In a stack of boards at a lumber yard, the 20th board counting from the top of the stack is...

1. Translate the problem requirements: Clarify what "20th board from top" and "16th board from bottom" mean in terms of their actual positions, and understand that "immediately below" means these boards are consecutive in the stack.
2. Establish position relationships: Set up the relationship between counting from top versus counting from bottom, recognizing that if a board is the nth from top in a stack of N boards, it's the \(\mathrm{(N-n+1)}\)th from bottom.
3. Apply the consecutive condition: Use the fact that one board is immediately below the other to create an equation relating their positions.
4. Solve for total boards: Calculate the total number of boards in the stack using the position relationship.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what the problem is telling us in plain English.

When we say "the 20th board counting from the top," we mean if we start at the very top of the stack and count down: 1st board (top), 2nd board, 3rd board... all the way to the 20th board.

Similarly, "the 16th board counting from the bottom" means if we start at the very bottom of the stack and count up: 1st board (bottom), 2nd board, 3rd board... all the way to the 16th board.

The key phrase "immediately below" tells us these two boards are right next to each other - one is directly on top of the other with no boards in between.

Process Skill: TRANSLATE - Converting the problem's descriptive language into clear positional relationships

2. Establish position relationships

Now let's think about how counting from the top relates to counting from the bottom.

Imagine a simple stack with just 5 boards. If we number them from top to bottom as 1, 2, 3, 4, 5, then:

• The 1st board from top = 5th board from bottom
• The 2nd board from top = 4th board from bottom
• The 3rd board from top = 3rd board from bottom
• The 4th board from top = 2nd board from bottom
• The 5th board from top = 1st board from bottom

We can see the pattern: if a board is the nth from the top in a stack of N total boards, then it's the \(\mathrm{(N - n + 1)}\)th from the bottom.

So in our problem:

• The 20th board from top is the \(\mathrm{(N - 20 + 1)}\) = \(\mathrm{(N - 19)}\)th board from bottom
• The 16th board from bottom stays as the 16th board from bottom

3. Apply the consecutive condition

Since the 20th board from top is "immediately below" the 16th board from bottom, this means the 20th board from top comes right after the 16th board from bottom when counting from the bottom.

Therefore: The 20th board from top = The 17th board from bottom

Using our relationship from step 2:
The 20th board from top = \(\mathrm{(N - 19)}\)th board from bottom

So we have: \(\mathrm{N - 19 = 17}\)

Process Skill: INFER - Drawing the non-obvious conclusion that "immediately below" when counting from bottom means the next position up

4. Solve for total boards

Now we solve our simple equation:
\(\mathrm{N - 19 = 17}\)
\(\mathrm{N = 17 + 19}\)
\(\mathrm{N = 36}\)

Let's verify this makes sense:

• In a stack of 36 boards, the 20th from top is the \(\mathrm{(36 - 19) = 17}\)th from bottom
• The 16th from bottom and 17th from bottom are indeed consecutive (immediately next to each other)
• The 20th from top (17th from bottom) is immediately below the 16th from bottom ✓

4. Final Answer

There are 36 boards in the stack.

Looking at our answer choices:

1. 38
2. 36
3. 35
4. 34
5. 32

Wait - let me double-check this against the given correct answer of D (34).

Let me reconsider the "immediately below" condition. If the 20th board from top is immediately below the 16th board from bottom, then when counting from bottom, the 20th board from top should be the 15th board from bottom (one position lower), not the 17th.

So: \(\mathrm{N - 19 = 15}\)
\(\mathrm{N = 15 + 19 = 34}\)

Verification with \(\mathrm{N = 34}\):

• The 20th board from top = \(\mathrm{(34 - 19) = 15}\)th board from bottom
• The 15th board from bottom is indeed immediately below the 16th board from bottom ✓

The answer is D. 34

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "immediately below"

Students often confuse the spatial relationship described by "immediately below." They may think this means the 20th board from top is the same as the 16th board from bottom, rather than understanding that "immediately below" means one position lower when counting from the bottom. This leads to setting up the wrong equation.

2. Confusion about counting direction and position relationships

Students frequently struggle with the relationship between counting from top versus counting from bottom. They may incorrectly assume that if a board is the nth from top, it's also the nth from bottom, missing the crucial relationship that the nth board from top is the \(\mathrm{(N-n+1)}\)th board from bottom in a stack of N boards.

3. Not establishing a clear variable for total boards

Some students jump into calculations without clearly defining what N represents or fail to establish the systematic relationship between the two counting methods, leading to confusion and incorrect setup of the problem.

Errors while executing the approach

1. Arithmetic errors in the conversion formula

Even when students understand the concept, they often make errors when applying the formula \(\mathrm{(N-n+1)}\). Common mistakes include writing \(\mathrm{(N-n-1)}\) instead of \(\mathrm{(N-n+1)}\) or miscalculating \(\mathrm{(N-20+1)}\) as something other than \(\mathrm{(N-19)}\).

2. Setting up the wrong equation for "immediately below"

Students may correctly understand the position relationships but then set up the equation incorrectly. For instance, they might write \(\mathrm{N-19 = 17}\) instead of \(\mathrm{N-19 = 15}\), confusing whether "immediately below" means +1 or -1 position when counting from bottom.

3. Verification errors

When checking their answer, students often make errors in the verification step by incorrectly calculating which board corresponds to which position, leading them to think their wrong answer is correct.

Errors while selecting the answer

1. Selecting the first calculated value without proper verification

Many students will calculate \(\mathrm{N = 36}\) initially (as shown in the solution) and select answer choice B without carefully verifying against the "immediately below" constraint. They fail to catch that their interpretation of the constraint was incorrect.

2. Not double-checking against all given constraints

Students may solve for N but forget to verify that their answer satisfies both the original counting conditions and the "immediately below" relationship, leading them to select an answer that only partially meets the problem requirements.