diff --git a/Recursive-Grid-Tiling/README.md b/Recursive-Grid-Tiling/README.md new file mode 100644 index 0000000..079fa5d --- /dev/null +++ b/Recursive-Grid-Tiling/README.md @@ -0,0 +1,157 @@ +# Coding 2: Recursive Grid Tiling + +## Grid Tiling + +Here's an interesting theorem: consider a square grid of squares with sides +given by some power of (i.e. a grid of size $2^n \times 2^n$) like this example +of a $2^3 \times 2^3$ grid: + +![t0](./images/t0.png) + +If you select exactly one grid square to remove (which is colored black below), +then the remaining squares can be tiled by 3-square L-shape tiles, like this. + +![tiling](./images/tiling.png) + +The proof of this fact is inductive and leads to a simple recursive algorithm +for producing a tiling. Assume that you can tile any $2^{n-1} \times 2^{n-1}$ +grid with a single square removed with L-shaped tiles. Suppose we are given a +$2^n \times 2^n$ grid with one square removed. + +Here's an example of what we've got: + +![t1](./images/t1.png) + +How can we use our inductive hypothesis to show that the entire grid can be +tiled? Well, let's start by splitting our grid vertically and horizontally in +half: + +![t2](./images/t2.png) + +Notice that this splits our $2^n \times 2^n$ sized grid into four +$2^{n-1} \times 2^{n-1}$ sized sub-grids, and one of them (the top left in our +example above) already has a square removed _but notice that three of them do +not yet have a square removed_. This means we can use our inductive hypothesis +to tile whichever sub-grid contains the square we already wanted removed: + +![t3](./images/t3.png) + +But now we're a bit stuck, because the conditions of the inductive hypothesis +apply if we have a $2^{n-1} \times 2^{n-1}$ grid with one square removed, but +instead we have three $2^{n-1} \times 2^{n-1}$ grids with no squares removed. +But now notice the following trick. We can place a single L-tile that covers +exactly one grid square in each of three sub-grids that we haven't yet tiled! + +![t4](./images/t4.png) + +But notice that since this new L-tile covers exactly one square from each of the +three untiled sub-grids, we now have three $2^{n-1} \times 2^{n-1}$ each with a +single square removed and so the inductive hypothesis now applies! We can tile +each quadrant individually using the inductive hypothesis. + +![t5](./images/t5.png) + +Then + +![t6](./images/t6.png) + +Then + +![t7](./images/t7.png) + +And finally, with the red dividing lines removed: + +![t8](./images/t8.png) + +The base case for this induction are the 2x2 grids, which become L-tiles if we +remove anyone grid square. + +## Your Task + +You are going to write a program to compute L-tilings of $2^n \times 2^n$ square +grids. You will assign each tile a number starting with 00. The first tile you +generate should have all of its grid cells marked with a 00, the next tile 01, +etc. + +### Input + +The first line of input will give the grid size parameter n as an integer. The +second line will contain the index (i, j) of the removed tile. + +### Output + +Your code should output $2^n$ lines, each containing $2^n$ 2-digit numbers +separated by spaces (and formatted so that single digit numbers have a leading +0). The value of each grid location should be the tile index you generated. (You +may assume you won't need 3 digits). + + + + + + + + + + + + + + + + + + +
Sample InputSample Output
3
0 0
 + 
+-1 02 03 03 07 07 08 08
+02 02 01 03 07 06 06 08
+04 01 01 05 09 09 06 10
+04 04 05 05 00 09 10 10
+12 12 13 00 00 17 18 18
+12 11 13 13 17 17 16 18
