Project Euler problem 44

Problem 44:

Pentagonal numbers are generated by the formula, P_n=n(3n−1)/2. The first ten pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, …

It can be seen that P₄ + P₇ = 22 + 70 = 92 = P₈. However, their difference, 70 − 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, P_j and P_k, for which their sum and difference is pentagonal and D = |P_k − P_j| is minimised; what is the value of D?

I’d originally made a start on this in Python, then gave up on it. Came back to it much later and tried doing it in ColdFusion. My solution there took too long to run, so I went back to my Python version and finished it off.

Here’s the code:

import math

def pentagonal(x):
	return x * (3 * x - 1) / 2

def isPentagonal(x):
        n = (math.sqrt((24 * x) + 1) + 1) / 6

        if n == int(n) and n > 0:
                return 1
        else:
                return 0

        
i = 0
pentagonals = []
solution = 0

while 1:
        i += 1
        p = pentagonal(i)

        for j in pentagonals:
                diff = abs(p - j)
                
                if isPentagonal(diff) == 1:
                        intSum = p + j
                        
                        if isPentagonal(intSum) == 1:
                            print("solution:", int(diff))
                            solution = 1
                            break

        if solution:
                break

        pentagonals.append(p)

The formula for calculating the pentagonal numbers is given to us. I already had a formula for the reverse, checking if a number is pentagonal, from problem 45.
To calculate the square root, I needed to import the math library.

Basically I loop until I find a solution. I calculate the pentagonal value of i. Then I subtract each pentagonal number smaller from it (but use the abs() function to get the absolute value).
If that difference is pentagonal, then I do a sum as well.
If that is also pentagonal, then our solution is the difference.
As I go along I append each pentagonal number into the array.
This solution runs in about 12 seconds for me.

Project Euler problem 55

Problem 55:

If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.

Not all numbers produce palindromes so quickly. For example,

349 + 943 = 1292,
1292 + 2921 = 4213
4213 + 3124 = 7337

That is, 349 took three iterations to arrive at a palindrome.

Although no one has proved it yet, it is thought that some numbers, like 196, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: 4668731596684224866951378664 (53 iterations, 28-digits).

Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is 4994.

How many Lychrel numbers are there below ten-thousand?

Yet another problem which was easy to code in ColdFusion, but it couldn’t deal with the large numbers generated. So rewrote it in Python.

Just for interest, here’s what I did in CFML. NB: there’s a logic error here, which I fixed once I moved to Python:

<cfscript>
function isPalindrome(x)
{
	if (x EQ Reverse(x))
		return true;
	else
		return false;
}
</cfscript>

<cfset Lychrel = ArrayNew(1)>

<cfloop index="i" from="1" to="9999">
	<cfset k = i>
	
	<cfloop index="j" from="1" to="50">
		
		<cfset temp = k + Reverse(k)>
		
		<cfif IsPalindrome(temp)>
			<cfoutput>#k# + #Reverse(k)# = #temp#</cfoutput><br>
			<cfset ArrayAppend(Lychrel, i)>
			<cfbreak>
		</cfif>
		
		<cfset k = temp>
	</cfloop>	
</cfloop>

<cfoutput><strong>#ArrayLen(Lychrel)#</strong></cfoutput>

Running this it quickly throws an error:
‘The value 210+E11200002108.1 cannot be converted to a number.’

Python doesn’t seem to have a built-in Reverse function like ColdFusion, so you have to write your own.

def Reverse(x):
        num = str(x)
        newnum = ''
        
        for i in range(len(num)-1, 0, -1):
                newnum += num[i]

        newnum += num[0]
        
        return newnum


Lychrel = []
limit = 10000

for i in range(1, limit):
	k = i

	for j in range(50):		
		temp = k + int(Reverse(k))

		if str(temp) == str(Reverse(temp)):
			break
		
		k = temp

	if j == 49:
	      Lychrel.append(i)
              
print(len(Lychrel))

This code is pretty fast, but it could be better. For instance I’m casting everything to a string or an int as required, but that could maybe be tidied up somehow.
Also in my Reverse function, I somehow couldn’t work out how to properly loop backwards through the string, so ended up having to add on the first character outside of the loop.

