Answers to questions

1. How many taps did Clive receive for his question? — 7

From total taps 40 and Alia’s received = 9 plus Badal/Dilshan/Ehsaan equal (call R), we have 9 + C + 3R = 40. The only integer solution in the feasible range is R = 8 and C = 7. So Clive’s column sum (taps received) is 7, matching the final table.

2. Which two people tapped an equal number of times in total? — Alia and Badal (Option 1)

Row totals come from known totals: A = 6, D = 11, E = 9 and B + C = 14 (since rows must sum to 40). With Clue “Clive tapped more than Badal” the unique split is B = 6, C = 8. Thus Alia and Badal each made 6 taps.

3. What was Clive’s response to Ehsaan’s question? — No (Option 2)

In the filled table C→E = 2. That assignment satisfies column/row totals, the reciprocity constraints for A and E (no matching answers), and the requirement that each column contain at least one 1, one 2, and one 3. So Clive answered “No” to Ehsaan.

4. How many “Yes” responses were received across all the questions? — 6

Count all 1-tap entries in the final table: A row has two 1s, B row two 1s, C row one 1, D row zero 1s, E row one 1 → total 2+2+1+0+1 = 6. This satisfies the constraints and the column diversity requirement.

Steps to make table: CAT 2025 DILR Slot 1

Step 1: Total taps = 40 → sum of column totals = 40 → with A = 9 and B,D,E equal (call each R) we get 9 + C + 3R = 40 → solve → R = 8 and C = 7. (Uses: total = 40; Alia received 9; B,D,E equal.)

Step 2: Therefore columns (taps received) are A=9, B=8, C=7, D=8, E=8. (Arithmetic result.)

Step 3: Row totals given: A tapped 6, D tapped 11, E tapped 9; remaining rows B and C must sum to 40 − (6+11+9) = 14 → B+C = 14. (Uses: row totals A,D,E and global total.)

Step 4: Clue “Clive tapped more than Badal” → C > B and B+C = 14 → unique integer solution is B=6, C=8. (Uses: C > B and B+C constraint.)

Step 5: Now row totals are A=6, B=6, C=8, D=11, E=9 — these match the table’s row sums. (Consequence.)

Step 6: Alia responded “Yes” to Clive and Dilshan → cells A→C = 1 and A→D = 1. (Uses: clue 1.)

Step 7: Fill remaining Alia row taps: A row must total 6 with no diagonal; remaining taps sum = 6 − (1+1) = 4 → split as A→B = 2 and A→E = 2. (Uses: row A total and step 6.)

Step 8: Dilshan responded “No” to Badal → cell D→B = 2. (Uses: clue 2.)

Step 9: Reciprocity mismatch for Alia and Ehsaan: for each X ≠ A, X→A ≠ A→X, and similarly for E → for every X ≠ E, X→E ≠ E→X. Use this to eliminate symmetric choices. (Uses: clue 5.)

Step 10: Limit “No one responded ‘Yes’ more than twice” bounds each row to at most two 1s; combine with known row totals to force specific entries (e.g., D must have many 3s to reach 11 while respecting at least one 1, one 2, one 3 in its column). (Uses: clue 4 and column diversity requirement.)

Step 11: Ehsaan’s row (total 9) must include exactly one 1 toward C (to meet column C = 7 and ensure A→C =1 already) → set E→C = 1; remaining E taps become E→A = 3, E→B = 3, E→D = 2 to reach 9 while satisfying reciprocity with A and column targets. (Combines: column sums, reciprocity with A, row total E.)

Step 12: Clive’s row (total 8) then fits as C→A = 2, C→B = 1, C→D = 3, C→E = 2 to meet row total 8, column targets (especially C column = 7), and the A/E reciprocity constraints. (Fills C row consistent with previous steps.)

Step 13: Badal’s row (total 6) becomes B→A = 1, B→C = 2, B→D = 2, B→E = 1 to satisfy B row sum 6, column sums, and the requirement that each question received at least one 1, one 2, and one 3. (Fills B row consistent with clues.)

Step 14: Dilshan’s row (total 11) completes as D→A = 3, D→C = 3, D→E = 3 (plus D→B = 2 set earlier) to reach 11 and provide the necessary 3s to meet column diversity and totals. (Fills D row consistent with step 8 and column sums.)

Step 15: Final verification → all off-diagonal cells are 1/2/3; each column has at least one 1, one 2, one 3; row/column totals equal the values found in Steps 2 and 5; reciprocity constraints for A and E are satisfied; Clive tapped more than Badal. (Global check — table validated.)

Working table — Tapping game CAT 2025 DILR Slot 1

Key: Rows = who tapped (respondents). Columns = whose question they answered (questions asked).

1 tap = Yes, 2 taps = No, 3 taps = Maybe.