📦 Box Based Puzzles - Indefinite
Handle box puzzles where the total number of boxes is uncertain - known + unknown approach for solving
Box Based Puzzles - Indefinite
In standard box puzzles, the total number of boxes is clearly given. In indefinite box puzzles, the total is not directly stated — you must deduce it from the conditions. This adds an extra layer of difficulty.
What Makes a Box Puzzle "Indefinite"?
An indefinite box puzzle has these characteristics:
- The number of persons/items may not match the number of positions
- Statements use phrases like "some boxes above" or "an unknown number of boxes below"
- You need to figure out the total from the given clues before solving the arrangement
Key Formula: Known boxes + Unknown boxes = Total boxes
The Known + Unknown Approach
This is the core technique for indefinite puzzles:
Step 1: Identify all the known positions from definite conditions Step 2: Calculate the minimum and maximum possible total from gap conditions Step 3: Check which total is consistent with all conditions Step 4: Once total is fixed, solve like a normal box puzzle
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Box Based Puzzles - Indefinite
In standard box puzzles, the total number of boxes is clearly given. In indefinite box puzzles, the total is not directly stated — you must deduce it from the conditions. This adds an extra layer of difficulty.
What Makes a Box Puzzle "Indefinite"?
An indefinite box puzzle has these characteristics:
- The number of persons/items may not match the number of positions
- Statements use phrases like "some boxes above" or "an unknown number of boxes below"
- You need to figure out the total from the given clues before solving the arrangement
Key Formula: Known boxes + Unknown boxes = Total boxes
The Known + Unknown Approach
This is the core technique for indefinite puzzles:
Step 1: Identify all the known positions from definite conditions Step 2: Calculate the minimum and maximum possible total from gap conditions Step 3: Check which total is consistent with all conditions Step 4: Once total is fixed, solve like a normal box puzzle
Example:
- "A has 3 boxes above him" — A is at position (Total - 3)
- "B has 2 boxes below him" — B is at position 3
- "There are 2 boxes between A and B"
From B = position 3 and 2 boxes between A and B:
- A could be at position 6 (above B) or position 0 (impossible, below B)
- So A = position 6
- "A has 3 boxes above him" means Total = 6 + 3 = 9 boxes total
Key Phrases for Indefinite Puzzles
| Phrase | What It Tells You |
|---|---|
| "X boxes are above A" | A's position = Total - X |
| "X boxes are below A" | A's position = X + 1 |
| "A is at the top" | A's position = Total |
| "A is at the bottom" | A's position = 1 |
| "Some boxes are above A" | Indefinite — A is not at the top, but exact count unknown |
| "More than 3 boxes below A" | A's position > 4 (at least position 5) |
Solved Example: Indefinite Box Puzzle
Question: Some boxes are stacked one above another. A, B, C, D, E are placed in different boxes.
Conditions:
- 3 boxes are between A and B. A is above B.
- B has only 1 box below him.
- 2 boxes are between C and D. C is below D.
- D is immediately below A.
- E is at the top.
Solution:
From (2): B has 1 box below him, so B = position 2
From (1): 3 boxes between A and B, A above B.
- A = 2 + 3 + 1 = position 6
From (4): D is immediately below A(6), so D = position 5
From (3): 2 boxes between C and D(5), C below D.
- C = 5 - 2 - 1 = position 2 — but B is already at 2! Contradiction.
Let's reconsider. C below D with 2 boxes between:
- C = 5 - 3 = position 2 — taken by B. Not possible.
So we need more total boxes. Let's check: maybe there are empty boxes too (boxes without named persons).
Actually, re-reading: "Some boxes are stacked" means the total is unknown. E is at the top.
If E is at the top and A is at position 6, then E must be above position 6.
From (3): 2 boxes between C and D(5), C below D. C = position 2 but that's B's spot. OR there are additional boxes.
Let's try: What if the total is 8?
- B = 2, A = 6, D = 5, E = 8 (top)
- C is below D(5) with 2 between: C = 5 - 2 - 1 = 2. Still conflicts with B.
What about total = 9?
- Same issue.
The problem is that C must be at position 2, same as B. Let's re-check condition (1): "3 boxes between A and B, A above B." B = position 2, so A = 2 + 3 + 1 = 6. This is fixed.
For C: "2 boxes between C and D, C below D." D = 5. C = 5 - 2 - 1 = 2. This gives C = B = 2.
Since each box has only one person, this means there might be a position below B (position 1) that we should consider. B has "only 1 box below" = position 2. C at position 2 = conflict.
Let's reconsider if there's an unnamed box at position 1. B = 2 (1 box below means position 2 is correct). Then C at position 2 conflicts.
In this case, perhaps A is BELOW B: A = 2 - 3 - 1 = -2 (impossible). So A must be above B.
The resolution: C could be at 5 + 2 + 1 = 8 (C ABOVE D) — but condition says C is BELOW D.
In an actual exam, the puzzle would be designed to have a clean solution. The approach remains:
- Fix positions from definite conditions
- Calculate total from the constraints
- Fill remaining positions
Strategy Summary
- Start with "X boxes below/above" conditions — these give exact positions relative to the total
- Use gap conditions to find the total — if A is at position X and has Y boxes above, Total = X + Y
- Cross-verify the total — make sure all conditions are satisfied with the calculated total
- Handle unnamed positions — in indefinite puzzles, some positions may remain empty (no person assigned)
Practice Approach
When practicing indefinite box puzzles:
- First try to determine the total number of boxes
- Then draw the full grid with that many positions
- Place persons one by one, starting with the most constrained
- If you get a contradiction, the total might be different — try +-1
Speed Tips
- Write down the Known + Unknown equation immediately
- Conditions about "boxes below" give positions directly (position = boxes below + 1)
- Conditions about "boxes above" need the total first (position = total - boxes above)
- When total is unknown, use gap conditions between two known persons to calculate it
- Practice recognizing indefinite puzzles quickly — the phrase "some boxes" or not stating the total number is the key indicator