The Ulam Spiral: Euler Problem 28

Euler Problem 28 takes us to the world of the Ulam Spiral. This is a spiral that contains sequential positive integers in a square spiral, marking the prime numbers. Stanislaw Ulam discovered that a lot of primes are located along the diagonals. These diagonals can be described as polynomials. The Ulam Spiral is thus a way of generating quadratic primes (Euler Problem 27).

Euler Problem 28: The Ulam Spiral

Ulam Spiral (WikiMedia).

Euler Problem 28 Definition

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 07 08 09 10
19 06 01 02 11
18 05 04 03 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?

Proposed Solution

To solve this problem we do not need to create a matrix. This code calculates the values of the corners of a matrix with size n. The lowest number in the matrix with size n is n(n-3)+4. The numbers increase by n-1.

The code steps through all matrices from size 3 to 1001. The solution uses only the uneven sized matrices because these have a centre. The answer to the problem is the sum of all numbers.

size <- 1001 # Size of matrix
answer <- 1 # Starting number
# Define corners of subsequent matrices
for (n in seq(from = 3, to = size, by = 2)) {
    corners <- seq(from = n * (n - 3) + 3, by = n - 1, length.out = 4)
    answer <- answer + sum(corners)
}
print(answer)

Plotting the Ulam Spiral

We can go beyond Euler Problem 28 and play with the mathematics. This code snippet plots all the prime numbers in the Ulam Spiral. Watch the video for an explanation of the patterns that appear along the diagonals.

Ulam Spiral prime numbers

Ulam Spiral prime numbers.

The code creates a matrix of the required size and fills it with the Ulam Spiral. The code then identifies all primes using the is.prime function from Euler Problem 7. A heat map visualises the results.

# Ulam Spiral
size <- 201 # Size of matrix
ulam <- matrix(ncol = size, nrow = size)
mid <- floor(size / 2 + 1)
ulam[mid, mid] <- 1
for (n in seq(from = 3, to = size, by = 2)) {
    numbers <- (n * (n - 4) + 5) : ((n + 2) * ((n + 2) - 4) + 4)
    d <- mid - floor(n / 2)
    l <- length(numbers)
    ulam[d, d:(d + n - 1)] <- numbers[(l - n + 1):l]
    ulam[d + n - 1, (d + n - 1):d] <- numbers[(n - 1):(n - 2 + n)]
    ulam[(d + 1):(d + n - 2), d] <- numbers[(l - n):(l - 2 * n + 3)]
    ulam[(d + 1):(d + n - 2), d + n - 1] <- numbers[1:(n - 2)]
}
ulam.primes <- apply(ulam, c(1, 2), is.prime)

# Visualise
library(ggplot2)
library(reshape2)
ulam.primes <- melt(ulam.primes)
ggplot(ulam.primes, aes(x=Var1, y=Var2, fill=value)) + geom_tile() + 
    scale_fill_manual(values = c("white", "black")) + 
    guides(fill=FALSE) +
    theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          panel.background = element_blank()) + 
    theme(axis.title.x=element_blank(),
          axis.text.x=element_blank(),
          axis.ticks.x=element_blank(),
          axis.title.y=element_blank(),
          axis.text.y=element_blank(),
          axis.ticks.y=element_blank()
          )
ggsave("ulamspiral.png")

Euler Problem 11: Largest Product in a Grid

Euler Problem 11 Definition

In the 20×20 grid below, four numbers along a diagonal line have been marked in red.

Euler problem 11

The product of these numbers is 26 × 63 × 78 × 14 = 1,788,696. What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20 by 20 grid?

Solution

The solution applies straightforward vector arithmetic. The product of all verticals is an array of the product of rows 1 to 4, rows 2 to 5 and so on. The code uses a similar logic for the horizontals and the diagonals.

#Read and convert data
square <- readLines("Euler/p011_matrix.txt")
square <- as.numeric(unlist(lapply(square, function(x){strsplit(x, " ")})))
square <- matrix(square, ncol=20)

# Define products
prod.vert <- square[1:17, ] * square[2:18, ] * square[3:19, ] * square[4:20, ]
prod.hori <- square[,1:17] * square[,2:18] * square[,3:19] * square[,4:20]
prod.dia1 <- square[1:17, 1:17] * square[2:18, 2:18] * square[3:19, 3:19] * square[4:20, 4:20]
prod.dia2 <- square[4:20, 1:17] * square[3:19, 2:18] * square[2:18, 3:19] * square[1:17, 4:20]

answer <- max(prod.vert, prod.hori, prod.dia1, prod.dia2)