Highly Divisible Triangular Number — Project Euler (Problem 12)

All n numbers are Triangle Numbers. They’re called so, because they can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one.


Problem 12 of Project Euler asks for the first triangle number with more than 500 divisors.

These are the factors of the first seven triangle numbers:

1 = 1: 1
2 = 3: 1,3
3 = 6: 1,2,3,6
4 = 10: 1,2,5,10
∑5 = 15: 1,3,5,15
∑6 = 21: 1,3,7,21
∑7 = 28: 1,2,4,7,14,28

Here’s how I proceeded:

First Step: Find the smallest number with 500 divisors. Seems like a good starting point to begin our search.
Second Step: Starting at the number found in the previous step, search for the next triangle number. Check to see whether this number has 500+ divisors. If yes, this is the number we were looking for, else…
Third Step: Check n for which ∑n = triangle number found in the previous step
Fourth Step: Add (n+1) to the last triangle number found, to find the next triangle number. Check whether this number has 500+ divisors. If yes, this number is the answer. If not, repeat Fourth Step till the process terminates.

Now for the details:

The First Step isn’t exactly a piece of cake, but necessary to reduce computation time. I solved this with a bit of mental math. The main tool for the feat is the prime number decomposition theorem:

Every integer N is the product of powers of prime numbers

N = pαqβ· … · rγ
Where p, q, …, r are prime, while α, β, …, γ are positive integers. Such representation is unique up to the order of the prime factors.
If N is a power of a prime, N = pα, then it has α + 1 factors:
1, p, …, pα-1, pα
The total number of factors of N equals (α + 1)(β + 1) … (γ + 1)

500 = 2 x 2 x 5 x 5 x 5
So, the number in question should be of the form abq4r4s4 where a, b, q, r, s are primes that minimize abq4r4s4. This is satisfied by 7x11x24x34x54 = 62370000. This marks the end of the First Step which is where we start our search for our magic number.

The next 3 steps would need helper functions defined as below:

from math import *
# Function to calculate the number of divisors of integer n
def divisors(n):
limit = int(sqrt(n))
divisors_list = []
for i in range(1, limit+1, 1):
if n % i == 0:
if i != n/i:
return len(divisors_list)
# Function to check for triangle number
def isTriangleNumber(n):
a = int(sqrt(2*n))
return 0.5*a*(a+1) == n
# Function to calculate the last term of the series adding up to the triangle number
def lastTerm(n):
if isTriangleNumber(n):
return int(sqrt(2*n))
return None

As can be seen from the above code, the algorithm to calculate divisors of an integer is as follows:
1. Start by inputting a number n
2. Let an int variable limit = √n
3. Run a loop from i = 1 to  i = limit
    3.1 if n is divisible by i
3.1.1 Add i to the list of divisors
3.1.2 if i and n/i are unequal, add n/i to the list too.
4. End

Finally, executing the 4 steps mentioned earlier can be done like so (the code took less than 2s to arrive at the answer):

# First Step
# First number 'check' to have 500 divisors
check = 2**4 * 3**4 * 5**4 * 7 * 11
# Second Step
# Starting from 'check', iterate sequentially checking for the next 'triangle' number
while not isTriangleNumber(check):
check += 1
# Third and Fourth Steps
# Calculate the last term of the series ('seriesLastTerm') that adds up to the newly calculated triangle number 'check'
seriesLastTerm = lastTerm(check)
# Iterate over triangle numbers checking for divisors > 500
while divisors(check) <= 500:
# add the next term to check to get the next triangle number
check += (seriesLastTerm + 1)
seriesLastTerm += 1
print check

Ans: 76576500


3 thoughts on “Highly Divisible Triangular Number — Project Euler (Problem 12)

  1. Great write up, but I spent the longest time trying to figure out how the “isTriangleNumber” function actually did its work. After a bit of googling, I found that it’s just a known formula for exactly that test. It may be intuitive for some, but it was completely lost on me, and with the bar set so high with the theory behind Step 1, I thought I had to be missing something.

    It might be helpful for the next guy to add in a bit about the theory behind Step 2 as well. Thanks!


  2. Your first step looks bizarre but works correctly in this case since 500 is a composite number with many primes and low power. But what if I rephrase the original question to “find the first triangle number with more than 499 divisors”? Now you realize that 499 is a prime number, so your first starting point would be:

    N = p^(499-1) = 2^498 = some very large number with 149 digits in base 10.

    Which is already way too large for the answer.

    The error here is that you miss to see that a total number of divisors of N is not an increasing order as N increased, e.g., 12 is the first number to have 6 divisors, while 16 is the first number to have 5 divisors.

    So in order to find a number X that has larger number of divisors than N, you can’t make an assumption that you can start at number Y, where Y < X, and Y has number of divisors less than or equal to N.

    Besides, this very first problem on Project Euler doesn’t require you to reduce a starting point at all! I run your code against my normal iterative method which finds number of divisors starts from 1,3,6,10,15,… the differ in result time is insignificant.

    Also the majority time of your code is spent on “divisors(n)” function, which could be a lot faster by factorize a number into a list of prime numbers first, count occurrence for each prime, then use your mention theorem to find a number of divisors. Instead of listing every possible numbers lesser than N and check if that number is a divisor.

    Liked by 1 person

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