The Viral Recurring Decimal: Euler Problem 26

A few years ago a fraction broke the internet. What happens when you divide 1 by 998001?

\frac{1}{998001} = 0.000001002003004005006007008009010011012013014015 \ldots

What is special about this fraction is that it lists every three-decimal number except for 998. Look carefully at the sequence to see that is 000, 001, 0002, 003, 004, 005 and so on. After it has reached 999, the sequence continues from the start. This fraction thus has 2997 recurring decimals. James Grime from Numberphile explains this mathematical oddity with his usual enthusiasm.

The decimal fraction of 1/998001 is a recurring decimal. These are decimal numbers with periodic digits (repeating its values at regular intervals). Euler problem 26 asks us to analyse recurring decimals (reciprocal cycles).

Euler Problem 26 Definition

A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:

1/2 = 0.5
1/3 = 0.(3)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1(6)
1/7 = 0.(142857)
1/8 = 0.125
1/9 = 0.(1)
1/10 = 0.1

Where 0.1(6) means 0.166666…, and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle. Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part.


A051626 describes the length of the recurring numbers in 1/n in the On-Line Encyclopaedia of Integer Sequences. To solve Euler Problem 26, we need to generate the first 1000 numbers of this sequence and find out which number has the longest recurring cycle.

R can only display up to 22 decimals by using options(digits=22). The base R capability is unsuitable for solving this problem, so I wrote some code to perform long division the old-fashioned way.

The recur function divides 1 by any arbitrary integer. The code continues until the decimal terminates, for example 1/4 = 0.25, or when a recurring pattern emerges, e.g. 1/7 = 0.(142857).

The function has two arguments: n is the input number. The output argument determines the outcome of the function: “len” for the length of the recurring decimals. Any other value shows the result of the calculation. The output of the function is a string. Using the European notation, the recurring part of the decimals is shown between brackets, e.g. 1/14 = 0.0(714285).

recur <- function(x, output = "") {
    # Prepare variable
    if (x == 0) return(NaN)
    if (x == 1) return(0)
    x <- floor(abs(x))
    # Initiate vectors to store decimals and remainders
    dec <- vector()
    rem <- vector()
    # Initiate values
    i <- 1
    r <- 10
    rem <- r
    # Long division
    repeat {
        dec[i] <- floor(r / x)
        r <- 10 * (r %% x)
        # Test wether the number is terminating or repeating
        if (r == 0 | r %in% rem) break
        rem[i + 1] <- r
        i <- i + 1 
    # Determine number of recurring digits
    rep <- ifelse(r != 0, length(rem) - which(r == rem) + 1, 0)
    # Output
    if (output == "len")
    else {
        if (rep != 0) {
            if (rep == length(dec)) 
                l <- "("
                l <- c(dec[1:(length(dec) - rep)], "(")
            dec <- c(l, dec[(length(dec) - rep + 1):length(dec)], ")")
        return(paste0("0.", paste0(dec, collapse = "", sep = "")))

A051626 <- sapply(1:1000, recur, "len")
answer <- which.max(A051626)


Using the iGraph package to Analyse the Enron Corpus

This blog post analyzes an extensive collection of e-mails from former Enron employees. The Enron corpus is analysed using network analysis tools provided by the iGraph package. Network analysis is a versatile technique that can be used to add value to a lot of different data sets, including the complex corporate relationships of Donald Trump.

The Enron scandal is one of the most famous corporate governance failures in the history of capitalism. The case itself is interesting for its sake, but in this post, I am more interested in one of the data sets that the subsequent investigation has provided.

The Enron Corpus

As part of their inquiries, The Federal Energy Regulatory Commission used an extensive collection of emails from Enron employees. The Enron corpus is one of the only publicly available collections of emails available for research. This dataset also provides a fascinating playground for citizen data scientists.

The set has privacy issues as it contains messages from living people. When analysing this data set, we need to keep in mind that the majority of former Enron employees were innocent people who lost their jobs due to the greed of their overlords. The code in this post only analyses the e-mail headers, ignoring the content.

