IPMAT Indore 2022 · Quantitative Ability · Question 42

Build the roster once and all five questions are read off it.

Step 0 — check the arithmetic. Six days need two staff and Sunday needs three:

6\times 2+3=15=5\text{employees}\times 3\text{days each}

Every slot is accounted for, so nobody has spare capacity — a useful rigidity.

Step 0a — how “consecutive” is meant. The showroom runs “on all seven days of the week throughout the year”, so the week is a repeating cycle and Sunday and Monday are consecutive days. This matters: read as a flat Monday-to-Sunday list the puzzle has a different unique answer, and the published answers follow the cyclical reading. Treat the seven days as a circle.

Step 1 — place Alex. Alex works Tuesday and Wednesday (given), which is his consecutive pair. His third day must not be adjacent to either, so it cannot be Monday or Thursday. That leaves Friday, Saturday or Sunday.

Step 2 — place Cathy. Cathy works neither Saturday nor Monday, and never alongside Alex — so she avoids Tuesday and Wednesday too. That leaves only

\text{Thursday, Friday, Sunday}

which is exactly three days, so Cathy takes all of them. Check the pattern: Thursday–Friday are consecutive, and Sunday is adjacent to Saturday and Monday — neither of which she works ✔

Step 3 — Alex’s third day. Alex cannot share a day with Cathy, who now holds Friday and Sunday. So Alex’s third day is Saturday:

\text{Alex}=\{\text{Tue}, \text{Wed}, \text{Sat}\}

Step 4 — Dilip’s third day. Dilip works Tuesday and Wednesday with Alex, and his third day must differ from Alex’s, so it is not Saturday. It also cannot be Monday or Thursday (adjacent to his pair). That leaves Friday or Sunday.

Step 5 — count Monday. Monday needs two staff. Alex, Cathy and Dilip are all unavailable on Monday, so Monday must be Bhabha and Ethan.

Step 6 — finish Bhabha and Ethan. Bhabha never works with Alex, so he avoids Tuesday, Wednesday and Saturday. With Monday fixed, his remaining options are Thursday, Friday and Sunday. His pattern needs exactly one consecutive pair including Monday’s neighbour Sunday — so Bhabha takes Sunday (adjacent to Monday) plus one isolated day, which must be Thursday (Friday would sit next to Saturday? no — Friday is adjacent to Thursday and Saturday, neither held, but Friday and Sunday are not adjacent, so both remain possible at this stage; the day counts settle it below).

Ethan must cover Saturday’s second slot (Alex is the only other Saturday worker) and so takes Friday–Saturday as his consecutive pair, with Monday isolated.

Step 7 — balance the days. Working through the remaining counts:

The completed roster.

Every condition checks out: each employee works three days with exactly one consecutive pair ✔; every day has the right headcount ✔; Alex and Dilip share Tuesday and Wednesday with different third days (Sat vs Sun) ✔; neither Bhabha nor Cathy ever shares a day with Alex ✔; Cathy avoids Saturday and Monday ✔

An exhaustive search over all valid patterns confirms this roster is the only one consistent with the constraints.

Check each pair against the roster.

Dilip and Ethan never overlap.

(For completeness, the full list of non-overlapping pairs is Alex–Bhabha, Alex–Cathy and Dilip–Ethan; only the last appears among the options.)

Answer: Option 4 — Dilip and Ethan

Shortcut. Sunday eliminates two options at once. Three people work Sunday — Bhabha, Cathy and Dilip — so any pair drawn from those three does work together. That removes “Bhabha and Dilip” (a) and “Cathy and Dilip” (c) immediately.

Between the survivors, Bhabha and Ethan both cover Monday, which leaves Dilip and Ethan.

The general tactic: find the busiest day and use it as an eliminator. In any scheduling set the day with the largest headcount rules out the most pairings for the least work.