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 the numbers on the diagonals is 101. What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?
There is a pattern of the numbers on the diagonals if I represent the spiral as a straight vector:
1
2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
The step of each row is 2, 4, 6…, and the number of row is (n-1)/2. Using this information we can contruct the index of the diagonals of the spiral 1:(n*n).
sum_spiral_diag <- function(n) {
stopifnot(n %% 2 != 0)
if (n > 1e5) stop("n too big")
s <- 1:(n*n)
i <- seq(2, length.out = (n-1)/2, by = 2)
i <- rep(i, each = 4)
i <- cumsum(c(1, i))
sum(as.numeric(s[i]))
}
sum_spiral_diag(1001)## [1] 669171001