Laid-off Enron employees outside Enron headquarters as the company collapsed in 2001 - Enron corpus analysis

Laid-off Enron employees outside Enron headquarters as the company collapsed in 2001.

The Enron Corpus is a large database of half a million of emails generated by more than 100 Enron employees. You can download the corpus from the Carnegie Mellon School of Computer Science. The first code snippet downloads the 7 May 2015 version of the dataset (about 423Mb, tarred and gzipped) and untars it to your working directory.

# Enron Email Dataset:
download.file("", destfile = "enron_mail_20150507.tgz")

Preparing the Data

The main folder is maildir, which holds all the personal accounts. Our first task is to load the required libraries and create a list of available emails. This code results in 517,401 e-mail files with 44,859 emails in the inboxes of users.

# E-mail corpus consists of nested folders per user with e-mails as text files
# Create list of all available e-mails
emails <- list.files("maildir/", full.names = T, recursive = T)
# Filter by inbox only
emails <- emails[grep("/inbox", emails)]

The bulk of the code creates a list of emails between Enron employees. The code performs a lot of string manipulations to extract the information from the text files. The content of the e-mails is ignored, the code only retrieves the sender and the receiver. The analysis is limited to e-mails between employees in the corpus. Only those receivers whose inbox forms part of the analysis are included. The result of this code is a data frame with the usernames of the sender and receiver for each email. The data frame contains 2779 emails that meet the criteria.

# Create list of sender and receiver (inbox owner)
inboxes <- data.frame(
    from = apply(, 1, function(x){readLines(x, warn = F)[3]}),
    to = emails,
    stringsAsFactors = F

# Keep only and strip all but username
library(stringr) # String manipulation
inboxes <- inboxes[grepl("", inboxes$from),]
inboxes$from <- str_sub(inboxes$from, 7, nchar(inboxes$from) - 10)
to <- str_split(inboxes$to, "/")
inboxes$to <- sapply(to, "[", 3)

# Create list of usernames
users <- data.frame(user = paste0("maildir/", unique(inboxes$to)))

# Remove those without sent mails
sent <- apply(users, 1, function(x){sum(grepl("sent", dir(x)))})
users <- subset(users, sent != 0)

# Replace username with e-mail name
users$mailname <- NA
for (i in 1:nrow(users)){
sentmail <- dir(paste0(users$user[i], "/sent_items/"))
name <- readLines(paste0(users$user[i], "/sent_items/", sentmail[1]), warn = F)[3]
name <- str_sub(name, 7, nchar(name)-10)
users$mailname[i] <- name
users$user <- str_sub(users$user, 9)
inboxes <- merge(inboxes, by.x="to", users, by.y="user")
inboxes <- data.frame(from = inboxes$from, to = inboxes$mailname)

inboxes$from <- as.character(inboxes$from)
inboxes$to <- as.character(inboxes$to)

# Only e-mails between inbox users
inboxes <- inboxes[inboxes$from %in% inboxes$to,]

# Remove no.address
inboxes <- subset(inboxes, from != "no.address" & to != "no.address")

# Remove emails to self
inboxes <- subset(inboxes, inboxes$from != inboxes$to)

Network Analysis

The last code snippet defines a graph from the table of e-mails. Each employee is a node in the network, and each e-mail is an edge (line). The iGraph package is a powerful tool to analyse networks. The graph_from_edgelist function creates a network object that can be analysed using the iGraph package. The graph is directed because the information flows from the sender to the receiver.

In the next step, the Spingglass algorithm finds community structure within the data. A community is a group of nodes that are more connected with each other than with any other node.

The last step visualises the network. The diagram is drawn using the Fruchterman-Reingold algorithm, which places the most connected nodes at the centre of the picture. The background colours in the diagram indicate the eight communities.

The graph tells us a lot about the group of employees in the Enron corpus and how they relate to each other. Each of the communities represents a tightly connected group of employees that mainly e-mail each other. Any connections between communities are shown in red. When the vertex.label = NA line is removed, the usernames are displayed at each node.

