Fitting Fifteen Examination Papers Into a Fixed March Window
The examination cell of a residential school must complete the annual examination for Classes VI to XII between 2 March and 20 March. This year 1 March falls on a Sunday.
School rules and orders in force: the school does not work on Sundays; the second Saturday of the month is a holiday; Holi falls on 14 March; and on 12 March the district administration has deputed six staff members for election duty, so no examination can be held that day. One working day, 2 March, must be kept free for verification and distribution of question papers.
Each class needs 15 papers, and the school can hold only one paper per class per day, because there is a single examination hall block and one set of invigilators. The invigilation register must show a different invigilator for every session. The lab assistant has separately asked for two consecutive working days in the physics and chemistry laboratories for the Class XII practical examination, and has proposed 8 and 9 March.
1. Count exactly how many usable examination days exist in the 2 to 20 March window, naming each excluded date and the reason for excluding it.
Read the worked answer
Step 1: calendar span. 2 March to 20 March inclusive = 19 calendar days.
Step 2: weekdays, from the anchor that 1 March is a Sunday. Sundays in the window are 8 March and 15 March (2 days). Saturdays are 7, 14 and 21 March; the second Saturday is 14 March.
Step 3: subtract each cause once.
- Sundays 8 and 15 March: −2
- Second Saturday 14 March: −1
- Holi, also 14 March: no further deduction - it coincides with the second Saturday, so the school loses one day, not two. This is the trap in the question.
- Election duty 12 March: −1
- 2 March reserved for paper verification: −1 Total deduction = 2 + 1 + 1 + 1 = 5 days.
Step 4: usable days = 19 − 5 = 14 examination days.
Usable dates are 3, 4, 5, 6, 7, 9, 10, 11, 13, 16, 17, 18, 19 and 20 March. Note that 7 March, the first Saturday, is a working day and is rightly counted.
2. Fifteen papers are needed and only fourteen days are available. Compare the three remedies available to the cell and recommend one.
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The gap: 15 papers needed against 14 single-paper days = a shortfall of one session.
Option A - hold a paper on Sunday 8 March. Rejected. In a residential school Sunday is the only rest and visiting day, several students go home, and no reserve invigilator is available. It also creates a precedent that any shortfall can be met by cancelling rest.
Option B - extend the window to 21 March. 21 March is the third Saturday and a working day, so it is legally possible, but it pushes the last paper back by a day just when evaluation, tabulation and the 31 March result deadline begin. It buys one day at the cost of compressing the evaluation chain.
Option C - run two shifts on one existing working day (recommended). Take 7 March and hold one paper 9:30 a.m. to 12:30 p.m. and another 1:30 p.m. to 4:30 p.m. This is safe only if: (i) the two papers chosen have small enrolment and can be seated in separate halls; (ii) no student sits both papers; (iii) a fresh set of invigilators is posted for the afternoon session, as the register requires; (iv) question papers for both are verified on 2 March. Choose the two smallest-enrolment papers, not two core papers, so that the risk of a clash is minimal.
Recommendation: Option C, with the invigilation register showing two distinct panels, and Option B held in reserve if a further day is lost to weather or an unscheduled holiday.
3. Evaluate the lab assistant's proposal of 8 and 9 March for the Class XII practicals, and state a correct alternative.
Read the worked answer
Test the proposal against the condition. The requirement is two consecutive working days.
Using the anchor that 1 March is Sunday, the weekday of any date d is found by (d − 1) mod 7, where 0 = Sunday. For 8 March: 7 mod 7 = 0, so 8 March is a Sunday - a non-working day. For 9 March: 8 mod 7 = 1, so 9 March is a Monday.
Therefore 8 and 9 March are not two consecutive working days; only 9 March is available. The proposal fails on its own terms, and the practical cannot begin on a holiday when lab staff and students are not on duty.
Correct alternatives
- 9 and 10 March (Monday–Tuesday): both working, and 12 March election duty does not touch them.
- 16 and 17 March (Monday–Tuesday): also both working, and further away from the Holi break.
Recommendation: 9 and 10 March. The laboratories are in a separate block from the examination hall, so a practical there does not consume one of the 14 theory examination days and does not compete with invigilators, provided the lab technician and one internal examiner are released from invigilation duty on those two days. Avoid 15 and 16 March, since 15 March is a Sunday.