IPMAT Indore 2025 · Quantitative Ability · Set 1: Five expedition teams – biologists, geologists and explorers

Five teams of 15 members each, every member a biologist, a geologist or an explorer.

Four rules stay on the board throughout:

Every cell is at least 2

The three skills are shared equally across all 75 members

Explorer counts are all different and strictly decrease from A to E

Teams B, C, D and E each have more biologists than explorers

Step 1 — each column totals 25

75 members shared equally among 3 skills gives 75 ÷ 3 = 25 per column.

Step 2 — Team A caps the explorers at 7

Row A totals 15 and 6 of them are geologists, so biologists + explorers = 9. Since biologists must be at least 2, Team A can hold at most 7 explorers.

Team A holds the largest explorer count, so this cap is really a cap on the whole explorer column.

Step 3 — the explorer column is forced

We need five different values, decreasing, smallest at least 2, largest at most 7, adding to 25. Only {2, 3, 4, 5, 6, 7} are available and they add to 27, so exactly one of them is dropped — and dropping the 2 is the only way to land on 25.

Team A's biologists come free here: 9 − 7 = 2.

Step 4 — Team B is squeezed from both sides

Two separate rules push Team B's biologists to the same number from opposite directions.

Both arrows point at 7, so Team B has 7 biologists and 2 geologists.

Step 5 — Team E's biologists by subtraction

The biologist column must reach 25, and four of its five entries are now known: 2 + 7 + 6 + 6 = 21, leaving 4 for Team E.

Check the rule: 4 biologists beats 3 explorers, so Team E still satisfies the biologists > explorers condition.

Step 6 — the last geologists

Each remaining geologist cell is whatever the row needs to reach 15.

Final table and answer

Team E has 4 biologists.

Verification sweep:

Every cell is at least 2

All three columns total 25

Explorers run 7, 6, 5, 4, 3 — distinct and strictly decreasing

B, C, D and E each have more biologists than explorers: 7 > 6, 6 > 5, 6 > 4, 4 > 3

IMPORTANT NOTE — 3 moves

In the exam, only three moves actually matter. Everything else is arithmetic.

Move 1 is the whole puzzle. Teach students to hunt for the row with a given value (Team A's 6 geologists) and turn it into a cap on the constrained column.

Answer: 4

Five teams of 15 members each, every member a biologist, a geologist or an explorer.

Four rules stay on the board throughout:

Every cell is at least 2

The three skills are shared equally across all 75 members

Explorer counts are all different and strictly decrease from A to E

Teams B, C, D and E each have more biologists than explorers

Step 1 — each column totals 25

75 members shared equally among 3 skills gives 75 ÷ 3 = 25 per column.

Step 2 — Team A caps the explorers at 7

Row A totals 15 and 6 of them are geologists, so biologists + explorers = 9. Since biologists must be at least 2, Team A can hold at most 7 explorers.

Team A holds the largest explorer count, so this cap is really a cap on the whole explorer column.

Step 3 — the explorer column is forced

We need five different values, decreasing, smallest at least 2, largest at most 7, adding to 25. Only {2, 3, 4, 5, 6, 7} are available and they add to 27, so exactly one of them is dropped — and dropping the 2 is the only way to land on 25.

Team A's biologists come free here: 9 − 7 = 2.

Step 4 — Team B is squeezed from both sides

Two separate rules push Team B's biologists to the same number from opposite directions.

Both arrows point at 7, so Team B has 7 biologists and 2 geologists.

Step 5 — Team E's biologists by subtraction

The biologist column must reach 25, and four of its five entries are now known: 2 + 7 + 6 + 6 = 21, leaving 4 for Team E.

Check the rule: 4 biologists beats 3 explorers, so Team E still satisfies the biologists > explorers condition.

Step 6 — the last geologists

Each remaining geologist cell is whatever the row needs to reach 15.

Final table and answer

Team E has 4 biologists.

Verification sweep:

Every cell is at least 2

All three columns total 25

Explorers run 7, 6, 5, 4, 3 — distinct and strictly decreasing

B, C, D and E each have more biologists than explorers: 7 > 6, 6 > 5, 6 > 4, 4 > 3

IMPORTANT NOTE — 3 moves

In the exam, only three moves actually matter. Everything else is arithmetic.

Move 1 is the whole puzzle. Teach students to hunt for the row with a given value (Team A's 6 geologists) and turn it into a cap on the constrained column.

Answer: 2

Five teams of 15 members each, every member a biologist, a geologist or an explorer.

Four rules stay on the board throughout:

Every cell is at least 2

The three skills are shared equally across all 75 members

Explorer counts are all different and strictly decrease from A to E

Teams B, C, D and E each have more biologists than explorers

Step 1 — each column totals 25

75 members shared equally among 3 skills gives 75 ÷ 3 = 25 per column.

Step 2 — Team A caps the explorers at 7

Row A totals 15 and 6 of them are geologists, so biologists + explorers = 9. Since biologists must be at least 2, Team A can hold at most 7 explorers.

Team A holds the largest explorer count, so this cap is really a cap on the whole explorer column.

Step 3 — the explorer column is forced

We need five different values, decreasing, smallest at least 2, largest at most 7, adding to 25. Only {2, 3, 4, 5, 6, 7} are available and they add to 27, so exactly one of them is dropped — and dropping the 2 is the only way to land on 25.

Team A's biologists come free here: 9 − 7 = 2.

Step 4 — Team B is squeezed from both sides

Two separate rules push Team B's biologists to the same number from opposite directions.

Both arrows point at 7, so Team B has 7 biologists and 2 geologists.

Step 5 — Team E's biologists by subtraction

The biologist column must reach 25, and four of its five entries are now known: 2 + 7 + 6 + 6 = 21, leaving 4 for Team E.

Check the rule: 4 biologists beats 3 explorers, so Team E still satisfies the biologists > explorers condition.

Step 6 — the last geologists

Each remaining geologist cell is whatever the row needs to reach 15.

Final table and answer

Team E has 4 biologists.

Verification sweep:

Every cell is at least 2

All three columns total 25

Explorers run 7, 6, 5, 4, 3 — distinct and strictly decreasing

B, C, D and E each have more biologists than explorers: 7 > 6, 6 > 5, 6 > 4, 4 > 3

IMPORTANT NOTE — 3 moves

In the exam, only three moves actually matter. Everything else is arithmetic.

Move 1 is the whole puzzle. Teach students to hunt for the row with a given value (Team A's 6 geologists) and turn it into a cap on the constrained column.

Answer: 6