For how many employees can the scores at the end of Quarter 3 be determined with certainty?

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E?	E?	N?	N?	N?
Q2
Q3

Filled: the Q1 groups.

From clue 1 the starting Elite trio is Asha, Bunty, Chintu, and the starting Novice trio is Dolly, Eklavya, Falguni. Ratings are still unknown, marked with $?$.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N?	N?	N?
Q2		N?
Q3		N?

Filled: Bunty's group through the quarters and his Q1 rating.

Clue 3 gives Bunty a Lalu (Novice) rating in Q2, so Bunty is in Novice during Q2 — he was demoted at the end of Q1.

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A demotion removes the lowest-scoring Elite member. After one quarter scores equal the Q1 ratings, so Bunty's Q1 rating is the minimum, $1$.

Since clue 1 puts all of Asha, Bunty, Chintu in Novice at the start of Q4, and only one swap occurs per quarter, none of them is re-promoted. Bunty therefore stays in Novice for Q2 and Q3.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N?	N?	N?
Q2	E?	N?
Q3	N?	N?

Filled: Asha's group in Q2 and Q3.

Clue 3 gives Asha a Lalu rating in Q3, so Asha is in Novice during Q3. She was Elite in Q1, so she was demoted at the end of Q2, which means she is still Elite in Q2.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N?	N?	N?
Q2	E?	N?	E?
Q3	N?	N?	E?

Filled: Chintu's group in Q2 and Q3.

The three demotions across Q1–Q3 must be exactly Asha, Bunty, Chintu, since all three are in Novice by the start of Q4.

Bunty left at the end of Q1 and Asha at the end of Q2, so Chintu is demoted only at the end of Q3 — he stays Elite throughout Q1, Q2, Q3.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N?	N?
Q2	E?	N?	E?	N2
Q3	N?	N?	E?	N2

Filled: Dolly in Novice for all three quarters, with rating $2$.

The three employees promoted into Elite must be Dolly, Eklavya, Falguni, since they form the Elite at the start of Q4. Clue 3 places Dolly in Novice during Q3, so she is promoted only at the end of Q3 — she sits in Novice for all of Q1, Q2, Q3.

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Clue 2 says Dolly's rating never changes, and clue 3 fixes it at $2$ in Q3. So Dolly $= 2$ in every quarter.

Eklavya and Falguni take the end-Q1 and end-Q2 promotions in some order.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N?	N3
Q2	E?	N?	E?	N2	N?	E3
Q3	N?	N?	E?	N2	E?	E3

Filled: Falguni's rating $3$ throughout, and the promotion order.

Falguni also keeps one fixed rating. In Q1 she shares Novice with Dolly $(=2)$; ratings in a group are distinct, so Falguni $\neq 2$, leaving $\{1,3\}$.

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Suppose Falguni $= 1$. The end-Q1 promotion goes to the rating-$3$ Novice, which would then be Eklavya, pushing Falguni's promotion to the end of Q2 — placing her in Novice during Q2 beside Bunty, whose Q2 rating is $1$. Two $1$'s in one group is impossible.

So Falguni $= 3$, making her the top Novice in Q1: she is promoted at the end of Q1, and Eklavya at the end of Q2. Hence Eklavya is Novice in Q1, Q2 and Elite in Q3; Falguni is Novice in Q1 and Elite in Q2, Q3.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N1	N3
Q2	E?	N?	E?	N2	N?	E3
Q3	N?	N?	E?	N2	E?	E3

Filled: Eklavya's Q1 rating.

Novice in Q1 is Dolly $(2)$, Eklavya, Falguni $(3)$. The only remaining rating, $1$, goes to Eklavya.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N1	N3
Q2	E?	N1	E?	N2	N3	E3
Q3	N?	N?	E?	N2	E?	E3

Filled: the Q2 Novice ratings.

Novice in Q2 is Bunty, Dolly $(2)$, Eklavya. Clue 3 gives Bunty $= 1$, so Eklavya takes the remaining $3$.

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Scores after Q2 in this group:

Bunty $= 1+1 = 2$

Dolly $= 2+2 = 4$

Eklavya $= 1+3 = 4$

Dolly and Eklavya tie at $4$; the higher Q2 rating, Eklavya's $3$ over Dolly's $2$, ranks Eklavya on top. So Eklavya is promoted, which matches the order found earlier.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N1	N3
Q2	E1	N1	E2	N2	N3	E3
Q3	N?	N?	E?	N2	E?	E3

Filled: the Q2 Elite ratings.