We can see groups that never mail each other, lonely hangers-on and tightly knit cliques within Enron. In the centre fo the graph we see w few individuals who are connectors between groups because they send a lot of emails to people outside their community. On the fringes of the graph are the hangers-on who only once or twice emailed somebody in the corpus but were still included in the investigation.

g <- graph_from_edgelist(as.matrix(inboxes), directed = T)
coms <-

# Plot network
par(mar = c(0,0,2,0))
plot(coms, g, 
     layout = layout.fruchterman.reingold,
     vertex.size = 3

This analysis provides only a quick glimpse into the knowledge contained in the Enron corpus. An extensive range of tools is available to analyse such networks. An interesting exercise would be to overlap this network with the organisation chart to see the relationships between teams. Have fun playing with this fantastic data set!

Enron corpus network with communities.


Euler Problem 29: Distinct Powers

Euler Problem 29 is another permutation problem that is quite easy to solve using brute force. The MathBlog site by Kristian Edlund has a nice solution using only pen and paper.

Raising number to a power can have interesting results. The video below explains why this pandigital formula approximates e to billions of decimals:

(1 + 9^{-4^{6 \times 7}})^{3^{2^{85}}} \approx e

Euler Problem 29 Definition

Consider all integer combinations of: a^b for 2 \leq a \leq 5 and \leq b \leq 5 .

2^2=4, \quad 2^3 = 8,\quad 2^4 = 16,\quad 2^5 = 32

3^2 = 9,\quad 3^3 = 27,\quad 3^4 = 81,\quad 3^5 = 243

4^2 = 16,\quad 4^3 = 64,\quad 4^4 = 256, \quad 4^5 = 1024

5^2 = 25,\quad 5^3 = 125,\quad 5^4 = 625,\quad 5^5 = 3125

If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:

4, \ 8, \ 9, \ 16, \ 25, \ 27, \ 32, \ 64, \ 81, \ 125, \ 243, \ 256,\ 625, \ 1024, \ 3125

How many distinct terms are in the sequence generated by a^b for 2 \leq a \leq 100 and 2 \leq b \leq 100 ?

Brute Force Solution

This code simply calculates all powers from 2^2 to 2^{1000} and determines the number of unique values. Since we are only interested in their uniqueness and not the precise value, there is no need to use Multiple Precision Arithmetic.

# Initialisation
target <- 100
terms <- vector()
i <- 1
# Loop through values of a and b and store powers in vector
for (a in 2:target) {
   for (b in 2:target) {
     terms[i] <- a^b
     i <- i + 1
# Determine the number of distinct powers
answer <- length(unique(terms))

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

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 function from Euler Problem 7. A heat map visualises the results.

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

# Visualise
ulam.primes &lt;- melt(ulam.primes)
ggplot(ulam.primes, aes(x=Var1, y=Var2, fill=value)) + geom_tile() + 
    scale_fill_manual(values = c(&quot;white&quot;, &quot;black&quot;)) + 
    guides(fill=FALSE) +
    theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          panel.background = element_blank()) + 

Large integers in R: Fibonacci number with 1000 digits, Euler Problem 25

The Fibonacci Sequence in nature: The nautilus shell. Euler problem 25

The Fibonacci Sequence occurs in nature: The nautilus shell.

Euler Problem 25 takes us back to the Fibonacci sequence and the problems related to working with very large integers.

The Fibonacci sequence follows a simple mathematical rule but it can create things of great beauty. This pattern occurs quite often in nature, like to nautilus shell shown in the image. The video by Arthur Benjamin at the end of this post illustrates some of the magic of this sequence.

Large Integers in R

By default, numbers with more than 7 digits are shown in scientific notation in R, which reduces the accuracy of the calculation. You can change the precision of large integers with the options function but R struggles with integers with more than 22 digits. This example illustrates this issue.

[1] 6.204484e+23
> options(digits=22)
> factorial(24)
[1] 620448401733239409999872

However, finding a number of 1000 digits is a problem with using special functions.

