IPMAT Indore 2022 · Quantitative Ability · Set 1: Showroom staffing – five employees across seven days

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).

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.

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.

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 Ethan.

\text{Ethan}=\{\text{Mon}, \text{Fri}, \text{Sat}\}

His consecutive pair is Friday and Saturday, with Monday as the isolated day.

Why it is forced. Saturday needs two workers. Alex is one of them, and Alex never works with Bhabha or Cathy — and Cathy is explicitly barred from Saturday anyway. Dilip’s third day differs from Alex’s, so Dilip is not on Saturday. That leaves only Ethan for Saturday’s second slot.

Ethan’s consecutive pair must therefore include Saturday, so it is Friday–Saturday or Saturday–Sunday. Sunday already has its three workers (Bhabha, Cathy, Dilip), so the pair is Friday–Saturday.

Why the others fail:

(a) Thursday and Friday — Thursday is taken by Bhabha and Cathy, filling it.

(b) Saturday and Sunday — Sunday’s three slots are already filled.

(d) Sunday and Monday — same problem with Sunday, and it would leave Saturday a worker short.

Answer: Option 3 — Friday and Saturday

Shortcut. Saturday is the bottleneck. Alex holds one of its two slots, and Bhabha, Cathy and Dilip are each excluded from the other — so Ethan must work Saturday.

That single deduction cuts the options to (b) and (c). Sunday is already full with three names, so Ethan’s pair must be Friday–Saturday.

In scheduling sets, always look for the day with the fewest eligible people. Here Saturday has only one candidate left after the exclusions, which forces Ethan’s whole row in one step.

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.