Initially I was appending to my Lychrel array at the point where I break out of the loop. However on reading the question again I realised the Lychrel numbers are the numbers which aren’t solved within 50 iterations, not the other way around. I don’t even need an array, I could just have a counter that I increment. But having it in the array gives us the advantage of being able to output the array contents to make sure we’re on the right track.

Project Euler problem 56

Problem 56:

A googol (10¹⁰⁰) is a massive number: one followed by one-hundred zeros; 100¹⁰⁰ is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1.

Considering natural numbers of the form, a^b, where a, b < 100, what is the maximum digital sum?

So, let’s do nested loops, both going 1 to 100. Inside the inner loop, calculate a^b (or i^j in this case). Then loop through the digits of that value, adding them up. Keep track of which is the largest value this produces.

Running time about 3 seconds in Python:

import time
tStart = time.time()

maxsum = 0

for i in range(100):
    for j in range(100):
        num = i ** j
        total = 0

        for k in str(num):
            total += int(k)

        if total > maxsum:
            maxsum = total

print(maxsum)
print("time:" + str(time.time() - tStart))        

Project Euler problem 53

Problem 53:

There are exactly ten ways of selecting three from five, 12345:

123, 124, 125, 134, 135, 145, 234, 235, 245, and 345

In combinatorics, we use the notation, ⁵C₃ = 10.

In general,
ⁿC_r = n! / (r!(n-r)!)
where r <= n, n! = n*(n-1)*…*3*2*1, and 0! = 1.

It is not until n = 23, that a value exceeds one-million: ²³C₁₀ = 1144066.

How many, not necessarily distinct, values of ⁿC_r, for 1 <= n <= 100, are greater than one-million?

At first I looked at this and thought it was all about iterating through the combinations of digits, just like how it lists 123, 124, etc. Which sounds tricky. But it’s much simpler than that. In fact they give you the hardest part of the algorithm, the equation needed:
n! / (r!(n-r)!)

In Python this solution takes about 0.36 seconds. I’m going to start using the time module to calculate running times.

import time
tStart = time.time()

values = 0

def factorial(x):
    y = 1
    for i in range(1,x+1):
        y = y * i
        
    return y

def calc(n, r):
    return  factorial(n) / (factorial(r) * factorial(n-r))

for i in range(1,101):
    for j in range(1, i):
        if calc(i, j) > 1000000:
            values += 1
           

print(values)
print("time:" + str(time.time() - tStart))

So, two functions, one calculates the factorial of x, the other just calculates our equation. The second function could instead have just been expressed as one line in our procedural code.

Then loop from 1 to 100. Within that, do an inner loop from 1 to whatever our current outer loop is on, passing the two loop counters into our function. We don’t really care which numbers produce values greater than a million; we only care how many numbers there are. So value is just a simple counter.

Project Euler problem 40

Problem 40:

An irrational decimal fraction is created by concatenating the positive integers:

0.123456789101112131415161718192021…

It can be seen that the 12^th digit of the fractional part is 1.

If d_n represents the n^th digit of the fractional part, find the value of the following expression.

d₁ * d₁₀ * d₁₀₀ * d₁₀₀₀ * d₁₀₀₀₀ * d₁₀₀₀₀₀ * d_1000000

Firstly I completed this in Coldfusion. I got the correct answer by brute-forcing it, but the solution was slow. This took about 11 minutes on a CF 5 server; it would undoubtedly be a lot quicker on any CFMX server. I then rewrote it in Python, which spits out the correct answer within a couple of seconds.

