Is there a way in which all the members can be seated?

Question

Here's a puzzle where it is very difficult to find a possible seating arrangement of all the members. (Image source)

Representatives from eight different Banks viz. A, B, C, D, E, F, G and H are sitting around a circular table facing the center but not necessarily in the same order. Each one of them is from a different Bank viz. UCO Bank, Oriental Bank of Commerce, Bank of Maharashtra, Canara Bank, Syndicate Bank, Punjab National Bank, Bank of India and Dena Bank.

F sits second to [the] right of the representative from Canara Bank. [The] representative from Bank of India is an immediate neighbour of the representative from Canara Bank. Two people sit between the representative of Bank of India and B. C and E are immediate neighbours of each other. Neither C nor E is an immediate neighbour of either B or the representative from Canara Bank. [The] representative of Bank of Maharashtra sits second to [the] right of D. D is neither the representative of Canara Bank nor Bank of India. G and the representative from UCO Bank are immediate neighbours of each other. B is not the representative of UCO Bank. Only one person sits between C and the representative from Oriental Bank of Commerce. H sits third to [the] left of the representative from Dena Bank. [The] representative from Punjab National Bank sits second to [the] left of the representative from Syndicate Bank.

Year: IBPS PO -2011

Source: This is a question from an aptitude and reasoning based exam conducted in India for recruitment purposes and Banking and Financial sectors, also sometimes to top Management Institutes (Institute of Banking Personnel Selection Probationary Officers Exam, 2011).

Answer 1

Here's a solution which explains the logic...

First, you might find it useful to draw up a diagram like this, where every block of 16 letters represents a seat at the table (numbered 1 to 8 for ease of explanation). The top two rows of each represent all the possible banks (abbreviating their names to a single initial); the bottom two rows of each represent all the possible representatives (labelled A-H in bold). When we work out that a particular bank or representative CANNOT occupy a certain seat we will cross them out...

So let's begin!

First, use the following rules:
- F sits second to right of the representative from Canara Bank (C).
- Representative from Bank of India (I) is an immediate neighbour of the representative from Canara Bank (C).

Let's fix the positions of Bank C and Rep F; we will then be able to anchor all the logic which follows off them. Arbitrarily, let Rep F occupy seat #1. This means Bank C must be in seat #3 (since Rep F sits second to right of them). Moreover, Bank I can then only occupy either seat #2 or seat #4 (since they sit next to Bank C).

Additionally, using the next rule (Two people sit between the representative of Bank of India (I) and B), we know that B can only be in seat #5 or seat #7, since these are the only unoccupied seats which might be three spaces away from I (thereby having two people between them).

Next:

Use the following rules:

- Neither C nor E is an immediate neighbour of either B or the representative from Canara Bank (C).
- C and E are immediate neighbours of each other.

This tells you that neither C nor E can occupy seat #2 or #4 (would be adjacent to Bank C). Neither can therefore occupy seat #3, since C and E must be adjacent to each other. Also, regardless of whether B occupies seat #5 or #7 (its two options), C and E cannot border it in seat #6. This in turn means C and E cannot occupy seat #5 (again since C and E must be adjacent to each other), which means that C and E must occupy seats #7 and #8 in some order. This has the knock-on effect that we now only have one option left for B: seat #5 - which also means (from the earlier clue) that Bank I must be in seat #2 (two people between Bank I and B).

Turn to the next rules:

- Representative of Bank of Marahashtra (M) sits second to right of D.
- D is neither the representative of Canara Bank (C) nor Bank of India (I).

Straightaway we know that D does not sit in seat #2 (Bank I) or #3 (Bank C). D also cannot be in seat #4, since the space two to the right is already occupied by Bank I (not Bank M). D must therefore be in seat #6 and Bank M in seat #4.

Now, use the next two rules:

- G and the representative from UCO Bank (U) are immediate neighbours of each other.
- B is not the representative of UCO Bank (U).

Firstly, Bank U is not in seat #5 (B). Since G must neighbour Bank U it cannot be in seats #4 or #3 (which borders Banks I and M). Only one space remains for G: seat #2. Since Bank U must neighbour G, it must be in seat #1.

Skip the next rule for now and focus on:

- H sits third to the left of the representative from Dena Bank (D).

Straightaway we can place H in seat #3, since if it were in seat #4 it would sit third to the left of Bank U (not Bank D). This also places A by deduction in seat #4 and Bank D in seat #8.

Now use this rule:

- Representative from Punjab National Bank (P) sits second to left of the representative from Syndicate Bank (S).

The only possible arrangement remaining which satisfies this rule is for Bank P to be in seat #7 and Bank S to be in seat #5. By deduction, Bank O is in the last remaining space: #6.

Return now to the rule we skipped earlier:

- Only one person sits between C and the representative from Oriental Bank of Commerce (O).
Immediately we can deduce that C must be in seat #8, since there must be one person between it and Bank O (which is in seat #6). Finally, this means E must occupy seat #7. Our final diagram thus looks like this:

Starting with our arbitrarily-numbered seat #1 we therefore have, going clockwise:

#1 F (UCO Bank)
#2 G (Bank of India)
#3 H (Canara Bank)
#4 A (Bank of Maharashtra)
#5 B (Syndicate Bank)
#6 D (Oriental Bank of Commerce)
#7 E (Punjab National Bank)
#8 C (Dena Bank)

And the puzzle is complete!

Answer 2

My answer is

Going anti-clockwise from some chair

E Punjab National Bank
D Oriental Bank of Commerce
B Syndicate Bank
A Bank of Maharashtra
H Canara Bank
G Bank of India
F UCO Bank
C Dena Bank

I could not solve this by logic, so I wrote a C program.
Total permutations of banks and their representatives $= 8! \times 8! = 1625702400$.
But given that the table is circular there are $7!$ ways that the banks can be placed.
And within the two constraints, only $192$ arrangements of the banks are possible.
So the total perms is $8! \times 192 = 7741440$ arrangements, not many.
I permuted them, checking the conditions, and found exactly one arrangement.

If someone can solve it by logic, I bow to their superior skill.