Five-Period Timetable with a Fixed Slot
At a government higher secondary school in Kota, the timetable committee must place five subjects-Hindi, English, Mathematics, Science, and Social Science-in the five periods of a single day, one subject per period. The committee has fixed four conditions: Mathematics must come immediately after Hindi; Science cannot be the first or the last period; English must come before Social Science; and Hindi cannot be in period 4 or period 5. A new teacher argues that the timetable cannot be finalised because several arrangements satisfy the conditions. The head teacher asks for a definite answer: is there any period whose subject is fixed regardless of which valid arrangement is chosen? If Science is placed in period 2, can the committee produce a unique timetable? The committee must also explain its reasoning to the principal in two minutes, using no trial-and-error table.
1. Which subject, if any, must occupy period 5 in every valid timetable? Justify your answer without listing every possible arrangement.
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Social Science must occupy period 5.
Reasoning from the Hindi–Mathematics block:
- Hindi can be in period 1, 2, or 3 only, because Mathematics immediately follows Hindi and Hindi cannot be in period 4 or 5.
- If Hindi is in period 1, Mathematics is in period 2. The remaining periods are 3, 4, and 5 for Science, English, and Social Science. Science cannot be period 5, so Science is 3 or 4. English must come before Social Science, so Social Science is forced into period 5.
- If Hindi is in period 2, Mathematics is in period 3. Science cannot be period 1 or 5, and period 2 and 3 are already taken, so Science must be period 4. English then goes to period 1 and Social Science to period 5.
- If Hindi is in period 3, Mathematics is in period 4. Science must be period 2, English period 1, and Social Science period 5.
In all three branches, Social Science is in period 5. Therefore period 5 is fixed.
2. If Science is placed in period 2, give the complete period-wise timetable and explain why no other arrangement works.
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With Science in period 2, the only valid timetable is:
| Period | Subject |
|---|---|
| 1 | English |
| 2 | Science |
| 3 | Hindi |
| 4 | Mathematics |
| 5 | Social Science |
Why other placements fail:
- If Hindi were in period 1, Mathematics would be in period 2, conflicting with Science in period 2.
- If Hindi were in period 2, period 2 would be Hindi, again conflicting with Science.
- Therefore Hindi must be in period 3, which puts Mathematics in period 4.
- The remaining periods are 1 and 5; Science is already fixed at 2, so English and Social Science fill 1 and 5. English must come before Social Science, so English is period 1 and Social Science is period 5.
3. A teacher proposes moving Hindi to period 1 to “settle” the timetable. Does that make the timetable unique? Explain.
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No. Putting Hindi in period 1 makes Mathematics period 2, but it does not settle the rest. Two valid arrangements remain:
- Hindi, Mathematics, Science, English, Social Science
- Hindi, Mathematics, English, Science, Social Science
In the first, Science is period 3 and English period 4; in the second, English is period 3 and Science period 4. Both satisfy Science not being first or last, and both keep English before Social Science. So the proposal narrows the possibilities but does not produce a unique timetable.