IPMAT Indore 2022 · Quantitative Ability · Question 41

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 Bhabha and Cathy.

\text{Bhabha}=\{\text{Mon}, \text{Thu}, \text{Sun}\}, \text{Cathy}=\{\text{Thu}, \text{Fri}, \text{Sun}\}

They overlap on Thursday and Sunday — two days.

Answer: Option 4 — 2

Shortcut. Do this question last, not first. It asks for an overlap, which needs the full roster; the other four questions each turn on a single deduction.

Once the grid is built, the count is immediate: Bhabha and Cathy share Thursday and Sunday.

A partial shortcut if you are short of time: Cathy’s three days are forced early (Thu, Fri, Sun — she is barred from Sat, Mon, and from Alex’s Tue and Wed), and Sunday needs three people. Since Alex is never with Cathy and Ethan ends up on Saturday, Bhabha is almost certainly on Sunday with her — pointing to at least one shared day, which already eliminates option (a).