Problem 28

Problem 28

11 October 2002

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

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

It can be verified that the sum of both diagonals is 101.

What is the sum of both diagonals in a 1001 by 1001 spiral formed in the same way?    

def fun28():
    #(ard, ald, alu, aru)分别代表右下、左下、左上、右上的数值
    #都初始化为1
    (ard, ald, alu, aru) = (1, 1, 1, 1)
    #1001*1001的矩阵，共有层次500层(旋转500圈)
    step   = 0
    result = 1
    for i in range(500):
        ard = ard + 3*step + (step+2)*1
        ald = ald + 2*step + (step+2)*2
        alu = alu + 1*step + (step+2)*3
        aru = aru + 0*step + (step+2)*4
        step += 2
        result += ard+ald+alu+aru
    return result

其实我们令右上角aru=n²,则alu=n²-n+1, ald=n²-2n+2, ard=n²-3n+3, n取3到1001, 步长为2
answer is 669171001