gpt4 August 3 version

---

You answer questions about the logic puzzles.


Use the below article on a logic puzzle to answer the subsequent question. If the answer cannot be found, write "I don't know."

Focus on the `MISSING`, this is what I want you to fill in.

Grids
- The `input` and `output` dictionaries contain key-value pairs.
- The `key` is the group number, starting with 0 and is incremented for each group.
- The `value` is a string that represent a 2D grid.
- The first two fields of the string specify the grid width and height. Example: `width30,height19` specifies a grid with 30 columns and 19 rows.
- The remaining part of the string is a one-dimensional representation of a 2D grid.

Article start marker
```python
input = {}
output = {}
# Group0
input[0] = 'width12,height12,800000000000,220000000000,008000000000,002200000000,000080000000,000022000000,000000800000,000000220000,000000008000,000000002200,000000000080,000000000022'
output[0] = 'width12,height12,000000000080,000000000022,000000008000,000000002200,000000800000,000000220000,000080000000,000022000000,008000000000,002200000000,800000000000,220000000000'
# Group1
input[1] = 'width12,height12,000000000022,000000000032,000000002200,000000003200,000000220000,000000320000,000022000000,000032000000,002200000000,003200000000,220000000000,320000000000'
output[1] = 'width12,height12,220000000000,320000000000,002200000000,003200000000,000022000000,000032000000,000000220000,000000320000,000000002200,000000003200,000000000022,000000000032'
# Group2
input[2] = 'width10,height10,2100000000,1000000000,0021000000,0010000000,0000210000,0000100000,0000002100,0000001000,0000000021,0000000010'
output[2] = 'width10,height10,0000000021,0000000010,0000002100,0000001000,0000210000,0000100000,0021000000,0010000000,2100000000,1000000000'
# Group3
input[3] = 'width14,height14,00000000000018,00000000000081,00000000001800,00000000008100,00000000180000,00000000810000,00000018000000,00000081000000,00001800000000,00008100000000,00180000000000,00810000000000,18000000000000,81000000000000'
output[3] = 'MISSING'
```
Article end marker

Question: Write 10 bullet points with observations about input and output.
Don't repeat text from the prompt that is already well specified.

Question: Fill in the `MISSING` piece into this python code block.
```python
output[3] = 'MISSING'
```

---

### Observations:

1. The `input` and `output` grid sizes are the same for each group. This means that the transformation from input to output doesn't change the dimensions of the grid.
  
2. The numbers in the grids appear to move in a pattern towards the bottom right corner in the output, based on the input grid for each group.

3. The first non-zero number in the `input` of each group appears at the bottom-right corner in the corresponding `output`.

4. The second non-zero number in the `input` of each group appears at one row above and one column to the left of its position in the `output`, relative to the first non-zero number.

5. This pattern of shifting up and to the left continues for all non-zero numbers in the `input` grid, mapping them to new locations in the `output` grid.

6. Zeros in the `input` grid seem to fill in the space left after the non-zero numbers have been relocated in the `output`.

7. The non-zero numbers maintain their relative orientation to each other. That is, if a number is above and to the right of another number in the `input`, it will be above and to the right of that number in the `output`.

8. Each group has a unique set of non-zero numbers, but the transformation rule appears to be consistent across groups.

9. All numbers are moved, including both digits in double-digit numbers (e.g., 18 and 81 in Group3), based on the observed pattern.

10. No new numbers are introduced in the `output` grids; all numbers from the `input` are present in the `output`, just rearranged.

### Filling in the MISSING Piece:

Based on the observations and patterns identified, the `MISSING` piece for `output[3]` would involve rotating the numbers in `input[3]` such that the first non-zero number (18) would end up at the bottom-right corner, and the second non-zero number (81) would end up one row above and one column to the left of the first number. The pattern continues for the remaining non-zero numbers.

```python
output[3] = 'width14,height14,18000000000000,81000000000000,00180000000000,00810000000000,00001800000000,00008100000000,00000018000000,00000081000000,00000000180000,00000000810000,00000000001800,00000000008100,00000000000018,00000000000081'
```

