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In a stack of boards at a lumber yard, the 20th board counting from the top of the stack is immediately below the 16th board counting from the bottom of the stack. How many boards are in the stack?
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
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:
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:
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
Now we solve our simple equation:
\(\mathrm{N - 19 = 17}\)
\(\mathrm{N = 17 + 19}\)
\(\mathrm{N = 36}\)
Let's verify this makes sense:
There are 36 boards in the stack.
Looking at our answer choices:
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 answer is D. 34
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.
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.
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.
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 \(\mathrm{(N-19)}\).
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.
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.
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.
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.