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New Source to Overdose with Problems

Past columns recommended websites that provide interesting collections of problems. The problems vary as much as the motivation behind the offering of the problems. Sometimes, the problems are offered by university math departments trying to recruit students and sometimes by individuals who just like to share their mathematical gems.

This week, the focus is on the problems offered by Project Euler, created by by Colin Hughes in October 2001. It openly states its motivation motivation: "provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context." And, what is unusual about the problem sets is their associated claim that "Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems."

Two sample problems are:

  • Problem 1: If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.

    My Comments: Obviously one could use brute force to solve this problem by listing the desired numbers by hand and adding them, or one could use the brute force of a computer algorithm to both identify and add these numbers. But, how can one use some mathematical insight to simplify both of the "brute force" processes? This is why the website has a "one-minute" rule, "which means that although it may take several hours to design a successful algorithm with more difficult problems, an efficient implementation will allow a solution to be obtained on a modestly powered computer in less than one minute."

  • Problem 26: The 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.

    My Comments: Again, one could use brute force to compute all of the recurring cycles by hand for the designated number range, or one could use the brute force of a computer algorithm to both compute the recurring cycles (which becomes an interesting process in itself) and identify the desired number d. But, how can one use some mathematical insight to simplify both of the "brute force" processes? And, one may wonder, is there a theorem that "predicts" the length of a recurring cycle for any number d?

Presently, more than 300 problems are waiting for your exploration. Some you will like and some you may want to avoid. Remember that the website's "intended audience include students for whom the basic curriculum is not feeding their hunger to learn, adults whose background was not primarily mathematics but had an interest in things mathematical, and professionals who want to keep their problem solving and mathematics on the edge." So, share the problems as well.

Oh, you might be wondering if the website offer hints or solutions? You will need to visit the website to check on that "wonderment."