# 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


# 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)])
return(a)
}

numbers <- 0:9
for (i in 1:(1E6 - 1)) numbers <- nextPerm(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 $a_i$.

## Combinatorics

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)
remain <- 1E6 - 1
for (i in 1:n) {
j <- floor(remain / factorial(n - i))