Euler Problem 25 Definition

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_{n-1} + F_{n-2} , where F_1 = 1 and F_2 = 1 . The 12th term, F_{12} , is the first term to contain three digits.

What is the index of the first term in the Fibonacci sequence to contain 1000 digits?

Proposed Solutions

The first solution uses the GMP library to manage very large integers. This library also contains a function to generate Fibonacci numbers. This solution cycles through the Fibonacci sequence until it finds a number with 1000 digits.

library(gmp) # GNU Multiple Precision Arithmetic Library
n <- 1
fib <- 1
while (nchar(as.character(fib)) < 1000) {
   fib <- fibnum(n) # Determine next Fibonacci number
   n <- n + 1
answer <- n

This is a very fast solution but my aim is to solve the first 100 Project Euler problems using only base-R code. The big.add function I developed to solve Euler Problem 13.

t <- proc.time()
fib <- 1 # First Fibonaci number
cur <- 1 # Current number in sequence
pre <- 1 # Previous number in sequence
index <- 2
while (nchar(fib) < 1000) {
    fib <- big.add(cur, pre) # Determine next Fibonacci number
    pre <- cur
    cur <- fib
    index <- index + 1
answer <- index

This code is much slower than the GMP library but is was fun to develop.

The Magic of the Fibonacci Numbers

Lexicographic Permutations: Euler Problem 24

Euler Problem 24 asks to develop lexicographic permutations which are ordered arrangements of objects in lexicographic order. Tushar Roy of Coding Made Simple has shared a great introduction on how to generate lexicographic permutations.

Euler Problem 24 Definition

A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:

012 021 102 120 201 210

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Brute Force Solution

The digits 0 to 9 have 10! = 3628800 permutations (including combinations that start with 0). Most of these permutations are, however, not in lexicographic order. A brute-force way to solve the problem is to determine the next lexicographic permutation of a number string and repeat this one million times.

nextPerm <- function(a) {
    # Find longest non-increasing suffix
    i <- length(a) while (i > 1 && a[i - 1] >= a[i])
        i <- i - 1
    # i is the head index of the suffix
    # Are we at the last permutation?
    if (i <= 1) return (NA)
    # a[i - 1] is the pivot
    # Find rightmost element that exceeds the pivot
    j <- length(a)
    while (a[j] <= a[i - 1]) 
        j <- j - 1
    # Swap pivot with j
    temp <- a[i - 1]
    a[i - 1] <- a[j]
    a[j] <- temp
    # Reverse the suffix
    a[i:length(a)] <- rev(a[i:length(a)])

numbers <- 0:9
for (i in 1:(1E6 - 1)) numbers <- nextPerm(numbers)
answer <- numbers

This code takes the following steps:

  1. Find largest index i such that a_{i-1} < a_i.
    1. If no such index exists, then this is already the last permutation.
  2. Find largest index j such that j \geq i and a_j > a_{i-1}.
  3. Swap a_j and a_{i-1}.
  4. Reverse the suffix starting at a_i.


A more efficient solution is to use combinatorics, thanks to MathBlog. The last nine digits can be ordered in 9! = 362880 ways. So the first 9! permutations start with a 0. By extending this thought, it follows that the millionth permutation must start with a 2.

\lfloor (1000000 - 1) / 9! \rfloor  = 2

From this rule, it follows that the 725761st permutation is 2013456789. We now need 274239 more lexicographic permutations:

(1000000 - 1) - (2 \times 9!) = 274239

We can repeat this logic to find the next digit. The last 8 digits can be ordered in 40320 ways. The second digit is the 6th digit in the remaining numbers, which is 7 (2013456789).

\lfloor 274239 / 8! \rfloor  = 6

274239 - (6 \times 7!) = 32319

This process is repeated until all digits have been used.

numbers <- 0:9
n <- length(numbers)
answer <- vector(length = 10)
remain <- 1E6 - 1
for (i in 1:n) {
    j <- floor(remain / factorial(n - i))
    answer[i] <- numbers[j + 1]
    remain <- remain %% factorial(n - i)
    numbers <- numbers[-(j + 1)]
answer <- paste(answer, collapse = "")

R blogger Tony’s Bubble Universe created a generalised function to solve this problem a few years ago.