I thought it might be interesting to do a code comparison between the two languages.
CFML:

<cfset fraction = "">
<cfset total = 1>

<cfloop index="i" from="1" to="1000000">
<!--- we're going to break out of the loop sooner than that --->
	<cfset fraction = fraction & i>
	
	<cfif Len(fraction) GTE 1000000>
		<cfbreak>
	</cfif>
</cfloop>

<cfset total = total *	Mid(fraction, 1, 1) * 
	Mid(fraction, 10, 1) * 
	Mid(fraction, 100, 1) * 
	Mid(fraction, 1000, 1) * 
	Mid(fraction, 10000, 1) * 
	Mid(fraction, 100000, 1) * 
	Mid(fraction, 1000000, 1)>

<cfoutput>#total#</cfoutput>

Python:

fraction = ""
total = 1

for i in range(1,1000000):
    fraction = fraction + str(i)

    if len(fraction) >= 1000000:
        break

total = total * \
        int(fraction[0]) * \
        int(fraction[9]) * \
        int(fraction[99]) * \
        int(fraction[999]) * \
        int(fraction[9999]) * \
        int(fraction[99999]) * \
        int(fraction[999999])

print(total)

Almost identical code. I’m still not anywhere near writing pythonic code, but it’s early days.

Basically I loop from 1 to 1000000 (not really that far, because I break out of the loop once our string has length of 1000000). As I go I append each loop counter to the string.

Then I just multiply together the 1st, 10th, 100th etc values from the string. That part could also have been done in a loop to simplify the code slightly. The whole thing’s not particularly clever, but it works.

If you’re coming to this from a Coldfusion background, you’re maybe wondering why I’m doing a DIV between each of the values before multiplying them. Because whitespace is important in Python, the \ can be used to join two lines together where the line length gets too long.

Project Euler problem 30

Problem 30:

Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:

1634 = 1⁴ + 6⁴ + 3⁴ + 4⁴
8208 = 8⁴ + 2⁴ + 0⁴ + 8⁴
9474 = 9⁴ + 4⁴ + 7⁴ + 4⁴

As 1 = 1⁴ is not a sum it is not included.

The sum of these numbers is 1634 + 8208 + 9474 = 19316.

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Another problem where I’d worked out the logic in Coldfusion, but had to rewrite it in Python to get the solution.

We’re going to loop through numbers testing them out. We know we can skip 1. But how far do we need to loop up to? Let’s look at 4 digit numbers first; the lowest 4-digit number is 1000.  1^5 + 0^5 + 0^5 + 0^5 =1.  The highest 4-digit number is 9999.  9^5 + 9^5 + 9^5 + 9^5 = 236196.  Somewhere in between there is scope for the 5th powers of a 4 digit number to add up to a 4 digit number.

What about 5 digit numbers?  The highest sum of the 5th powers is 295,245. So all 5 digit numbers fall within that range.

And with 6 digit numbers?  The highest sum of the 5th powers is 354294.  So we can cover 6 digit numbers up to that far and no further.

With 7 digit numbers, the highest value is 413343, i.e. the highest sum of the 5th powers of a 7 digit number only adds up to a 6 digit number.  So we only need to loop from 2 to 354294.

powers = []
grandtotal = 0

for i in range(2,354294):
        total = 0
        for j in str(i):
                total += int(j) ** 5

        if total == i:
                powers.append(i)

for i in powers:
        grandtotal += i

print("powers:", powers)
print("total:", grandtotal)

So we loop through our numbers. On each number, loop through the digits, adding up the values of each digit to the power of 5. If the total of those values = our number, then store that number in a list.
At the end of the loop, add up all the values in the list to give the solution to the problem.

Project Euler: problem 48

Problem 48:

The series, 1¹ + 2² + 3³ + … + 10¹⁰ = 10405071317.

Find the last ten digits of the series, 1¹ + 2² + 3³ + … + 1000¹⁰⁰⁰.

