Puzzles — Explained
Detailed Explanation
A puzzle is a set of simultaneous constraints
A reasoning puzzle gives objects, possible positions or categories, and clues that restrict how those objects can be arranged. Solving it means finding arrangements that satisfy all the clues together.
The task may ask for the complete arrangement, a relationship that must hold, or something that could hold in at least one valid arrangement. These requests are different. A single workable arrangement can establish possibility, but it cannot establish necessity unless the remaining alternatives have been ruled out.
Choose a representation that matches the clues
For people seated in a row, use numbered positions. For people assigned different days and subjects, use a grid connecting each person to each attribute. For an ordering problem, arrows can record before-and-after relationships without inventing adjacency.
A is before B does not mean A is immediately before B. In a circular arrangement, choose a reference person to remove irrelevant rotational duplicates, but preserve reflection differences when left and right matter.
The representation should expose the constraint rather than add an unstated condition.
Consider four students A, B, C and D assigned one each to Monday, Tuesday, Wednesday and Thursday. The clues say A is before C, B is immediately after D, and C is not on Thursday. The adjacent block D–B can occupy Monday–Tuesday, Tuesday–Wednesday or Wednesday–Thursday. Testing each block is more efficient than listing all 24 permutations. The exercise is original and hypothetical; it is intended to teach constraint propagation, not reproduce a named examination question.
Follow the implications of a strong clue
If D–B occupies Monday–Tuesday, the remaining days are Wednesday and Thursday. A must precede C, placing C on Thursday, which is forbidden. If D–B occupies Tuesday–Wednesday, A and C must take Monday and Thursday, again putting C on Thursday.
The block must therefore occupy Wednesday–Thursday, leaving A on Monday and C on Tuesday. The solution is A, C, D, B. Notice that the adjacency clue reduced the search, while the final exclusion supplied uniqueness.
Neither clue alone would have produced the complete answer.
This method illustrates a useful order of work: enter fixed positions, form required blocks, note exclusions, and then test the few remaining placements. Do not erase the original clues after finding a plausible arrangement. Recheck each one against the completed grid, because an early transcription error can produce a neat but invalid solution. When several arrangements survive, retain them rather than choosing the one that looks most symmetrical.
Distinguish local and global deductions
A local deduction uses a clue with a small part of the arrangement. A global deduction depends on the full allocation, such as one remaining person being forced into the last vacant slot. In the scheduling example, D immediately before B is local, while the conclusion that A occupies Monday emerges from the entire set.
Good working notes identify which deductions are unconditional and which belong to a temporary branch. Writing a tentative placement as a permanent fact can contaminate every later step.
One way to control branching is to label alternatives Case 1 and Case 2, carrying only the consequences of each within its own small table. If a branch violates a clue, cross it out and state the conflicting condition. Do not reject it merely because it differs from an answer option you expected. Options can help test a proposition, but the clues are the authority for whether an arrangement is valid.
Translate the question's logical force
Suppose two valid arrangements survive and A occupies the first position in both, while B occupies the second in one and the third in the other. A being first is a must-be-true statement. B being second is a could-be-true statement. B being fourth is impossible under the surviving cases. A question asking which statement need not be true can be answered by finding a valid counterexample. It does not require the statement to be false in every arrangement.
The same care applies to either-or clues. In ordinary logic, an inclusive or allows both alternatives; some puzzles explicitly mean exactly one. Use the wording supplied. Similarly, different means unequal values, not necessarily consecutive values, and at least two permits three or more. These small distinctions often matter more than a clever shortcut. If the wording is genuinely ambiguous, state the interpretation instead of silently treating a preferred convention as a fact.
Avoid common representation traps
Left and right depend on the direction a seated person faces. For people facing north in a straight row, their left matches the viewer's conventional west-to-east diagram; for south-facing people it reverses.
In circular seating, facing the centre and facing outward reverse the relevant orientation. A direction arrow beside the diagram is more reliable than remembering a verbal rule without reference to the picture.
When the question does not involve facing directions, adding them can create unnecessary complexity.
Also distinguish indistinguishable empty slots from named objects. Counting arrangements of identical items as though they were different overcounts possibilities. Conversely, merging two distinct people because they share an attribute can erase a valid distinction. Write the domain of each variable before counting or eliminating.
Make speed depend on disciplined checking
The efficient solution is usually the one with the fewest uncontrolled assumptions. Start with the most restrictive clue, propagate its consequences, and branch only when necessary. Keep the final response tied to the actual question rather than solving an entire grid that is not needed.
Under time pressure, a clean partial table can support elimination better than an elaborate mental picture. Written reasoning practice should explain the constraint that makes an answer necessary, while the examination task remains objective Preliminary Paper II reasoning.