Question

Which of the following cannot be seated at either end?

• A. C
• B. D
• C. F
• D. G
• A. B & D
• B. C & F
• C. D & G
• D. E & A
• A. F & D
• B. D & E
• C. E & G
• D. C & F

Answer 1

The Correct Option is C

Step 1: Fix the seat positions:
Number the seven seats 1 to 7 from left to right. The players sit in one row, so every player gets exactly one seat.
B must be in the centre. In a row of seven the centre is seat 4, so B sits in seat 4.
A and G must be at the extreme right. The two seats at the extreme right are seat 6 and seat 7, so A and G fill these two seats in some order.

Step 2: Place C and D:
The seats still free are 1, 2, 3 and 5. These four seats are for C, D, E and F.
C and D must be as far apart as possible. Find the gap for each pair of free seats:
\[ (1,2)=1,\ (1,3)=2,\ (1,5)=4,\ (2,3)=1,\ (2,5)=3,\ (3,5)=2 \]
The largest gap is 4 and it comes only from seats 1 and 5. So C and D take seats 1 and 5 in some order, and E and F take seats 2 and 3 in some order.

Step 3: Look at the two ends:
The two ends are seat 1 and seat 7. Seat 7 holds A or G. Seat 1 holds C or D.
So the people who can sit at an end are A, G, C and D. The people E, F and B never sit at an end.

Step 4: Check each option:
Option 1 (C): C can be in seat 1. So C can be at an end.
Option 2 (D): D can be in seat 1. So D can be at an end.
Option 4 (G): G can be in seat 7. So G can be at an end.
Option 3 (F): F is forced into seat 2 or seat 3, which are not ends. So F cannot be seated at either end.

Final Answer:
Only F is stuck in the middle part of the row. Every other listed player can reach an end seat, so the answer is option 3. \[ \boxed{3} \]

Answer 2

Step 1: Build the fixed arrangement:
Number the seats 1 to 7 from left to right. B is in the centre, seat 4. A and G take the extreme right, seats 6 and 7.
C and D must be as far apart as possible. Among the free seats 1, 2, 3 and 5 the largest gap is 4, between seats 1 and 5. So C and D take seats 1 and 5 in some order.
E and F then take seats 2 and 3 in some order. The row looks like this:
\[ \{C/D\},\ \{E/F\},\ \{E/F\},\ B,\ \{D/C\},\ \{A/G\},\ \{G/A\} \]

Step 2: Check B and D (option 1):
B is in seat 4. D can be in seat 5, which is next to B. Example: C E F B D A G. So B and D can sit together.

Step 3: Check C and F (option 2):
C can be in seat 1 and F in seat 2, which are next to each other. Example: C F E B D A G. So C and F can sit together.

Step 4: Check D and G (option 3):
D can be in seat 5 and G in seat 6. These seats are next to each other. Example: C E F B D G A. So D and G can sit together.

Step 5: Check E and A (option 4):
E must be in seat 2 or 3. A must be in seat 6 or 7. The smallest gap between these seats is from seat 3 to seat 6, which is 3 seats.
So E and A can never be next to each other, whatever order we choose.

Final Answer:
The pair that can never sit together is E and A, which is option 4. \[ \boxed{4} \]

Answer 3

Step 1: Recall the forced arrangement:
Number the seats 1 to 7 from left to right. B is in seat 4, A and G are in seats 6 and 7, and C and D are in seats 1 and 5 (this is the only way to keep them as far apart as possible). E and F fill seats 2 and 3.

Step 2: Find who sits on either side of B:
The seats on either side of B (seat 4) are seat 3 and seat 5.
Seat 3 holds E or F. Seat 5 holds C or D.
So the two neighbours of B are always one person from {E, F} and one person from {C, D}.

Step 3: Check each option:
Option 1 (F and D): F in seat 3 and D in seat 5 is possible. Example: C E F B D A G. This is possible.
Option 2 (D and E): E in seat 3 and D in seat 5 is possible. Example: C F E B D G A. This is possible.
Option 4 (C and F): F in seat 3 and C in seat 5 is possible. Example: D E F B C A G. This is possible.
Option 3 (E and G): G must be in seat 6 or 7, but the neighbours of B are in seats 3 and 5. G can never be next to B, so this pair is impossible.

Final Answer:
The pair E and G cannot be on the two sides of B because G is always at the far right. This is option 3. \[ \boxed{3} \]