We have two kind of tiles : 1 x 1 and 2 x 2, and use them to cover the 8 x 3 road without overlapping, how many different ways to arrange tiles there?
46
It is not the right answer, unfortunately.
well, using 24 1x1 tiles is one way.
Using 2x2 tiles, only one tile will fit across. It can be on the left or right side.
So, a single 2x2 tile can be placed in 2x7=14 places.
Now consider using more than one 2x2 tile. Figure how many places it can slide around into. I'm sure you can work it out.
Only a maximum of 4 2x2 tiles can fit into the road.
Ok.
So for 2x2 2 tiles I got 16
for 2x2 3 tiles I got 24
and for 2x2 4 tiles I got 10
Finally, I counted up 65 ways. But it is still the wrong answer. Where am I wrong?
must use recurrence equation.
To solve this problem, we can break it down into smaller parts and build our solution step by step.
First, let's consider the possible arrangements for the leftmost column of the road (3 x 1).
1. If we use a 1 x 1 tile, then we have two remaining columns of road (3 x 2) to cover. The total number of arrangements for this case can be found by solving the problem for the remaining road, which is now an 8 x 2 road.
2. If we use a 2 x 2 tile, then we have just one remaining column of road (3 x 1) to cover. The total number of arrangements for this case can be found by solving the problem for the remaining road, which is now a 6 x 3 road.
To find the total number of arrangements for the original 8 x 3 road, we need to consider both cases and sum their arrangements.
Case 1 (Using a 1 x 1 tile on the first column):
- The number of arrangements for the remaining 8 x 2 road can be found by repeating the same process.
- We start by considering the leftmost column (2 x 1) of the remaining road.
- Again, we have two cases: using a 1 x 1 or a 2 x 2 tile.
- If we use a 1 x 1 tile, we now have a 2 x 2 road left to cover.
- If we use a 2 x 2 tile, we now have a 6 x 2 road left to cover.
Case 2 (Using a 2 x 2 tile on the first column):
- The number of arrangements for the remaining 6 x 3 road can be found by repeating the same process.
- We start by considering the leftmost column (3 x 1) of the remaining road.
- Again, we have two cases: using a 1 x 1 or a 2 x 2 tile.
- If we use a 1 x 1 tile, we now have a 3 x 2 road left to cover.
- If we use a 2 x 2 tile, we now have a 5 x 3 road left to cover.
We continue this process until we reach the base case, which is when the remaining road has dimensions of 1 x 1 (1 column and 1 row).
Finally, we sum up all the arrangements obtained from each case to find the total number of different ways to arrange the tiles on the 8 x 3 road without overlapping.
Let's walk through the steps for better clarity:
Step 1: Start with the original 8 x 3 road
Step 2: Choose a tile for the first column (leftmost column):
a) If a 1 x 1 tile is chosen, we have a 7 x 3 road remaining.
b) If a 2 x 2 tile is chosen, we have a 6 x 3 road remaining.
Step 3: For each case from Step 2, repeat Steps 2-3 until reaching the base case (1 x 1 road):
a) For the 7 x 3 road:
- Choose a tile for the next column (left to right).
- Based on the chosen tile, determine the remaining road dimensions, and repeat Step 3.
- Continue this process until reaching the base case.
b) For the 6 x 3 road:
- Choose a tile for the next column (left to right).
- Based on the chosen tile, determine the remaining road dimensions, and repeat Step 3.
- Continue this process until reaching the base case.
Step 4: Sum up all the arrangements obtained from each case to find the total number of different ways to arrange the tiles on the 8 x 3 road without overlapping.