The Over-Constrained Timetable
The academic coordinator of a coaching centre must place five doubt-clearing sessions - Physics, Chemistry, Mathematics, Biology and Logical Reasoning - on Monday to Friday, exactly one subject per day. The constraints circulated to the faculty are: (i) Chemistry is neither on Monday nor on Friday. (ii) Biology is on Tuesday. (iii) Biology is on the day immediately before or immediately after Chemistry. (iv) Mathematics is later in the week than Physics. (v) Physics is not on Friday. (vi) Logical Reasoning is not on the day immediately before or immediately after Mathematics. A faculty member claims the list is over-constrained and that at least two different timetables will satisfy all six clues. The coordinator wants to know whether the clues force a single timetable, and whether any clue in the list can be deleted without changing the answer.
1. Deduce the timetable, showing the elimination at each step.
Read the worked answer
Step 1 - fix Chemistry from (ii) and (iii). Biology is on Tuesday. For Biology to be adjacent to Chemistry, Chemistry must be Monday or Wednesday. Clue (i) forbids Monday, so Chemistry = Wednesday.
Step 2 - the remaining three days. Monday, Thursday and Friday are left for Physics, Mathematics and Logical Reasoning, with Mathematics after Physics (iv).
Step 3 - test the three possible Physics days.
- Physics Monday → Mathematics can be Thursday or Friday.
- Maths Thursday, LR Friday: LR is adjacent to Maths → breaks (vi).
- Maths Friday, LR Thursday: LR is adjacent to Maths → breaks (vi). Both fail.
- Physics Thursday → Mathematics must be Friday, leaving LR Monday. LR (Mon) is not adjacent to Maths (Fri) → valid.
- Physics Friday → forbidden by (v), and impossible anyway because nothing can come after Mathematics.
Result:
| Day | Subject |
|---|---|
| Monday | Logical Reasoning |
| Tuesday | Biology |
| Wednesday | Chemistry |
| Thursday | Physics |
| Friday | Mathematics |
The timetable is unique, so the faculty member's claim is wrong.
2. Which single clue in the list is logically redundant, and would deleting it change the answer?
Read the worked answer
Redundant clue: (v) 'Physics is not on Friday.'
It adds nothing, because clue (iv) already says Mathematics is later in the week than Physics. If Physics were on Friday there would be no later day available for Mathematics, so (iv) alone already excludes Physics-on-Friday. In logic terms, (v) is implied by (iv) plus the fact that the week has five days.
Effect of deleting it: none. The deduction in the previous task never used (v) except as a final check; the elimination of Physics-on-Friday follows from (iv). The timetable stays Monday LR, Tuesday Biology, Wednesday Chemistry, Thursday Physics, Friday Mathematics.
Why this matters in an exam. Redundant clues are placed deliberately to see whether you test each statement or just consume them all. A useful habit is to ask, for each clue, 'what does this rule out that no other clue already rules out?' A clue that rules out nothing new is a distractor, not extra information - and it also means 'using all six clues' is not what makes the answer unique; four of them do all the work.
3. Suppose clue (vi) is withdrawn instead of (v), with the other five clues kept. How many valid timetables now exist, and what does that tell you about which clue is doing the real work?
Read the worked answer
Fixed part unchanged. (ii) and (iii) with (i) still force Chemistry = Wednesday and Biology = Tuesday. Monday, Thursday and Friday go to Physics, Mathematics and Logical Reasoning, with Mathematics after Physics.
Enumerate the three orders that satisfy (iv):
- Physics Monday, Mathematics Thursday, LR Friday
- Physics Monday, Mathematics Friday, LR Thursday
- Physics Thursday, Mathematics Friday, LR Monday
All three respect (i)–(v): Chemistry is on Wednesday, Physics is never on Friday. Removing (vi) removes the only rule that killed options 1 and 2.
Answer: three valid timetables.
What this shows.
- Clues (i), (ii) and (iii) do the structural work: they pin two of the five days outright.
- Clue (iv) only orders the remaining three subjects.
- Clue (vi) is the discriminating constraint - it is the one that converts three candidates into one. Remove a structural clue and you usually get many solutions; remove the discriminating clue and you get exactly the number of orderings it was filtering.
Practical use. If a puzzle has too many solutions, look for the adjacency-type clue that eliminates specific orderings rather than for more subject-placement clues, which are often already implied.