IPMAT Indore 2022 · Quantitative Ability · Question 43

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 Alex.

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

Of the four options, only Saturday appears.

The direct argument, without the full grid. Alex works Tuesday and Wednesday, so his third day cannot be Monday or Thursday (both adjacent to that pair). That leaves Friday, Saturday or Sunday. Now:

Cathy is barred from Saturday and Monday, and cannot work with Alex, so her three days must be Thursday, Friday and Sunday.

Alex cannot share a day with Cathy, which rules out Friday and Sunday.

Therefore Alex’s third day is Saturday.

Answer: Option 4 — Saturday

Shortcut. Two of the options die on the adjacency rule alone. Alex’s consecutive pair is Tuesday–Wednesday, so his third day must be isolated — Monday (a) and Thursday are both adjacent and impossible. That removes option (a) at once.

Then pin Cathy: barred from Saturday, Monday, and from Alex’s Tuesday and Wednesday, she has exactly three days left (Thu, Fri, Sun) and must take all of them. Since Alex never works with Cathy, Friday (b) and Sunday (c) are gone.

Saturday by elimination — and this is the question to solve first in the set, because forcing Cathy’s row is the key that unlocks everything else.