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Problem Description

Here are 10 small programs that solve the first 10 even numbered problems from the  Project Euler programming challenge at educational website mathschallenge.net.

#2.  Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...  Find the sum of all the even valued terms  below one million in the sequence.  

#4.  A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.  Find the largest palindrome made from the product of two 3-digit numbers.


#6. The sum of the squares of the first ten natural numbers is, 1² + 2² + ... + 10² = 385.  The square of the sum of the first ten natural numbers is, (1 + 2 + ... + 10)² = 55² = 3025.  Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 – 385 = 2640.  Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

#8.  Find the greatest product of five consecutive digits in the 1000-digit number.    (See program for the number)


#10.  The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.  Find the sum of all the primes below one million. 

#12.  The sequence of triangle numbers is generated by adding the natural numbers.  So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first  ten terms would be: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...    The 7th triangle number, 28, is the first triangle number to have over five factors (1,2,4,7,14,28).   Which is the first triangle number to have over five-hundred factors?


#14.  The following iterative sequence is defined for the set of natural numbers:  n_i+1= n_i / 2   ( if n_i is even),  n_i+1 = 3n_i + 1 (if n_i is odd).  Using this rule and starting with 13, we generate the following sequence:  13, 40, 20, 10,  5, 16, 8, 4, 2, 1.   It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.   Which starting number, under one million, produces the longest chain?    NOTE: Once the chain starts the terms are allowed to go above one million.

#16.  Work out the first 10 digits of the sum of the one-hundred 50-digit numbers defined below.  (See program for numbers.)


#18.  By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total sum from top to bottom is 23  (3 + 7 + 4 + 9).

3
7 5
2 4 6
8 5 9 3

Find the maximum total from top to bottom for the following triangle. 

64
45 63
75 09 91
....22 more rows (see program for values, last row has 25 values)

#20.   n! means n × (n – 1) × ... × 3 × 2 × 1.  Find the sum of the digits in the number 100!

Background & Techniques

I ran across this site several days ago and got hooked on writing the programs to solve the problems.  Some are harder than others, but they are all fun!   It is required to sign up to gain access to the problems  and scores are posted by the handle selected.  So you can check that ole  "delphiforfun" is tied for 1st place having solved all 21 problems that are currently available.   There are seven levels with 3 problems at each level.  

The programs posted here are all beginner level programs based on my code length criteria - they have only 20 to 36 lines of user written code.  The conceptual thinking required to solve them however is not beginner level.  But that's part of the fun.  Stretch your mind!   

As you can see, the problems cover quite a span of topics:  primes, factors, palindromes, factorials,  large number arithmetic, and graph-searching come to mind.

I decided to post even numbered problems to prevent cheaters from just posting the answers these programs provide.  (Even though in the long run such persons would only be cheating themselves.)   One of the most important characteristics of a good programmer is persistence.  If you don't give up, and the goal was reasonable in the first place, you will succeed!   These programs all represent reasonable goals.   

If  you just download and run these programs, you will not have learned much.  I would recommend that, if you plan to participate in the challenge, spend at least 8 hours on a problem before checking my solution.  Then figure out why you couldn't figure it out and you will have learned something either way.  

Running/Exploring the Program 

Suggestions for Further Explorations

These and the  rest of the problems on the Project Euler webpages at mathschallenge.net

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