square puzzle solution

Question

Question: given an integer number n, print the numbers from 1 up to n² like this:

n = 4

result is:

01 02 03 04
12 13 14 05
11 16 1         


        

Answer 1

Here's a different approach. It relies on spotting that the movements you make cycle between: right, down, left, up, right, .... Further, the number of times you move goes: 3 right, 3 down, 3 left, 2 up, 2 right, 1 down, 1 left. So without further ado, I will code this up in Python.

First, I will use some itertools and some numpy:

from itertools import chain, cycle, imap, izip, repeat
from numpy import array

The directions cycle between: right, down, left, up, right, ...:

directions = cycle(array(v) for v in ((0,1),(1,0),(0,-1),(-1,0)))

(I'm using numpy's arrays here so I can easily add directions together. Tuples don't add nicely.)

Next, the number of times I move counts down from n-1 to 1, repeating each number twice, and the first number three times:

countdown = chain((n-1,), *imap(repeat, range(n-1,0,-1), repeat(2)))

So now my sequence of directions can be created by repeating each successive direction by the paired number in countdown:

dirseq = chain(*imap(repeat, directions, countdown))

To get my sequence of indices, I can just sum this sequence, but (AFAIK) Python does not provide such a method, so let's quickly throw one together:

def sumseq(seq, start=0):
  v = start
  yield v
  for s in seq:
    v += s
    yield v

Now to generate the original array, I can do the following:

a = array(((0,)*n,)*n) # n-by-n array of zeroes
for i, v in enumerate(sumseq(dirseq, array((0,0)))):
  a[v[0], v[1]] = i+1
print a

Which, for n = 4, gives:

[[ 1  2  3  4]
 [12 13 14  5]
 [11 16 15  6]
 [10  9  8  7]]

and, for n = 5, gives:

[[ 1  2  3  4  5]
 [16 17 18 19  6]
 [15 24 25 20  7]
 [14 23 22 21  8]
 [13 12 11 10  9]]

This approach can be generalised to rectangular grids; I leave this as an exercise for the reader ;)