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

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

I have been struggling to solve it. I have devoted so much of my time trying to figure out solution.

Can anyone help?

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).

• I'm struggling on many bank titles... – Conifers Oct 23 '19 at 9:14
• Where are the "given questions" that must be answered? I presume the puzzle is as the title: find a possible seating arrangement, but was that necessary to answer the (unstated) questions? – Weather Vane Oct 23 '19 at 9:47
• @WeatherVane The questions can only be answered after arranging each and everything correctly in place. A definite arrangement exists in this case. Although I've edited it. – Hisenberg Oct 23 '19 at 9:57
• This should probably have source attribution? Googling the number at the end led to finding the question here, though not sure what the original source is: cracku.in/ibps-po-2011-question-paper-solved?page=2. I'm not convinced the full solution is necessary to answer the relevant questions. – Mohirl Oct 23 '19 at 13:28
• @Mohirl This is a question from an aptitude and reasoning based exam conducted in India for recruitment purpose and Banking and Financial sectors also sometimes to top Management Institutes. – Hisenberg Oct 23 '19 at 13:32

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!

• Nice charts~ and congrate for 10K! :D – Conifers Oct 24 '19 at 2:13

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.

• A friend of mine solved it by logic in 5 minutes. He is the most genius person I've ever seen in my entire life. He is extremely good at logic and picking up clues. Although I cannot vouch for the authenticity of his statement because I wasn't with him while he solved it but yes I can trust him because he is known to solve most unsolvable problems really quick. I know him since childhood. – Hisenberg Oct 23 '19 at 12:23
• I am not saying it can't be solved by logic - it probably can - just that I could not find a solution by logic. Although I started out with a truth chart and a circular table, using pencil and paper, I got lost. – Weather Vane Oct 23 '19 at 12:26
• I also got lost solving it using pen and paper. I'm looking for a logic to solve it and that's why I've put it here. – Hisenberg Oct 23 '19 at 12:30
• If the question had the "no-computers" tag I would not have gone down this route. As it is, I found it an interesting challenge, obtaining a result in less than 1 second of run time. – Weather Vane Oct 23 '19 at 12:32
• I forgot to put it there. Anyways thumbs up for your time and effort. – Hisenberg Oct 23 '19 at 12:34