How to solve sitting arrangement questions with 100% accuracy?

Question

There are questions of sitting arrangement like persons sitting in a circular arrangement facing inward/outward or both, or a linear arrangement or box arrangement etc. Sometimes a few data points are missing and we need to produce 2-3 different cases to solve the question. But making 2-3 different diagrams and solving them is quite hectic. Is there any straightforward approach which follows some specific procedure and when done will cover all possible cases and finally I'll get the solution with 100% accuracy?

Example question:

There are eight persons A,B,C,D,E,F,G and H are sitting around a square table. Four of them are sitting in the middle of the table facing outside and rest of them sitting on the corner of the table facing inside. Only one person sit between G and A. F and G sit opposite to each other and facing each other. D does not sit adjacent to B. E does not sit between A and F. B sits third to the left of A. G does not sit to the immediate left of B. H and D are immediate left of each other.

I want to solve such problems using pen and paper in a way which takes all possible cases so that finally I will come to a conclusion with 100% accuracy.

Answer 1

_{This answer does not user spoilers because what's being presented is not a solution that may be spoiled, but a strategy for reaching solutions. It is also (likely overly) long in an attempt to methodically demonstrate the strategy.}

You may be able to use a logic grid to do this. Label the seats as so (ignore the header row):

1	2	3
8		4
7	6	5

Those in positions 1, 3, 5, and 7 are facing in. Those in positions 2, 4, 6, and 8 are facing out. Now, set up a 2-member logic grid as so (see here for how to set logic grids up):

	1	2	3	4	5	6	7	8
A
B
C
D
E
F
G
H

Before we start, let's list all the clues given in the problem (with my minor grammar edits):

• Only one person sits between G and A.
• F and G sit opposite to each other and facing each other.
• D does not sit adjacent to B.
• E does not sit between A and F.
• B sits third to the left of A.
• G does not sit to the immediate left of B.
• H and D are immediately left of each other.

The first thing to note is that the numbering is entirely arbitrary: shifting all people two seats in either direction will create another valid solution. Thus we need to fix a reference point, arbitrarily assigning someone to a seat of the correct type (corner/middle). Then we can place everyone in relation to them. This arbitrary fixed point is a necessity whenever problems like this have circular arrangements where the clues provide only local positioning information, such as "on the left side of X" or "across from X", instead of global position information, such as "north of X" or "on the right side of the table". The only global information provided is that which distinguishes corner seats and middle seats.

Therefore, our first step is finding someone for whom we are confident on seat type. The key is this line:

F and G sit opposite to each other and facing each other.

To be "facing each other", these people must be at corners. We assign F to position 1 and G to position 5. Everyone else can now be put relative to them. The updated logic grid is:

	1	2	3	4	5	6	7	8
A	X				X
B	X				X
C	X				X
D	X				X
E	X				X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X				X

With some people fixed in place, this becomes a game of progressively narrowing down the possibilities. The first thing I looked for were clues that involved people whom we had already placed, making marking position deductions on the grid easier.

Only one person sits between G and A.

Since we already know where G is, A must be either position 3 or 7.

G does not sit to the immediate left of B.

Since we already know where G is, B cannot be in position 6.

The updated logic grid is:

	1	2	3	4	5	6	7	8
A	X	X		X	X	X		X
B	X				X	X
C	X				X
D	X				X
E	X				X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X				X

Having exhausted the use of those clues, let's look at what we have left to play with:

• D does not sit adjacent to B.
• E does not sit between A and F.
• B sits third to the left of A.
• H and D are immediately left of each other.

The first one is useless right now: with both B and D able to be nearly anywhere, it doesn't narrow anything down. The second is also out: without knowing where at least two of A, E, and F are, the third can't be blocked from an illegal location.

The third, however, means that B's seat number has to be A's + 3, (subtracting 8 if the number goes higher than 8). Since A can only be in 3 or 7, according to our current logic grid, this means B must be in either 6 or (10 - 8 = 2). But that first one is impossible, because we decided that B could not be 6 based off a previous clue! Thus we can place both A and B. The updated logic grid is:

	1	2	3	4	5	6	7	8
A	X	X	X	X	X	X	✓	X
B	X	✓	X	X	X	X	X	X
C	X	X			X		X
D	X	X			X		X
E	X	X			X		X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X	X			X		X

Half of the people placed! We can go back to a previous strategy: scour the clues for any involving people we have already placed. There are, again, two useful ones:

D does not sit adjacent to B.

Since we know where B is, D cannot be in position 3.

E does not sit between A and F.

Since we know where A and F are, E cannot be in position 8.

The updated logic grid is:

	1	2	3	4	5	6	7	8
A	X	X	X	X	X	X	✓	X
B	X	✓	X	X	X	X	X	X
C	X	X			X		X
D	X	X	X		X		X
E	X	X			X		X	X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X	X			X		X

There is only a single clue left unexhausted, so let's figure out how to use it!

H and D are immediately left of each other.

This means that H and D need to have consecutive numbers for their positions. The only consecutive numbers left untaken are 3 and 4; out of these, D cannot be 3 according to our grid. The updated logic grid is:

	1	2	3	4	5	6	7	8
A	X	X	X	X	X	X	✓	X
B	X	✓	X	X	X	X	X	X
C	X	X	X	X	X		X
D	X	X	X	✓	X	X	X	X
E	X	X	X	X	X		X	X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X	X	✓	X	X	X	X	X

We have no clues left! But the logic grid lets us make the last placements. According to it, E can only be in position 6, and position 8 can only be occupied by C. The final logic grid is:

	1	2	3	4	5	6	7	8
A	X	X	X	X	X	X	✓	X
B	X	✓	X	X	X	X	X	X
C	X	X	X	X	X	X	X	✓
D	X	X	X	✓	X	X	X	X
E	X	X	X	X	X	✓	X	X
F	✓	X	X	X	X	X	X	X
G	X	X	X	X	✓	X	X	X
H	X	X	✓	X	X	X	X	X

Which corresponds to this arrangement of seats:

F	B	H
C		D
A	E	G

(Remember that everyone shifting two places in either direction corresponds to another valid solution, but we arbitrarily fixed F to the top left corner.)

My hope is that this strategy will allow you to methodically apply the information from each clue, building from one to another until the final solution is found. The logic grid is helpful because it is a memory device: as long as you only make marks when you are sure of them, you can trust all the marks in the grid. Thus, if it says that a certain person cannot be in a certain spot, you can use that information as fact without having to hold it in your head or re-deduce it. I personally find this organization helpful; this answer is my attempt at laying out my thought process when using a logic grid for a problem of this type.

Answer 2

I don't know how the other similar problems you're referring to looks like, but for the example:

First,

you know that F and G are seated at opposite corners since they're facing each other.

second,

You know that A has to be seated at a corner since there is exactly one person between G and A.

thirdly,

B sits third to the left of A. G does not sit to the immediate left of B.

Now we can place:

A,B,F and G like this:

And now we also can deduct that,

H has to be seated at the final corner, since "D does not sit adjacent to B" AND since "H and D are immediate left of each other". So from this, we get this arrangement:

And finally,

E does not sit between A and F.

Which means we get this arrangement:

Note:

There are four total solutions if we just let each person shift two seats in the same direction.