Persons on Steps: Analyzing Placement Rules

This problem involves determining the possible position of person A on a set of steps, given certain conditions about the relative positions of different people.

Understanding the Conditions

We have five people (A, B, C, D, E) standing on six steps numbered 1 to 6 from the bottom. The rules governing their positions are:

• Each step can have at most one person.
• The step number where A is standing is two less than that of C. This can be written as:

  Step(A) = Step(C) - 2

  Or equivalently,

  Step(C) = Step(A) + 2

• The step number where B is standing is one more than that of D. This can be written as:

  Step(B) = Step(D) + 1

Applying the Given Scenario

We are told that person D is standing on step 1.

Using the second condition, Step(B) = Step(D) + 1:

• Since D is on step 1, B must be on step $1 + 1 = 2$.

So, steps 1 and 2 are occupied by D and B, respectively.

Determining Possible Steps for A

The steps available for the remaining people (A, C, and E) are 3, 4, 5, and 6.

Now, let's use the first condition, Step(C) = Step(A) + 2, to find the possible positions for A among the available steps {3, 4, 5, 6}. We need to find pairs of steps $(A, C)$ from this set where C's step number is exactly 2 greater than A's step number.

• Possibility 1: If A is on step 3.
  • Then C must be on step $3 + 2 = 5$.
  • Both step 3 and step 5 are available in the set {3, 4, 5, 6}.
  • This is a valid possibility. So, A could be on step 3.
• Possibility 2: If A is on step 4.
  • Then C must be on step $4 + 2 = 6$.
  • Both step 4 and step 6 are available in the set {3, 4, 5, 6}.
  • This is also a valid possibility. So, A could be on step 4.
• Possibility 3: If A is on step 5.
  • Then C must be on step $5 + 2 = 7$.
  • Step 7 is not available as there are only 6 steps.
  • This is not a valid possibility.
• Possibility 4: If A is on step 6.
  • Then C must be on step $6 + 2 = 8$.
  • Step 8 is not available.
  • This is not a valid possibility.

Therefore, the only possible steps on which A could be standing are step 3 or step 4.

Matching with Options

Based on our analysis, A can be on step 3 or step 4. Comparing this with the given options:

• Option 1: 2 or 4 only (Incorrect)
• Option 2: 3 or 5 only (Incorrect)
• Option 3: 3 or 4 only (Correct)
• Option 4: 4 or 5 only (Incorrect)

The correct option states that A could be standing on step 3 or step 4.