行交替顺序时如何将网格数转换为坐标

Question

7|8|9
6|5|4
1|2|3

1 -> (1,1)
2 -> (2,1)
3 -> (3,1)
4 -> (3,2)
5 -> (2,2)
6 -> (1,2)
7 -> (1,3)
8 -> (2,3)
9 -> (3,3)

In this grid, the mapping of the numbers to coordinates is shown above.

I'm struggling to come up with a formula where given the number of the grid and the number of rows and columns in the grid, it outputs the coordinates of the grid.

I tried following the logic in this question but in this question , the coordinate system starts from 0 and the rows are not alternating.

Answer 1

If there was no alternating and the numbers were all starting at 0 and not 1, then you could apply Euclidean division directly:

x = n % 3
y = n // 3

where // gives the quotient of the Euclidean division, and % gives the remainder of the Euclidean division.

If there was no alternating, but the numbers all start at 1 instead of 0, then you can fix the above formula by removing 1 from n to make it start at 0, then adding 1 to x and y to make them start at 1:

x = ((n - 1) % 3) + 1
y = ((n - 1) // 3) + 1

Now all we have to change to take the alternating into account is to flip the x values on the right-to-left rows.

y remains unchanged, and x remains unchanged on the left-to-right rows.

The right-to-left rows are the rows with an even y , and you can flip x symmetrically around 2 by removing it from 4 :

if y % 2 == 0:
    x = 4 - x

Putting it all together in a function and testing it, in python:

def coord(n):
    y = ((n-1) // 3) + 1
    x = ((n-1) % 3) + 1
    if y % 2 == 0:  # right-to-left row
        x = 4 - x   # flip horizontally
    return (x, y)

for n in range(1, 9+1):
    x, y = coord(n)
    print(f'{n} -> ({x},{y})')

Output:

1 -> (1,1)
2 -> (2,1)
3 -> (3,1)
4 -> (3,2)
5 -> (2,2)
6 -> (1,2)
7 -> (1,3)
8 -> (2,3)
9 -> (3,3)

Inverse function

The inverse operation of a Euclidean division is a multiplication and an addition:

if y % 2 == 1:
    n = 3 * (y-1) + x
else:
    n = 3 * (y-1) + 4 - x