Difference between revisions of "2023 IOQM/Problem 8"
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*Case III: If the 2 x 2 tile is on the 3rd column of the board then the number of ways of tiling the rest of the board will be <math>a_2a_3=6</math>. | *Case III: If the 2 x 2 tile is on the 3rd column of the board then the number of ways of tiling the rest of the board will be <math>a_2a_3=6</math>. | ||
Now we will add all the cases and multiply by 2 (Why? Because the cases will repeat again). <math>2(8+5+6)=38</math>, remember this is when we consider a 2 x 2 tile, without the 2 x 2 tile there are <math>a_7=21</math> ways. So the total number of ways of tiling a 2 x 7 board using a 2 x 2 tile and 2 x 1 dominos is <math>\boxed{38+21=\textbf{59}}</math> | Now we will add all the cases and multiply by 2 (Why? Because the cases will repeat again). <math>2(8+5+6)=38</math>, remember this is when we consider a 2 x 2 tile, without the 2 x 2 tile there are <math>a_7=21</math> ways. So the total number of ways of tiling a 2 x 7 board using a 2 x 2 tile and 2 x 1 dominos is <math>\boxed{38+21=\textbf{59}}</math> | ||
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Latest revision as of 06:26, 10 August 2025
Problem 8
Given a 2 x 2 tile and seven dominoes (2 x 1 tile), find the number of ways of tilling a 2 x 7 rectangle using some of these tiles.
Solution
Let 
be the total number of ways of tiling a 2 x n rectangle using only dominoes. So 
. Obviously 
and 
(remember right now we are not considering the 2 x 2 tile). This will give us 
. Now we introduce the 2 x 2 tile. Number of ways of tiling the 2 x 7 board will totally depend on the position of the 2 x 2 tile.
- Case I: If the 2 x 2 tile is on the 1st column of the 2 x 7 board then the numbers of ways for tiling the rest of the board would be
. - Case II: If the 2 x 2 tile is on the 2nd column of the 2 x 7 board then numbers of ways of tiling the rest of the board will be
. - Case III: If the 2 x 2 tile is on the 3rd column of the board then the number of ways of tiling the rest of the board will be
.
.
.
.Now we will add all the cases and multiply by 2 (Why? Because the cases will repeat again). 
, remember this is when we consider a 2 x 2 tile, without the 2 x 2 tile there are ways. So the total number of ways of tiling a 2 x 7 board using a 2 x 2 tile and 2 x 1 dominos is 
