# Logical Reasoning - Seating Arrangement problem

Source

P, Q, R, S, T, U, V and W are sitting around a square table in such a way that four of them sit at four corners of the square while four sit in the middle of each of the four sides. The ones who sit at the four corners face the centre while those who sit in the middle of the sides face outside. Each of them has a different profession viz. Professor, Architect, Businessman, Doctor, Chartered accountant, Trainer, Banker and Coder. (None of the information given is necessarily in the same order).

• The one who is a Coder does not face inside. Only two persons sit between R and W.
• The one who is a Professor sits on the immediate right of W. The one who is a Chartered accountant sits second to the right of V. V is an immediate neighbor of neither W nor R. V does not sit third to the right of R.
• R and U do not sit diagonally. Only one person sits between P and the one who is a Chartered accountant. R sits third to the left of the one who is a a Coder.
• The one who is a Chartered accountant is not an immediate neighbour of the one who is Coder. S sits on the immediate left of the one who is a Trainer. V is not a Trainer.
• T is a Banker. T is not an immediate neighbour of P.
• Architect is an immediate neighbour of T.
• Doctor is an immediate neighbour of U.

Questions:

1. Find the one who sits diagonally opposite to the Professor?
2. Who sits exactly between W and P?

I feel there is something missing in this question, as above is not leading to a well defined set up

• it is a practice question in bank clerk job in India. Commented May 2 at 20:39
• When you say sit diagonally, can you clarify if that can only be true for people sitting on the corners? If not what does it exactly mean for people sitting in the middles. Thanks! Commented May 2 at 21:33

That the Professor is sitting diagonally to R, and that we cannot tell who is between W and P

But it probably should have been:

That the Professor is sitting diagonally to R who is a Trainer, and U the Business man is sitting between W and P

Let denote the seats by 0 to 7 starting from the top left corner and going around:

0--1--2
|     |
7     3
|     |
6--5--4


We also denote the professions by the first two letters. Now for how to obtain the solution:

We can reformulate the problem with these notations, doing all calculations modulo 8 (i.e. we're looping from 7 to 0 if we add and from 0 to 7 if we substract):

1. Co ∈ {1,3,5,7}, the Coder is not facing inside, so he's not in a corner, meaning he is in a middle;
2. R = W ± 3, between R and W there is 2 seats so they need to be 3 positions apart from eachother;
3. Pr = W - 1, the Professor is sitting to the immediate right of W;
4. Ch = V - 2, the Chartered accountant sits second to the right of V;
5. V ≠ W ± 1, V is not an immediate neighbor W;
6. V ≠ R ± 1, V is not an immediate neighbor R;
7. V ≠ R - 3, V does not sit third to the right of R;
8. (R,U) ∉ {(0,4), (2,6), (4,0), (6,2)}, R and U does not sit diagonally;
9. Ch = P ± 2, only one person sits between P and the Chartered accountant;
10. R = Co + 3, R sits third to the left of the one who is the Coder;
11. Ch ≠ Co ± 1, the Chartered accountant is not an immediate neighbour of the Coder;
12. S = Tr + 1, S sits on the immediate left of the Trainer;
13. V ≠ Tr, V is not the Trainer;
14. T = Ba, T is the Banker;
15. T ≠ P ± 1 , T is not an immediate neighbour of P;
16. Ar = T ± 1, the Architect is an immediate neighbour of T;
17. Do = U ± 1 , Doctor is an immediate neighbour of U.

In addition we add that the names are all different (i.e. P, Q, R, S, T, U, V and W are all different) and the professions are all different.

I used a CSP solver to find all the solutions

and I got 8 possibilities, which after removing the symmetric ones gave 2 possibilities:

In both of them, the Professor was sitting diagonally to R, which is either a Trainer or a Chartered accountant. Unfortunately they don't give the same solutions for who sits between W and P. It is either no one and they are immediate neighbors, or its U who is a Business man. So my guess is that some missing info or oversight in the question writing was supposed to let us eliminate the world that doesn't allow us to respond to the second question. But I could also have made a mistake.

• With W facing outward, the Professor on W's right will be in the next position clockwise in your diagram. Commented May 3 at 2:26
• 12 is incorrect, inward/outward is not taken into account Commented May 3 at 5:51
• @DanielMathias oh maybe that's what's going on that OP and I missed, I'll edit as soon as I can Commented May 3 at 7:44
• But honestly this is partly poor phrasing, the source question could have been more precise about from which point of view was left and right. I interpreted it as the absolute left and right, the one of an outside observer. Commented May 3 at 7:50

Fluorine has a good point about this being poorly phrased. The question as written means "absolute left and right, the one of an outside observer" (Fluorine). Note that the originator of this problem should have used phrasing such as "V does not sit 3 spaces on R's right" if he meant direction with respect to R's orientation.

I coded this in Fortran as written (absolute left and right) and got multiple fits all of which confirmed Fluorine's results.

For completeness I ran this using relative directionality (person's right or left). In this case the banker (Ba) and T are opposite the professor (Pr) and S is between W and P. An example output starting from the upper left (0) and going clockwise:

q w s p r u t v ---- bu ch pr tr dr ar ba co