Trying to do this one in CFML, on a Coldfusion 5 server, I get #INF, i.e. the CF server decides the number is too close to infinity and gives up! 1000^1000 is a number with 3000 zeroes in it, which is a pretty big number by all accounts. Python spits it out within a second without any problem. I’m really glad I decided to look at Python for these Project Euler problems; there are several where I could only go so far in CFML, knowing it wouldn’t work with the large numbers, but I think I’ll be able to complete in Python.

limit = 1000
sum = 0

for i in range(1, limit):
        sum = sum + pow(i,i)

sumStr = str(sum)

length = len(sumStr)

lastTen = ''

for i in range(length-10, length):
        lastTen += sumStr[i]

print(lastTen)

I don’t think this Python code is particularly good. I’m still struggling a bit to get my head around the range() function, and just how you do the basic things with strings. In CFML, I’d simply have said Right(sum, 10) to get the 10 right-most digits. I expect Python has similar, but I couldn’t find it after a quick look, so used the Range to loop 10 times towards the end of the sum, appending to a string.

Project Euler: problem 25

Problem 25:

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_n-1 + F_n-2, where F₁ = 1 and F₂ = 1.

Hence the first 12 terms will be:

F₁ = 1
F₂ = 1
F₃ = 2
F₄ = 3
F₅ = 5
F₆ = 8
F₇ = 13
F₈ = 21
F₉ = 34
F₁₀ = 55
F₁₁ = 89
F₁₂ = 144

The 12th term, F₁₂, is the first term to contain three digits.

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

In Python, reusing most of the logic from way back in Problem 2:

fibonacci = 1
old1 = 0
old2 = 1
limit = 1000

i = 1

while len(str(fibonacci)) < limit:
	fibonacci = old1 + old2
	old1 = old2
	old2 = fibonacci
	i = i + 1

print(i)

Project Euler: problem 20

Problem 20:

n! means n x (n − 1) x … x 3 x 2 x 1

Find the sum of the digits in the number 100!

Another one that had previously foxed me in CFML (on Coldfusion 5), but was simple in Python:

limit = 100
total = 0
product = 1

for i in range(limit, 1, -1):
        product = product * i

prodStr = repr(product)
length = len(prodStr)

for i in range(length):
        total = total + int(prodStr[i])

print(total)

Similar code to problem 16. I work out 100!, then I turn the number into a string, work out the length of the string, loop through the string, adding up each digit.

This is only my second bit of Python ever, so I’d love some feedback from anyone with more experience of the language.

Project Euler: problem 16

Problem 16:

2¹⁵ = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 2¹⁰⁰⁰?

This was a problem I’d tried doing in Coldfusion a while ago, and gave up. The problem is that 2¹⁰⁰⁰ is a 302-digit number. Coldfusion supports 32-bit integers, between -2,147,483,648 and 2,147,483,647. Even a 64-bit integer wouldn’t be long enough.

I briefly tried something in Javascript, then turned to Python, which I’ve never used before. So the following is my first ever Python program! Probably a few lines longer than it needs to be; I later saw solutions expressed in one line.

x = 0
y = repr(pow(2,1000))
length = len(y)

for i in range(length)
	x = x + int(y[i])

print(x)

This was exactly what I’d tried and failed to do with CFML. After installing Python 3.0, skimming through the tutorial and reading a few other bits in the documentation, I was able to put the above together and get the solution pretty quickly.

Although Python is dynamically-typed, you still have to cast a numeric value to a string and a string to an int, using repr() and int() respectively. In Coldfusion there’s no need to do that; I can create a string, then perform arithmetic directly on it (if it’s a numeric value in the string), then check the length of the string, append another string, find the square root, etc. all without having had to cast or convert it at any stage.

Apart from that, I like what I see of Python so far. I’m a stickler for good indentation normally, so I don’t see the requirement to indent as being a problem. I’ll maybe see if I can use Python to solve some of the other Project Euler problems that I’ve not been able to finish due to Coldfusion’s limitations with long integers.