In an arithmetic reasoning puzzle, there are n boxes and m fruits. If 3 fruits are placed in each box, 3 fruits remain unused. If 4 fruits are placed in each box, then exactly 1 box remains empty. What are the numbers of boxes and fruits?

Introduction / Context:
This is a classic arithmetic reasoning and linear equations puzzle involving boxes and fruits. We are given two different ways of distributing fruits into boxes and told what happens in each case: in one arrangement some fruits are left over, and in another arrangement some boxes are left empty. The goal is to translate the English description into algebraic equations in terms of the number of boxes and the number of fruits, then solve these equations systematically to find both unknown quantities.

Given Data / Assumptions:
- Let n be the number of boxes.
- Let m be the total number of fruits.
- If 3 fruits are placed in each box, then 3 fruits remain extra, so the fruits used in boxes plus the extra equals m.
- If 4 fruits are placed in each box, then 1 box is left empty, meaning only n - 1 boxes are actually filled.
- Each box that is used in either arrangement contains the same number of fruits specified in that scenario.

Concept / Approach:
We convert each verbal condition into an algebraic equation. In the first arrangement, the number of fruits used is 3n and there are 3 fruits left over. Therefore the total fruit count m is 3n + 3. In the second arrangement, only n - 1 boxes are used, each containing 4 fruits, so the total number of fruits m is also 4(n - 1). Since both expressions describe the same total m, we can equate them, obtain a simple linear equation in n, and then compute m. Finally, we check which option matches the pair (n, m).

Step-by-Step Solution:
Step 1: From the first condition, write m = 3n + 3, because 3 fruits per box across n boxes accounts for 3n fruits and there are 3 extra fruits. Step 2: From the second condition, only n - 1 boxes are filled with 4 fruits each, so m = 4(n - 1). Step 3: Since both expressions equal m, set them equal: 3n + 3 = 4(n - 1). Step 4: Expand the right side: 4(n - 1) = 4n - 4, so we have 3n + 3 = 4n - 4. Step 5: Rearrange: 3n + 3 - 4n + 4 = 0 gives -n + 7 = 0, so n = 7 boxes. Step 6: Substitute n = 7 into m = 3n + 3 to get m = 3 * 7 + 3 = 21 + 3 = 24 fruits.

Verification / Alternative check:
Check the first condition: With 7 boxes and 24 fruits, placing 3 in each box uses 3 * 7 = 21 fruits, leaving 24 - 21 = 3 fruits extra, which matches the statement. Check the second condition: If we place 4 fruits in each box but have only enough fruits to fill 6 boxes, then 4 * 6 = 24 fruits are used and exactly 1 box is left empty. This matches the second condition perfectly. Hence (7, 24) satisfies both constraints.

Why Other Options Are Wrong:
Option 3 boxes and 9 fruits: With 3 fruits per box, all fruits are used and none are excess, so the first condition fails. The second condition with 4 fruits per box also cannot be satisfied.
Option 6 boxes and 19 fruits: 3 fruits per box would use 18 fruits and leave 1 extra, not 3. Thus it does not fit the first condition.
Option 5 boxes and 18 fruits: 3 fruits per box uses all 15 fruits, leaving 3 fruits short of 18, not 3 fruits extra. This again does not match the statement.

Common Pitfalls:
A common error is misreading excess as meaning extra boxes instead of extra fruits, or misinterpreting an excess box as unused fruits. Another mistake is to treat n and m as interchangeable or to forget that the same total number of fruits must satisfy both scenarios. Carefully defining variables and writing equations before solving prevents such confusion.

Final Answer:
The correct numbers are 7 boxes and 24 fruits.