Elite in Q2 is Asha, Chintu, Falguni $(3)$, so Asha and Chintu split $\{1,2\}$. Falguni's score after Q2 is $3+3 = 6$. Asha is the one demoted, so she must hold the minimum Elite score.

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Test Asha $= 2$ (Chintu $= 1$), with $\{$Asha, Chintu$\}$ Q1 ratings $= \{2,3\}$:

if Asha Q1 $= 2$: Asha $= 4$, Chintu $= 3+1 = 4$, tie, but Asha's higher Q2 rating ranks her above, so Chintu would be the minimum — contradiction.

if Asha Q1 $= 3$: Asha $= 5$, Chintu $= 2+1 = 3$, so Chintu would be the minimum — contradiction.

So Asha's Q2 rating $= 1$ and Chintu's $= 2$.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N1	N3
Q2	E1	N1	E2	N2	N3	E3
Q3	N1	N3	E?	N2	E?	E3

Filled: the Q3 Novice ratings.

Novice in Q3 is Asha, Bunty, Dolly. Clue 3 gives Asha $= 1$ and Dolly $= 2$, leaving Bunty $= 3$.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E?	E1	E?	N2	N1	N3
Q2	E1	N1	E2	N2	N3	E3
Q3	N1	N3	E1	N2	E2	E3

Filled: the Q3 Elite ratings.

Elite in Q3 is Chintu, Eklavya, Falguni $(3)$, so Chintu and Eklavya split $\{1,2\}$. Chintu is demoted, so he must be the minimum. Falguni's score is $3+3+3 = 9$, and Eklavya's is $1+3+(\text{Q3 rating})$.

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Test Chintu Q3 $= 2$ (Eklavya $= 1$): Eklavya $= 1+3+1 = 5$, while Chintu $=$ (Q1) $+2+2$ gives $6$ or $7$, so Eklavya would be the minimum — contradiction.

So Chintu's Q3 rating $= 1$ and Eklavya's $= 2$.

Check: Eklavya $= 6$; Chintu $=$ (Q1) $+3 =$ $5$ or $6$. If $5$, Chintu is clearly lowest; if $6$, the tie is broken by the Q3 rating, where Chintu's $1$ is below Eklavya's $2$, so Chintu is still the minimum. Either way Chintu is demoted.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E2/3	E1	E3/2	N2	N1	N3
Q2	E1	N1	E2	N2	N3	E3
Q3	N1	N3	E1	N2	E2	E3

Filled: the last two Q1 Elite ratings.

Elite in Q1 is Asha, Bunty $(1)$, Chintu, so Asha and Chintu split $\{2,3\}$. Both orders, Asha $=2$/Chintu $=3$ and Asha $=3$/Chintu $=2$, satisfy every demotion and the constancy rule of clue 2, so this pair stays interchangeable. It is shown as $2/3$ for Asha and $3/2$ for Chintu, linked so that the first value of one pairs with the first value of the other.

Quarter	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	E2/3	E1	E3/2	N2	N1	N3
Q2	E1	N1	E2	N2	N3	E3
Q3	N1	N3	E1	N2	E2	E3

All ratings are now fixed except the linked Asha/Chintu Q1 pair. Adding the ratings cumulatively gives the scores:

Score after	Asha	Bunty	Chintu	Dolly	Eklavya	Falguni
Q1	2/3	1	3/2	2	1	3
Q2	3/4	2	5/4	4	4	6
Q3	4/5	5	6/5	6	6	9

The membership flow is:

Q1 Elite $\{A,B,C\}$, Novice $\{D,E,F\}$

Q2 Elite $\{A,C,F\}$, Novice $\{B,D,E\}$

Q3 Elite $\{C,E,F\}$, Novice $\{A,B,D\}$

Start of Q4 Elite $\{D,E,F\}$, Novice $\{A,B,C\}$.

The only unresolved values are Asha's and Chintu's Q1 ratings (and the scores that depend on them): the pair is $(A,C)=(2,3)$ or $(3,2)$, giving end-Q3 scores $(4,6)$ or $(5,5)$. Bunty, Dolly, Eklavya and Falguni are fully determined.

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End-Q3 scores: Bunty $= 5$, Dolly $= 6$, Eklavya $= 6$, Falguni $= 9$ are all fixed. Asha is $4$ or $5$ and Chintu is $6$ or $5$ because their Q1 ratings ($2$/$3$) are interchangeable. So exactly 4 employees have a determinable Q3 score. Answer: 4.