+14 11 11 15 19 16 16 20
+14 14 15 15 19 19 20 20
+ 
3
1 2
 + 
+02 02 03 03 07 07 08 08
+02 01 -1 03 07 06 06 08
+04 01 01 05 09 09 06 10
+04 04 05 05 00 09 10 10
+12 12 13 00 00 17 18 18
+12 11 13 13 17 17 16 18
+14 11 11 15 19 16 16 20
+14 14 15 15 19 19 20 20
+ 
4
10 10
 + 
+03 03 04 04 08 08 09 09 24 24 25 25 29 29 30 30
+03 02 02 04 08 07 07 09 24 23 23 25 29 28 28 30
+05 02 06 06 10 10 07 11 26 23 27 27 31 31 28 32
+05 05 06 01 01 10 11 11 26 26 27 22 22 31 32 32
+13 13 14 01 18 18 19 19 34 34 35 35 22 39 40 40
+13 12 14 14 18 17 17 19 34 33 33 35 39 39 38 40
+15 12 12 16 20 17 21 21 36 36 33 37 41 38 38 42
+15 15 16 16 20 20 21 00 00 36 37 37 41 41 42 42
+45 45 46 46 50 50 51 00 66 66 67 67 71 71 72 72
+45 44 44 46 50 49 51 51 66 65 65 67 71 70 70 72
+47 44 48 48 52 49 49 53 68 65 -1 69 73 73 70 74
+47 47 48 43 52 52 53 53 68 68 69 69 64 73 74 74
+55 55 56 43 43 60 61 61 76 76 77 64 64 81 82 82
+55 54 56 56 60 60 59 61 76 75 77 77 81 81 80 82
+57 54 54 58 62 59 59 63 78 75 75 79 83 80 80 84
+57 57 58 58 62 62 63 63 78 78 79 79 83 83 84 84
+
+ +## Visualizing Your Results: + +You can use the [plot_tiles.py](./plot_tiles.py) program to visualize your +input. It will draw figures similar to the ones shown in this assignment and may +help you debug your program if it has issues. + +## Program Grading and Submission + +This assignment is to be submitted into Gradescope. Grading criterion is: + +- 10 points for the example problems shown in the assignment +- 8 points on hidden example problems +- 2 points for style diff --git a/Recursive-Grid-Tiling/images/t0.png b/Recursive-Grid-Tiling/images/t0.png new file mode 100644 index 0000000..b642021 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t0.png differ diff --git a/Recursive-Grid-Tiling/images/t1.png b/Recursive-Grid-Tiling/images/t1.png new file mode 100644 index 0000000..36b1a2b Binary files /dev/null and b/Recursive-Grid-Tiling/images/t1.png differ diff --git a/Recursive-Grid-Tiling/images/t2.png b/Recursive-Grid-Tiling/images/t2.png new file mode 100644 index 0000000..4feeaa9 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t2.png differ diff --git a/Recursive-Grid-Tiling/images/t3.png b/Recursive-Grid-Tiling/images/t3.png new file mode 100644 index 0000000..40442d8 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t3.png differ diff --git a/Recursive-Grid-Tiling/images/t4.png b/Recursive-Grid-Tiling/images/t4.png new file mode 100644 index 0000000..916f5c8 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t4.png differ diff --git a/Recursive-Grid-Tiling/images/t5.png b/Recursive-Grid-Tiling/images/t5.png new file mode 100644 index 0000000..0746caf Binary files /dev/null and b/Recursive-Grid-Tiling/images/t5.png differ diff --git a/Recursive-Grid-Tiling/images/t6.png b/Recursive-Grid-Tiling/images/t6.png new file mode 100644 index 0000000..c8496e4 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t6.png differ diff --git a/Recursive-Grid-Tiling/images/t7.png b/Recursive-Grid-Tiling/images/t7.png new file mode 100644 index 0000000..2027105 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t7.png differ diff --git a/Recursive-Grid-Tiling/images/t8.png b/Recursive-Grid-Tiling/images/t8.png new file mode 100644 index 0000000..896e6f4 Binary files /dev/null and b/Recursive-Grid-Tiling/images/t8.png differ diff --git a/Recursive-Grid-Tiling/images/tiling.png b/Recursive-Grid-Tiling/images/tiling.png new file mode 100644 index 0000000..e8c0f0c Binary files /dev/null and b/Recursive-Grid-Tiling/images/tiling.png differ