Tic Tac Toe Part 3: The Minimax Algorithm

Tic Tac Toe minimax alogrithmIn two previous posts, I presented code to teach R to play the trivial game of Tic Tac Toe. I thought this was an unbeatable algorithm. Alas, a comment from Alberto shattered my pride as he was able to beat my code.

The reason for the demise of my code was that I didn’t implement a full minimax algorithm, but instead looked only two moves ahead. I thought the common strategy rules (start in the centre and occupy the corners) would make the program unbeatable. When I simulated the game by instructing the computer to play against itself, Alberto’s strategy never arose because the code forces the centre field. Alberto’s code shows that you need to look at least three moves ahead for a perfect game. He has been so kind to share his code and gave me permission to publish it.

Alberto recreated two functions, for completeness I have added the complete working code that merges his improvements with my earlier work. The first two functions are identical to the previous post. These functions draw the game board and process the human player’s move by waiting for a mouse click.

# Draw the game board
draw.board <- function(game) {
    xo <- c("X", " ", "O") # Symbols
    par(mar = rep(1,4))
    plot.window(xlim = c(0,30), ylim = c(0,30))
    abline(h = c(10, 20), col="darkgrey", lwd = 4)
    abline(v = c(10, 20), col="darkgrey", lwd = 4)
    text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), xo[game + 2], cex = 4)
    # Identify location of any three in a row
    square <- t(matrix(game, nrow = 3))
    hor <- abs(rowSums(square))
    if (any(hor == 3)) 
        hor <- (4 - which(hor == 3)) * 10 - 5 
        hor <- 0
    ver <- abs(colSums(square))
    if (any(ver == 3)) 
        ver <- which(ver == 3) * 10 - 5 
        ver <- 0
    diag1 <- sum(diag(square))
    diag2 <- sum(diag(t(apply(square, 2, rev)))) 
    # Draw winning lines
    if (all(hor > 0)) 
        for (i in hor) lines(c(0, 30), rep(i, 2), lwd = 10, col="red")
    if (all(ver > 0)) 
        for (i in ver) lines(rep(i, 2), c(0, 30), lwd = 10, col="red")
    if (abs(diag1) == 3) 
        lines(c(2, 28), c(28, 2), lwd = 10, col = "red")
    if (abs(diag2) == 3) 
        lines(c(2, 28), c(2, 28), lwd = 10, col = "red")

# Human player enters a move
move.human <- function(game) {
    text(4, 0, "Click on screen to move", col = "grey", cex=.7)
    empty <- which(game == 0)
    move <- 0
    while (!move %in% empty) {
        coords <- locator(n = 1) # add lines
        coords$x <- floor(abs(coords$x) / 10) + 1
        coords$y <- floor(abs(coords$y) / 10) + 1
        move <- coords$x + 3 * (3 - coords$y)
    return (move)

Alberto rewrote the functions that analyse the board and determine the move of the computer. The ganador (Spanish for winning) function assesses the board condition by assigning -10 or + 10 for a winning game and 0 for any other situation.

ganador <- function(juego, player) {
    game <- matrix(juego, nrow = 3, byrow = T)
    hor <- rowSums(game)
    ver <- colSums(game)
    diag <- c(sum(diag(game)), sum(diag(apply(game, 1, rev))))
    if (-3 %in% c(hor, ver, diag))
    if (3 %in% c(hor, ver, diag))

The next function is the actual minimax algorithm. If the computer starts then the first move (9!= 362880 options to assess) takes a little while. The commented lines can be used to force a corner and make the games faster by forcing a random corner.

