IPMAT Indore 2022 · Quantitative Ability · Question 45

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.

Reading off Sunday. Sunday is the only three-person day, and the roster gives

\text{Sunday}=\{\text{Bhabha}, \text{Cathy}, \text{Dilip}\}

The direct argument.

Alex is out: his days are Tuesday, Wednesday and Saturday.

Cathy is in: her forced trio is Thursday, Friday and Sunday.

Ethan is out: he must take Saturday (the only remaining candidate there), and his consecutive pair is Friday–Saturday with Monday isolated — no Sunday.

That leaves Bhabha and Dilip to join Cathy, and both fit: Dilip’s third day is Sunday (it differs from Alex’s Saturday and is non-adjacent to Tue–Wed), and Bhabha’s Sunday pairs with Monday around the week.

Answer: Option 1 — Bhabha, Cathy and Dilip

Shortcut. Alex settles it in one step. His three days are Tuesday, Wednesday and Saturday, so he is not on Sunday — which removes option (c).

Cathy is forced onto Sunday (her only possible trio is Thu, Fri, Sun), so any option omitting her is wrong — but all three survivors contain her. The discriminator is Ethan versus Dilip, and Ethan is pinned to Saturday, making Friday–Saturday his pair.

Hence Bhabha, Cathy and Dilip.

Note how the five questions interlock: Q43 forces Cathy’s row, Q44 forces Ethan’s, and together they determine Sunday. Solve a scheduling set in the order the constraints bite, not the order the questions are printed.