Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Advanced level:

Greedy to Avoid Progressions

An arithmetic progression is a sequence of 3 or more integers whose terms differ by a constant, e.g., 20, 23, 26, 29 is an arithmetic progression. Question: does every increasing sequence of integers have to contain an arithmetic progression in it? For instance, the sequence of primes 2, 3, 5, 7, 11, 13,... contains an arithmetic progression: 3,7,11.

Somewhat surprisingly, there are sequences with no progressions! We'll construct an example:

Start with 0. Then for the next term in the sequence, be greedy: take the smallest possible integer that does not cause an arithmetic progression to form in the sequence constructed so far. (There must be such an integer because there are infinitely many integers beyond the last term, and only finitely many possible progressions the new term could complete.) This gives:

0, 1, 3, 4, 9, 10, 12, 13, 27, 28,...

Clearly this sequence has the required property. Now that we have a solution, is there a better way to understand what this sequence is, without having to rely on recursion?

Yes. The above sequence can also be obtained by writing all the positive integers in base 2, then interpreting them in base 3! The base 2 integers are:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001,...
and if you think of these as base 3 numbers, you get the above integers!

Presentation Suggestions:
Encourage students to think about why the binary trick works.

The Math Behind the Fact:
There are two important mathematical ideas here.

The trick used to construct the above sequence is an example of a greedy algorithm. Greedy algorithms arise as solutions to many problems in computer science and mathematics.

The second lesson is that often when a solution is developed, we can find a simpler one by insight: it is a nice exercise to show that the binary trick work because in base 3, if any two terms contain just 0's and 1's, then a third term that completes an arithmetic progression must contain a 2!

How to Cite this Page:
Su, Francis E., et al. "Greedy to Avoid Progressions." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Subjects:    number theory
Level:    Advanced
Fun Fact suggested by:   Jorge Aarao
Suggestions? Use this form.
4.07
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!