The minimax function returns a list with the move and its valuation through the ganador function. The function works recursively until it has filled the board and retains the best scoring move using the minimax method. To avoid the computer always playing the same move in the same situation random variables are added.

minimax <- function(juego, player) {
    free <- which(juego == 0)
    if(length(free) == 1) {
        juego[free] <- player
        return(list(move = free, U = ganador(juego, player)))
    poss.results <- rep(0, 9)
    for(i in free) {
        game <- juego
        game[i] <- player
        poss.results[i] <- ganador(game, player)
    mm <- ifelse(player == -1, "which.min", "which.max")
    if(any(poss.results == (player * 10))) {
        move <-, list(poss.results))
        return(list(move = move, U = poss.results[move]))
    for(i in free) {
        game <- juego
        game[i] <- player
        poss.results[i] <- minimax(game, -player)$U
    random <- runif(9, 0, 0.1)
    poss.results[-free] <- 100 * -player
    poss.results <- poss.results + (player * random)
    move <-, list(poss.results))
    return(list(move = move, U = poss.results[move]))

This final function stitches everything together and lets you play the game. Simply paste all functions in your R console and run them to play a game. The tic.tac.toe function can take two parameters, “human” and/or “computer”. The order of the parameters determines who starts the game.

# Main game engine
tic.tac.toe <- function(player1 = "human", player2 = "computer") {
    game <- rep(0, 9) # Empty board
    winner <- FALSE # Define winner
    player <- 1 # First player
    players <- c(player1, player2)
    while (0 %in% game & !winner) { # Keep playing until win or full board
        if (players[(player + 3) %% 3] == "human") # Human player
            move <- move.human(game)
        else { # Computer player
            move <- minimax(game, player)
            move <- move$move
        game[move] <- player # Change board
        winner <- max(, 1), abs(, -1))) == 6 # Winner, winner, chicken dinner?
        player <- -player # Change player


This is my last word on Tic Tac Toe but now that the minimax conundrum is solved I could start working on other similar games such as Connect Four, Draughts or even the royal game of Chess.

Euler Problem 23: Non-Abundant Sums

A demonstration of the abundance of the number 12 using Cuisenaire rods (Euler Problem 23)

A demonstration of the abundance of the number 12 using Cuisenaire rods (Wikipedia).

Euler problem 23 asks us to solve a problem with abundant or excessive numbers.

These are numbers for which the sum of its proper divisors is greater than the number itself.

12 is an abundant number because the sum of its proper divisors (the aliquot sum) is larger than 12: (1 + 2 + 3 + 4 + 6 = 16).

All highly composite numbers or anti-primes greater than six are abundant numbers. These are numbers that have so many divisors that they are considered the opposite of primes, as explained in the Numberphile video below.

Euler Problem 23 Definition

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis, even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.


This solution repurposes the divisors function that determines the proper divisors for a number, introduced for Euler Problem 21. The first code snippet creates the sequence of all abundant numbers up to 28123 (sequence A005101 in the OEIS). An abundant number is one where its aliquot sum is larger than n.

# Generate abundant numbers (OEIS A005101)
A005101 <- function(x){
    abundant <- vector()
    a <- 1
    for (n in 1:x) {
        aliquot.sum <- sum(proper.divisors(n)) - n 
        if (aliquot.sum > n) {
            abundant[a] <- n
            a <- a + 1

abundant <- A005101(28123)

The solution to this problem is also a sequence in the Online Encyclopedia of Integer Sequences (OEIS A048242). This page states that the highest number in this sequence is 20161, not 28123 as stated in the problem definition.

The second section of code creates a list of all potential numbers not the sum of two abundant numbers. The next bit of code sieves any sum of two abundant numbers from the list. The answer is determined by adding remaining numbers in the sequence.

# Create a list of potential numbers that are not the sum of two abundant numbers
A048242 <- 1:20161

# Remove any number that is the sum of two abundant numbers
for (i in 1:length(abundant)) {
    for (j in i:length(abundant)) {
        if (abundant[i] + abundant[j] <= 20161) {
            A048242[abundant[i] + abundant[j]] <- NA
A048242 <- A048242[!]
answer <- sum(A048242)

The Sierpinski Triangle: Visualising infinity in R

Sierpinski triangleWacław Sierpiński was a mathematical genius who developed several of the earliest fractals. The Sierpiński triangle is an easy to conceptualise geometrical figure but it hides a fascinating mathematical complexity. Start by drawing an equilateral triangle and draw another one in its centre. Then draw equilateral triangles in the four resulting triangles, and so on, ad infinitum.

