Alia, Badal, Clive, Dilshan, and Ehsaan played a game in which each asks a unique question to all the others and they respond by tapping their feet, either once or twice or thrice. One tap means “Yes”, two taps mean “No”, and three taps mean “Maybe”. A total of 40 taps were heard across the five questions. Each question received at least one “Yes”, one “No”, and one “Maybe.” The following information is known. 1. Alia tapped a total of 6 times and received 9 taps to her question. She responded “Yes” to the questions asked by both Clive and Dilshan. 2. Dilshan and Ehsaan tapped a total of 11 and 9 times respectively. Dilshan responded “No” to Badal. 3. Badal, Dilshan, and Ehsaan received equal number of taps to their respective questions. 4. No one responded “Yes” more than twice. 5. No one’s answer to Alia’s question matched the answer that Alia gave to that person’s question. This was also true for Ehsaan. 6. Clive tapped more times in total than Badal. How many taps did Clive receive for his question?

Answer: 7. Solution: Each of the five people asks exactly one question, answered by the other four. A response is 1 tap (Yes), 2 taps (No), or 3 taps (Maybe). The rows are the responders; the columns are the five questions, where QA is Alia's question, QB is Badal's, QC is Clive's, QD is Dilshan's, QE is Ehsaan's. A cell holds the taps the row-person gives to that question. The diagonal is blank — no one answers their own question. Each row total is that person's total taps; each column total (the Received row) is the taps that question collected. Both add to 40., Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 6, Badal, —, Clive, —, Dilshan, —, 11, Ehsaan, —, 9, Received, 40, From clues 1 and 2, the totals Alia =6, Dilshan =11, Ehsaan =9 are placed. Each question gets four answers and must include at least one Yes(1), one No(2), one Maybe(3). The minimum is 1+2+3=6, plus a fourth answer of 1 to 3. So every question receives 7, 8, or 9 taps. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 6, Badal, —, Clive, —, Dilshan, —, 11, Ehsaan, —, 9, Received, 9, 8, 7, 8, 8, 40, The Received row is filled. Alia's question received 9 taps (clue 1). By clue 3, Badal's, Dilshan's and Ehsaan's questions each received the same amount t; let Clive's be c. 9 + 3t + c = 40 3t + c = 31, with t, c \7,8,9\. Only t = 8,\ c = 7 works. So QB = QD = QE = 8 and QC = 7. For the row totals: 6 + B + C + 11 + 9 = 40 B + C = 14, where B and C are Badal's and Clive's total taps. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 1, 1, 6, Badal, —, Clive, —, Dilshan, 2, —, 11, Ehsaan, —, 9, Received, 9, 8, 7, 8, 8, 40, From clue 1, Alia answered Yes to Clive and Dilshan: (A,QC)=1 (A,QD)=1 From clue 2, Dilshan answered No to Badal: (D,QB)=2 —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, —, Clive, —, Dilshan, 2, —, 11, Ehsaan, —, 9, Received, 9, 8, 7, 8, 8, 40, Alia's row total is 6, and (A,QC)+(A,QD)=2, so (A,QB)+(A,QE)=4. Clue 4 caps Yes responses at two per person, and Alia already gave Yes to Clive and Dilshan. So neither remaining answer can be 1. Two values each at least 2 summing to 4 forces (A,QB)=2 (A,QE)=2 —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, —, Clive, —, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, —, 9, Received, 9, 8, 7, 8, 8, 40, Dilshan's row total is 11 with (D,QB)=2, so (D,QA)+(D,QC)+(D,QE)=9. Three answers, each at most 3, summing to 9 must all be 3: (D,QA)=3 (D,QC)=3 (D,QE)=3 Dilshan's row is complete. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, —, Clive, 2, —, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, —, 9, Received, 9, 8, 7, 8, 8, 40, Clue 5 (Alia's question): each person's answer to QA differs from Alia's answer to their own question. (A,QB)=2 (B,QA)\1,3\ (A,QC)=1 (C,QA)\2,3\ (A,QE)=2 (E,QA)\1,3\ Column QA sums to 9 with (D,QA)=3, so (B,QA)+(C,QA)+(E,QA)=6. If (C,QA)=3 then (B,QA)+(E,QA)=3, impossible with both in \1,3\. So (C,QA)=2, and \(B,QA),(E,QA)\=\1,3\. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, 1, —, Clive, 2, —, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, 3, —, 9, Received, 9, 8, 7, 8, 8, 40, Two options remain for column QA: either (B,QA)=1,(E,QA)=3 or (B,QA)=3,(E,QA)=1. Limits on Ehsaan's other answers: Column QB sum 8 with (A,QB)=(D,QB)=2 gives (C,QB)+(E,QB)=4; to keep a Yes and a Maybe these are \1,3\, so (E,QB)\1,3\. Column QC sum 7 with (A,QC)=1,(D,QC)=3 gives (B,QC)+(E,QC)=3, so (E,QC)\1,2\. Clue 5 (Ehsaan): (D,QE)=3 (E,QD)\1,2\. If (E,QA)=1, Ehsaan's other three answers must total 8, but their maximum is 3+2+2=7. Impossible. So (B,QA)=1 and (E,QA)=3. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, 1, —, Clive, 2, 1, —, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, 3, 3, —, 9, Received, 9, 8, 7, 8, 8, 40, Ehsaan's total is 9 with (E,QA)=3, so (E,QB)+(E,QC)+(E,QD)=6. Since (E,QC)≤ 2 and (E,QD)≤ 2, their sum is at most 4, forcing (E,QB)≥ 2. As (E,QB)\1,3\, this gives (E,QB)=3. Column QB then gives (C,QB)=4-3=1. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, 1, —, Clive, 2, 1, —, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, 3, 3, 1, 2, —, 9, Received, 9, 8, 7, 8, 8, 40, Now (E,QC)+(E,QD)=3, each in \1,2\. Test (E,QC)=2,(E,QD)=1: column QC gives (B,QC)=1; clue 5 (Ehsaan) needs (C,QE)≠(E,QC), forcing (C,QE)=1 and so (B,QE)=2. Column QD then needs (B,QD)+(C,QD)=6; Badal already has two Yes (QA, QC), so (B,QD)≠ 1, forcing (B,QD)=3,(C,QD)=3. This makes Badal = Clive = 7, violating clue 6 (C>B). So (E,QC)=1 and (E,QD)=2. Ehsaan's row is complete. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, 1, —, 2, 1, Clive, 2, 1, —, 2, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, 3, 3, 1, 2, —, 9, Received, 9, 8, 7, 8, 8, 40, Column QC: (B,QC)=3-(E,QC)=2. Clue 5 (Ehsaan): (C,QE)≠(E,QC)=1, and (C,QE)\1,2\, so (C,QE)=2. Column QE: (B,QE)=3-(C,QE)=1. —--, Responder, QA, QB, QC, QD, QE, Tapped, Alia, —, 2, 1, 1, 2, 6, Badal, 1, —, 2, 2, 1, 6, Clive, 2, 1, —, 3, 2, 8, Dilshan, 3, 2, 3, —, 3, 11, Ehsaan, 3, 3, 1, 2, —, 9, Received, 9, 8, 7, 8, 8, 40, Column QD: (B,QD)+(C,QD)+(E,QD)=7 with (E,QD)=2, so (B,QD)+(C,QD)=5. Badal already has two Yes (QA, QE), so (B,QD)≠ 1. (B,QD)=2 (C,QD)=3: Badal 6, Clive 8 — satisfies C>B. (B,QD)=3 (C,QD)=2: Badal 7, Clive 7 — fails C>B. So (B,QD)=2 and (C,QD)=3. The grid is complete and unique. Verification: Row totals 6,6,8,11,9 sum to 40. Column totals 9,8,7,8,8 sum to 40. Every question contains a 1, a 2, and a 3. Yes counts: Alia 2, Badal 2, Clive 1, Dilshan 0, Ehsaan 1 — none exceeds two. Clive (8) is greater than Badal (6), and B+C=14. Clue 5 holds for both Alia's and Ehsaan's questions. Alia's question received 9 taps. By clue 3, Badal's, Dilshan's and Ehsaan's questions each received the same amount t, and Clive's received c. Then 9+3t+c=40, i.e. 3t+c=31, with each value in \7,8,9\. The only solution is t=8,\ c=7. So Clive's question received 7 taps. Answer: 7.

Logical Reasoning

Math Puzzles

Hard

Alia, Badal, Clive, Dilshan, and Ehsaan played a game in which each asks a unique question to all the others and they respond by tapping their feet, either once or twice or thrice. One tap means “Yes”, two taps mean “No”, and three taps mean “Maybe”.

A total of 40 taps were heard across the five questions. Each question received at least one “Yes”, one “No”, and one “Maybe.”

The following information is known.

Alia tapped a total of 6 times and received 9 taps to her question. She responded “Yes” to the questions asked by both Clive and Dilshan.
Dilshan and Ehsaan tapped a total of 11 and 9 times respectively. Dilshan responded “No” to Badal.
Badal, Dilshan, and Ehsaan received equal number of taps to their respective questions.
No one responded “Yes” more than twice.
No one’s answer to Alia’s question matched the answer that Alia gave to that person’s question. This was also true for Ehsaan.
Clive tapped more times in total than Badal.

