = \text{R} + 2$), we can establish a consistent pattern for P, R, and S:

• If we assume P is on step $x$, then:
• R is on step $x+2$ (from Clue I).
• S is on step $(x+2)+2 = x+4$ (from Clue III).
• So, the positions of P, R, and S are consistently spread out at two-step intervals: $x$, $x+2$, and $x+4$.

Integrating Q's Position with P, R, S

Now, let's incorporate Clue II: Q is on the next step to S. Since S is on step $x+4$, Q can be on step $x+3$ (one step below S) or $x+5$ (one step above S). This leads to two potential arrangements for the four people (P, Q, R, S):

• Arrangement 1: Q is on step $x+3$ (i.e., $\text{Q} = \text{S} - 1$).
  • The relative steps for P, R, Q, S would be:
  • P on step $x$
  • R on step $x+2$
  • Q on step $x+3$
  • S on step $x+4$
  • This arrangement satisfies all initial clues: $\text{R}-\text{P}=2$, $\text{S}-\text{R}=2$, and $\text{S}-\text{Q}=1$.
• Arrangement 2: Q is on step $x+5$ (i.e., $\text{Q} = \text{S} + 1$).
  • The relative steps for P, R, S, Q would be:
  • P on step $x$
  • R on step $x+2$
  • S on step $x+4$
  • Q on step $x+5$
  • This arrangement also satisfies all initial clues: $\text{R}-\text{P}=2$, $\text{S}-\text{R}=2$, and $\text{Q}-\text{S}=1$.

Applying the New Information about T and Q

The problem introduces a fifth person, T, with additional specific conditions:

• T was on the third step ($\text{T}=3$).
• Q was on a higher step than T ($\text{Q} > \text{T}$, meaning $\text{Q} > 3$).

We will now evaluate both possible arrangements against these new conditions, keeping in mind the "no two people on the same step" rule:

Evaluating Arrangement 1: P($x$), R($x+2$), Q($x+3$), S($x+4$)

• The highest step in this arrangement is $x+4$. Since the staircase has only 6 steps, $x+4$ must be less than or equal to 6 ($\text{x}+4 \le 6$), which means $x \le 2$.
• Since step numbers start from 1, $x$ must be greater than or equal to 1 ($\text{x} \ge 1$).
• Therefore, possible values for $x$ are 1 or 2.
• Also, we know $\text{Q} = x+3$, and the condition is $\text{Q} > 3$. So, $x+3 > 3$, which simplifies to $x > 0$. Both $x=1$ and $x=2$ satisfy this.

Let's test each possible value for $x$:

• Case $x=1$:
  • P is on step 1.
  • R is on step $1+2=3$.
  • Q is on step $1+3=4$.
  • S is on step $1+4=5$.
  • T is on step 3 (given).
  • Conflict: Both R and T would be on step 3. This violates the rule that no two people can be on the same step. Thus, $x=1$ is not a valid solution for Arrangement 1.
• Case $x=2$:
  • P is on step 2.
  • R is on step $2+2=4$.
  • Q is on step $2+3=5$.
  • S is on step $2+4=6$.
  • T is on step 3 (given).
  • Let's check the conditions:
    • The occupied steps are 2, 3, 4, 5, 6. All are distinct and within the 1-6 range.
    • $\text{Q}=5$ and $\text{T}=3$. The condition $\text{Q} > \text{T}$ (5 > 3) is satisfied.
  • This arrangement is consistent with all given conditions.

Evaluating Arrangement 2: P($x$), R($x+2$), S($x+4$), Q($x+5$)

• The highest step in this arrangement is $x+5$. Since there are 6 steps, $x+5 \le 6$, which implies $x \le 1$.
• Since $x \ge 1$, the only possible value for $x$ is 1.
• We also know $\text{Q} = x+5$, and the condition is $\text{Q} > 3$. So, $x+5 > 3$, which simplifies to $x > -2$. $x=1$ satisfies this.

Let's test the only possible value for $x$:

• Case $x=1$:
  • P is on step 1.
  • R is on step $1+2=3$.
  • S is on step $1+4=5$.
  • Q is on step $1+5=6$.
  • T is on step 3 (given).
  • Conflict: Both R and T would be on step 3. This violates the rule that no two people can be on the same step. Thus, $x=1$ is not a valid solution for Arrangement 2.

Determining the Final Occupied Steps

Based on our detailed evaluation, only Arrangement 1 with $x=2$ proves to be valid under all given conditions. This gives us the following definitive positions for the five people on the staircase:

• P is on step 2.
• T is on step 3.
• R is on step 4.
• Q is on step 5.
• S is on step 6.

Staircase Step Number	Person Occupying the Step
1	Vacant
2	P
3	T
4	R
5	Q
6	S

Identifying the Vacant Step

The staircase consists of steps numbered 1 through 6. We have determined that steps 2, 3, 4, 5, and 6 are occupied by P, T, R, Q, and S respectively. Therefore, the only step remaining unoccupied is the first step.