Step-by-step solution

1. For the first pair, 13 and 170: 13² = 169. Adding 1 to this gives 169 + 1 = 170.
2. So the pattern is (Number)² + 1.
3. Applying this pattern to 17: 17² = 289.
4. Adding 1 to 289 gives 289 + 1 = 290.

Answer: 290

Step-by-step solution

1. 1. Start with 'B is between F and C'. We can represent this as F - B - C or C - B - F in the circle.
2. 2. Next, 'A is between E and D'. This gives E - A - D or D - A - E.
3. 3. Now, 'F is to the left of D'. Since everyone is facing the center, 'left' means counter-clockwise. So, the order is D, then F in counter-clockwise direction.
4. Let's combine these pieces of information. We have two groups: (F, B, C) and (E, A, D). We also know D ... F (counter-clockwise).
5. From (F - B - C) and (F is to the left of D), we can infer a sequence. If D is at a position, F must be at its left. So, D, then F. Since B is between F and C, the sequence becomes D, F, B, C. (Clockwise: C, B, F, D)
6. Now, let's incorporate (A is between E and D). We have D as a common point. So, the complete sequence (clockwise) around the circle would be C, B, F, D, E, A. (Counter-clockwise: A, E, D, F, B, C).
7. Let's verify all conditions with the clockwise arrangement: C, B, F, D, E, A.
8. - B is between F and C? Yes, (C, B, F).
9. - A is between E and D? Yes, (D, E, A).
10. - F is to the left of D? Yes, if facing center, D's left is F.
11. The arrangement is consistent.
12. 4. The question asks: Who is between A and F?
13. Looking at the arrangement (C, B, F, D, E, A), we can see that D and E are between A and F if we go clockwise (F -> D -> E -> A). If we go counter-clockwise (A -> E -> D -> F).
14. Let's draw the circle: A --- E --- D --- F --- B --- C --- A. Starting from A clockwise: A, then C, B, F, D, E. So F is C, B. Or from A counter-clockwise: A, E, D, F, B, C.
15. The question 'who is between A and F' implies the direct neighbor in one direction. From our derived order (C, B, F, D, E, A), F and A are separated by D and E in one direction, and by C and B in the other. This usually means the person directly adjacent to both on one side.
16. Let's draw it:
    A
    E C
    D B
    F
    
    Starting from D (left of F): D -> F -> B -> C. Then A is between E and D. So, D has E to its right (clockwise) and A to E's right. So, D -> E -> A. This links up: D -> F -> B -> C -> A -> E -> D (clockwise).
17. Clockwise sequence: D, F, B, C, A, E. (And E is next to D).
18. Now, who is between A and F? Going clockwise from F, we have B, C, A. Going counter-clockwise from F, we have D, E, A. So D is between A and F when moving counter-clockwise from A towards F.
19. The clockwise order is: D, E, A, C, B, F. (Connecting F back to D).
20. Who is between A and F? Going clockwise from A: C, B, F. Going counter-clockwise from A: E, D, F. So, C is between A and F (if we mean the person directly adjacent in a specific direction, usually the shorter path).
21. Let's re-read 'A is between E and D'. This means E-A-D. 'B is between F and C'. This means F-B-C. 'F is to the left of D'. This means D (clockwise) F. So D is to the right of F. Or D - F (counter-clockwise).
22. Using D ... F (counter-clockwise):
    _ _ _ D _ _ _ F _ _ _
    E A D (so D is to the right of A, E is to the left of A). E A D. So we have E A D F.
    
    Now F B C. So E A D F B C.
    Let's check the full circle: E A D F B C. (Clockwise)
    - B is between F and C? Yes. (F-B-C)
    - A is between E and D? Yes. (E-A-D)
    - F is to the left of D? No. D's left is A. D's right is F. So this clockwise order means F is to the right of D. This is incorrect.
23. Let's restart with 'F is to the left of D' assuming standard counter-clockwise left for facing center. So, if we trace counter-clockwise: D -> F.
    (D, F, _, _, _, _)
    
    B is between F and C: D, F, B, C.
    (D, F, B, C, _, _)
    
    A is between E and D: This implies D, E, A (clockwise) or A, E, D (counter-clockwise).
    Since we have D, F, B, C (counter-clockwise), and A is between E and D, the gap is between C and D. So, the sequence clockwise from D must be D, A, E. This means D-A-E (clockwise) is the missing part.
24. Let's put it together (clockwise): D, A, E, C, B, F. (Connecting F back to D).
    Let's verify:
    1. B is between F and C? (F, D, A, E, C, B) - No. B is between F and C. So F-B-C or C-B-F. In D, A, E, C, B, F, B is between C and F. This works.
25. 2. A is between E and D? (D, A, E) - Yes.
26. 3. F is to the left of D? (Clockwise from D is A. Counter-clockwise is F. So yes, F is to the left of D).
27. So the correct clockwise order is D, A, E, C, B, F.
28. Now, who is between A and F?
    From A clockwise: E, C, B, F. (E, C, B are between A and F).
    From A counter-clockwise: D, F. (D is between A and F).
29. The question implies the person directly adjacent to both when traversing the circle. If we follow the sequence D-A-E-C-B-F-D. A is between D and E. F is between B and D. So, if we ask who is between A and F, and we look at the segment D-A-E-C-B-F, we find E, C, B. If we look at the other segment, it's just D.
30. The question 'who is between A and F' usually implies the one adjacent to both, or the one that completes the shortest segment. In the sequence D, A, E, C, B, F:
31. A is next to D and E.
32. F is next to B and D.
33. The common neighbor for A and F is D. A is right of D, F is left of D. So D is between A and F.

Answer: C

Step-by-step solution

1. Analyze the first pair: 17 : 306. We observe that 17 × (17 + 1) = 17 × 18 = 306.
2. The rule is 'n : n × (n + 1)'.
   $n \rightarrow n(n+1)$
3. Apply the same rule to the second pair: 23 : ?. So, 23 × (23 + 1) = 23 × 24.
4. Calculate 23 × 24 = 552.

Answer: 552