Tiling a Rectangle with Domino's

I'm stuck on finding approach to eliminate undesirable combinations in the problem below, any help would be appreciated.

How many ways can a 8x2 rectangle be tiled with 2x1 dominoes such that exactly two of the dominoes are vertical?

Originally posted by joe shmo
I'm stuck on finding approach to eliminate undesirable combinations in the problem below, any help would be appreciated.

How many ways can a 8x2 rectangle be tiled with 2x1 dominoes such that exactly two of the dominoes are vertical?

Is that 8 columns (positions) where a vertical domino can go?

Or 2 columns of 8?

Originally posted by wolfgang59
Is that 8 columns (positions) where a vertical domino can go?

Or 2 columns of 8?

8 columns where a vertical domino can be placed.

Originally posted by joe shmo
8 columns where a vertical domino can be placed.

Leftmost domino in position
1 then second can go in 5 places (not 3 or 7)
3 then second can go in 3 places (not 5 or 7)
4 then second can go in 2 places (not 6 or 7)
5 then second can go in 2 places (not 7)
6 then second can go in 0 places (not 7 or 8)
7 then second can go in 1 place (only 8)


So I make that 13 in total.

Originally posted by wolfgang59
Leftmost domino in position
1 then second can go in 5 places (not 3 or 7)
3 then second can go in 3 places (not 5 or 7)
4 then second can go in 2 places (not 6 or 7)
5 then second can go in 2 places (not 7)
6 then second can go in 0 places (not 7 or 8)
7 then second can go in 1 place (only 8)


So I make that 13 in total.

The answer is 10, so you have a few more to find.

Correct me if i'm wrong, but your method is somewhat brute force? Is their a way that is mathematically succinct and generalized to any 2*n x 2 rectangle using combinatorics?

Originally posted by joe shmo
The answer is 10, so you have a few more to find.

Correct me if i'm wrong, but your method is somewhat brute force? Is their a way that is mathematically succinct and generalized to any 2*n x 2 rectangle using combinatorics?

OK. Sober now.
The answer is n(n+1)/2 for all n.
(triangle numbers)

The proof is obvious but difficult to put down .....

Start by considering that the left most domino is always odd and right most is always even..

Originally posted by wolfgang59
OK. Sober now.
The answer is n(n+1)/2 for all n.
(triangle numbers)

The proof is obvious but difficult to put down .....

Start by considering that the left most domino is always odd and right most is always even..

Ok.
n(n+1)/2 is true for n=1
Now to prove if true for n then true for m=(n+1)

When we have 2m columns there is ONE additional place for the right most tile for all n positions of the left most tile. In addition there is one more configuration with left most tile in column (m-1) and right most in position m,
Therefore there are an additional m, or (n+1) configurations.

Total = n(n+1)/2 + (n+1) = {n(n+1)+2(n+1)}/2
= (n+1)(n+2)/2 = m(m+1)/2
QED

Originally posted by joe shmo
I'm stuck on finding approach to eliminate undesirable combinations in the problem below, any help would be appreciated.

How many ways can a 8x2 rectangle be tiled with 2x1 dominoes such that exactly two of the dominoes are vertical?

Seven. Basically there are four 2x2 squares which each fit two dominoes. Bother must be either vertical or horizontal. So depending on which of those four squares you put the vertical dominoes you can have four possibilities.

You can also have one vertical one on each end. So five.

Or one vertical, two horizontal, one vertical, four horizontal. That's six.

Or one vertical, four horizontal, vertical, two horizontal. Seven.

Originally posted by AThousandYoung
Seven. Basically there are four 2x2 squares which each fit two dominoes. Bother must be either vertical or horizontal. So depending on which of those four squares you put the vertical dominoes you can have four possibilities.

You can also have one vertical one on each end. So five.

Or one vertical, two horizontal, one vertical, four horizontal. That's six.

Or one vertical, four horizontal, vertical, two horizontal. Seven.

The answer is 10...so your missing a few.

Originally posted by wolfgang59
Ok.
n(n+1)/2 is true for n=1
Now to prove if true for n then true for m=(n+1)

When we have 2m columns there is ONE additional place for the right most tile for all n positions of the left most tile. In addition there is one more configuration with left most tile in column (m-1) and right most in position m,
Therefore there are an additional m, or (n ...[text shortened]... configurations.

Total = n(n+1)/2 + (n+1) = {n(n+1)+2(n+1)}/2
= (n+1)(n+2)/2 = m(m+1)/2
QED

Are there any methods in combinatorics that you know of that can be used to eliminate combinations of the remaining un-chosen columns?; That is removing combinations for which any two of the un-chosen columns are not adjacent?