How many taps did Clive receive for his question?

Entered answer:

✅ Correct Answer: 7

Slide 1/11

Each of the five people asks exactly one question, answered by the other four. A response is 1 tap (Yes), 2 taps (No), or 3 taps (Maybe).

The rows are the responders; the columns are the five questions, where QA is Alia's question, QB is Badal's, QC is Clive's, QD is Dilshan's, QE is Ehsaan's. A cell holds the taps the row-person gives to that question. The diagonal is blank — no one answers their own question.

Each row total is that person's total taps; each column total (the Received row) is the taps that question collected. Both add to 40.

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—					6
Badal		—
Clive			—
Dilshan				—		11
Ehsaan					—	9
Received						40

From clues 1 and 2, the totals Alia $=6$ , Dilshan $=11$ , Ehsaan $=9$ are placed.

Each question gets four answers and must include at least one Yes(1), one No(2), one Maybe(3).

The minimum is $1+2+3=6$ , plus a fourth answer of $1$ to $3$ .

So every question receives $7$ , $8$ , or $9$ taps.

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—					6
Badal		—
Clive			—
Dilshan				—		11
Ehsaan					—	9
Received	9	8	7	8	8	40

The Received row is filled.

Alia's question received $9$ taps (clue 1).

By clue 3, Badal's, Dilshan's and Ehsaan's questions each received the same amount $t$ ; let Clive's be $c$ .

$9 + 3t + c = 40 \Rightarrow 3t + c = 31$ , with $t, c \in \{7,8,9\}$ .

Only $t = 8,\ c = 7$ works.

So QB $=$ QD $=$ QE $= 8$ and QC $= 7$ .

For the row totals:

$6 + B + C + 11 + 9 = 40 \Rightarrow B + C = 14$ , where $B$ and $C$ are Badal's and Clive's total taps.

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—		1	1		6
Badal		—
Clive			—
Dilshan		2		—		11
Ehsaan					—	9
Received	9	8	7	8	8	40

From clue 1, Alia answered Yes to Clive and Dilshan:

$(A,QC)=1$

$(A,QD)=1$

From clue 2, Dilshan answered No to Badal:

$(D,QB)=2$

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal		—
Clive			—
Dilshan		2		—		11
Ehsaan					—	9
Received	9	8	7	8	8	40

Alia's row total is $6$ , and $(A,QC)+(A,QD)=2$ , so

$(A,QB)+(A,QE)=4$ .

Clue 4 caps Yes responses at two per person, and Alia already gave Yes to Clive and Dilshan.

So neither remaining answer can be $1$ .

Two values each at least $2$ summing to $4$ forces

$(A,QB)=2$

$(A,QE)=2$

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal		—
Clive			—
Dilshan	3	2	3	—	3	11
Ehsaan					—	9
Received	9	8	7	8	8	40

Dilshan's row total is $11$ with $(D,QB)=2$ , so

$(D,QA)+(D,QC)+(D,QE)=9$ .

Three answers, each at most $3$ , summing to $9$ must all be $3$ :

$(D,QA)=3$

$(D,QC)=3$

$(D,QE)=3$

Dilshan's row is complete.

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal		—
Clive	2		—
Dilshan	3	2	3	—	3	11
Ehsaan					—	9
Received	9	8	7	8	8	40

Clue 5 (Alia's question): each person's answer to QA differs from Alia's answer to their own question.

$(A,QB)=2 \Rightarrow (B,QA)\in\{1,3\}$

$(A,QC)=1 \Rightarrow (C,QA)\in\{2,3\}$

$(A,QE)=2 \Rightarrow (E,QA)\in\{1,3\}$

Column QA sums to $9$ with $(D,QA)=3$ , so

$(B,QA)+(C,QA)+(E,QA)=6$ .

If $(C,QA)=3$ then $(B,QA)+(E,QA)=3$ , impossible with both in $\{1,3\}$ .

So $(C,QA)=2$ , and $\{(B,QA),(E,QA)\}=\{1,3\}$ .

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal	1	—
Clive	2		—
Dilshan	3	2	3	—	3	11
Ehsaan	3				—	9
Received	9	8	7	8	8	40

Two options remain for column QA: either $(B,QA)=1,(E,QA)=3$ or $(B,QA)=3,(E,QA)=1$ .

Limits on Ehsaan's other answers:

Column QB sum $8$ with $(A,QB)=(D,QB)=2$ gives $(C,QB)+(E,QB)=4$ ; to keep a Yes and a Maybe these are $\{1,3\}$ , so $(E,QB)\in\{1,3\}$ .