The original Sierpinski triangle will eventually disappear into Cantor dust, a cloud of ever shrinking triangles of infinitesimal size. The triangle is self-similar, no matter how far you zoom in, the basic geometry remains the same.

The Chaos Game

A fascinating method to create a Sierpinski Triangle is a chaos game. This method uses random numbers and some simple arithmetic rules. Sierpinski Triangles can be created using the following six steps:

  1. Define three points in a plane to form a triangle.
  2. Randomly select any point on the plane.
  3. Randomly select any one of the three triangle points.
  4. Move half the distance from your current position to the selected vertex.
  5. Plot the current position.
  6. Repeat from step 3.

This fractal is an implementation of chaos theory as this random process attracts to a complex ordered geometry. The game only works with random numbers and when selecting random vertices of the triangle.

Sierpinski Triangle Code

This code implements the six rules in R. The code first initializes the triangle, defines a random starting point and then runs a loop to place random dots. The R plot engine does not draw pixels but uses characters, which implies that the diagram is not as accurate as it could be but the general principle is clear. The x(11) and Sys.sleep() commands are used to plot during the for-loop.

# Sierpinsky Triangle

# Initialise triangle
p <- c(0, 500, 1000)
q <- c(0, 1000, 0)
par(mar = rep(0, 4))
plot(p, q, col= "red", pch = 15, cex = 1, axes = FALSE)

# Random starting point
x <- sample(0:1000, 1)
y <- sample(0:1000, 1)

# Chaos game
for (i in 1:10000) {
    n <- sample(1:3, 1)
    x <- floor(x + (p[n] - x) / 2)
    y <- floor(y + (q[n] - y) / 2)
    points(x, y, pch = 15, cex = 0.5)

This algorithm demonstrates how a seemingly chaotic process can result in order. Many other versions of chaos games exist, which I leave to the reader to play with. If you create your own versions then please share the code in the comment box below.

Euler Problem 22 : Names Scores

Euler Problem 22

R logo in ASCII art by

Euler problem 22 is another trivial one that takes us to the realm of ASCII codes. ASCII is a method to convert symbols into numbers, originally invented for telegraphs.

Back in the 8-bit days, ASCII art was a method to create images without using lots of memory. Each image consists of a collection of text characters that give the illusion of an image. Euler problem 22 is, unfortunately, a bit less poetic.

Euler Problem 22 Definition

Using names.txt, a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.

For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 × 53 = 49,714.

What is the total of all the name scores in the file?


This code reads and cleans the file and sorts the names alphabetically. The charToRaw function determines the numerical value of each character in each name. This output of this function is the hex ASCII code for each character. The letter A is number 65, so we subtract 64 from each value to get the sum total.

# ETL: reads the file and converts it to an ordered vector.
names <- readLines("", warn = F)
names <- unlist(strsplit(names, ","))
names <- gsub("[[:punct:]]", "", names)
names <- sort(names)

# Total Name scores
answer <- 0
for (i in names) {
    value <- sum(sapply(unlist(strsplit(i, "")), function(x) as.numeric(charToRaw(x)) - 64))
    value <- value * which(names==i)
    answer <- answer + value

We can have a bit more fun with this problem by comparing this list with the most popular baby names in 2016. The first section of the code extracts the list of popular names from the website. The rest of the code counts the number of matches between the lists.

# Most popular baby names
url <- ""
babynames <- url %>%
    read_html() %>%
    html_nodes(xpath = '//*[@id="babyNameList"]/table') %>%
babynames <- babynames[[1]]

# Convert Project Euler list and test for matches
proper=function(x) paste0(toupper(substr(x, 1, 1)), tolower(substring(x, 2)))
names <- proper(names)

sum(babynames$GIRLS %in% names)
sum(babynames$BOYS %in% names)