Six Registers on a Six-Slot Shelf
You are the attendant responsible for the record shelf in a subdivision office. Six bound registers must stand on a six-slot shelf, numbered from slot 1 on the left to slot 6 on the right. The registers are D, F, B, A, C and E. The office rule sheet states: (1) D must be at one of the two ends; (2) F must be immediately to the right of D; (3) B, A and C must stand together as a block in that left-to-right order, with nothing between them; (4) C must not be in the extreme right slot; (5) E must not be adjacent to B. A new inward register G has arrived, and the oldest register D is being sent permanently to the record room, so the shelf will still hold six registers: F, B, A, C, E and G. The head clerk directs that G be placed in the extreme left slot and that the relative order of the other five must not otherwise change. A junior staff member insists the original arrangement is impossible and that D must actually sit at the right end, because the rule only says D is at an end. Inspection begins at 4 p.m.
1. Work out the complete left-to-right order of the six registers. Show each deduction and name the clue that fixes each slot.
Read the worked answer
Deduction 1 - slots 1 and 2. Clue 1 places D at the left or right end. Clue 2 requires F immediately to the right of D, so D cannot be in slot 6, which has no slot to its right. Therefore D = slot 1 (clue 1 read together with clue 2) and F = slot 2 (clue 2).
Deduction 2 - slots 3, 4 and 5. The remaining registers are B, A, C and E, and the free slots are 3, 4, 5 and 6. Clue 3 requires B, A and C to be consecutive in that order. The only run of three consecutive free slots is 3-4-5, so B = 3, A = 4, C = 5, with E = slot 6 left over.
Deduction 3 - verification. Clue 4: C is in slot 5, not slot 6, so it is satisfied. Clue 5: E is in slot 6 and B in slot 3, so they are not adjacent, so it is satisfied.
Final order: D, F, B, A, C, E.
Checks asked at the counter: the register third from the right is A (slot 4), and the registers adjacent to C (slot 5) are A on its left and E on its right.
2. The junior says D could sit at the right end because the rule only says D is at an end. Explain precisely why that fails and what general mistake it illustrates.
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Why it fails: If D sat in slot 6, clue 2 would require F in slot 7, which does not exist. Clue 1 by itself is genuinely weak - it permits either end - but rules are not read one line at a time. The moment clue 2 is applied, the right end is eliminated and the left end becomes forced.
The general mistake: the junior treated a permissive clue (at one of the ends) as a complete specification and ignored a restrictive clue that interacts with it. In arrangement puzzles, clues fall into two classes:
- Strong or positional clues - immediately to the right of, immediately left of, exactly in the middle - which pin positions.
- Weak or negative clues - not at an end, not adjacent to - which only eliminate possibilities.
A weak clue can never decide a position on its own; it must be combined with a strong one. The junior's conclusion was also self-defeating in practice: acting on it would have produced an arrangement that violates clue 2, so the shelf would have failed inspection even though every other rule looked satisfied.
Practical lesson: before moving a single register, test the proposed order against every rule in the list, not only the rule being discussed.
3. After D goes to the record room and G is placed in the extreme left slot, state the new order and check each of the five original conditions against it.
Read the worked answer
New order: G, F, B, A, C, E.
Condition-by-condition check:
- Clue 1 (D at an end): inapplicable - D has left the shelf. A rule about a register that is no longer present cannot be satisfied or violated, and it must be struck off the working list rather than forced onto another register.
- Clue 2 (F immediately right of D): inapplicable for the same reason. This is the trap in the situation: anyone who mechanically applies all five rules will conclude the task is impossible, because there is no D for F to follow.
- Clue 3 (B, A, C consecutive in that order): holds - slots 3, 4 and 5, with nothing between them, and the relative order of the five surviving registers is unchanged as directed.
- Clue 4 (C not in the extreme right slot): holds - C is in slot 5, with E in slot 6.
- Clue 5 (E not adjacent to B): holds - E is in slot 6 and B in slot 3.
- New direction (G in the extreme left slot): holds - G is in slot 1.
Conclusion: the arrangement is valid, and only the two rules naming the departed register drop out. The inspection point to state aloud is that a rule tied to a removed item lapses, whereas a rule tied to a surviving item keeps its force.