Column QC sum $7$ with $(A,QC)=1,(D,QC)=3$ gives $(B,QC)+(E,QC)=3$ , so $(E,QC)\in\{1,2\}$ .

Clue 5 (Ehsaan): $(D,QE)=3 \Rightarrow (E,QD)\in\{1,2\}$ .

If $(E,QA)=1$ , Ehsaan's other three answers must total $8$ , but their maximum is $3+2+2=7$ . Impossible.

So $(B,QA)=1$ and $(E,QA)=3$ .

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal	1	—
Clive	2	1	—
Dilshan	3	2	3	—	3	11
Ehsaan	3	3			—	9
Received	9	8	7	8	8	40

Ehsaan's total is $9$ with $(E,QA)=3$ , so

$(E,QB)+(E,QC)+(E,QD)=6$ .

Since $(E,QC)\le 2$ and $(E,QD)\le 2$ , their sum is at most $4$ , forcing $(E,QB)\ge 2$ .

As $(E,QB)\in\{1,3\}$ , this gives $(E,QB)=3$ .

Column QB then gives $(C,QB)=4-3=1$ .

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal	1	—
Clive	2	1	—
Dilshan	3	2	3	—	3	11
Ehsaan	3	3	1	2	—	9
Received	9	8	7	8	8	40

Now $(E,QC)+(E,QD)=3$ , each in $\{1,2\}$ .

Test $(E,QC)=2,(E,QD)=1$ :

column QC gives $(B,QC)=1$ ; clue 5 (Ehsaan) needs $(C,QE)\neq(E,QC)$ , forcing $(C,QE)=1$ and so $(B,QE)=2$ .

Column QD then needs $(B,QD)+(C,QD)=6$ ; Badal already has two Yes (QA, QC), so $(B,QD)\neq 1$ , forcing $(B,QD)=3,(C,QD)=3$ .

This makes Badal $=$ Clive $= 7$ , violating clue 6 ( $C>B$ ).

So $(E,QC)=1$ and $(E,QD)=2$ . Ehsaan's row is complete.

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal	1	—	2		1
Clive	2	1	—		2
Dilshan	3	2	3	—	3	11
Ehsaan	3	3	1	2	—	9
Received	9	8	7	8	8	40

Column QC: $(B,QC)=3-(E,QC)=2$ .

Clue 5 (Ehsaan): $(C,QE)\neq(E,QC)=1$ , and $(C,QE)\in\{1,2\}$ , so $(C,QE)=2$ .

Column QE: $(B,QE)=3-(C,QE)=1$ .

Responder	QA	QB	QC	QD	QE	Tapped
Alia	—	2	1	1	2	6
Badal	1	—	2	2	1	6
Clive	2	1	—	3	2	8
Dilshan	3	2	3	—	3	11
Ehsaan	3	3	1	2	—	9
Received	9	8	7	8	8	40

Column QD: $(B,QD)+(C,QD)+(E,QD)=7$ with $(E,QD)=2$ , so $(B,QD)+(C,QD)=5$ .

Badal already has two Yes (QA, QE), so $(B,QD)\neq 1$ .

$(B,QD)=2 \Rightarrow (C,QD)=3$ : Badal $6$ , Clive $8$ — satisfies $C>B$ .

$(B,QD)=3 \Rightarrow (C,QD)=2$ : Badal $7$ , Clive $7$ — fails $C>B$ .

So $(B,QD)=2$ and $(C,QD)=3$ . The grid is complete and unique.

Verification:

Row totals $6,6,8,11,9$ sum to $40$ .

Column totals $9,8,7,8,8$ sum to $40$ .

Every question contains a $1$ , a $2$ , and a $3$ .

Yes counts: Alia $2$ , Badal $2$ , Clive $1$ , Dilshan $0$ , Ehsaan $1$ — none exceeds two.

Clive ( $8$ ) is greater than Badal ( $6$ ), and $B+C=14$ .

Clue 5 holds for both Alia's and Ehsaan's questions.

Alia's question received $9$ taps. By clue 3, Badal's, Dilshan's and Ehsaan's questions each received the same amount $t$ , and Clive's received $c$ . Then $9+3t+c=40$ , i.e. $3t+c=31$ , with each value in $\{7,8,9\}$ . The only solution is $t=8,\ c=7$ . So Clive's question received $7$ taps. Answer: 7.

Question 20

Which two people tapped an equal number of times in total?

Question 21

What was Clive’s response to Ehsaan’s question?

Question 22

How many “Yes” responses were received across all the questions?

How many taps did Clive receive for his question?

Solution

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