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!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    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:

All Horses are the Same Color

If you know how to prove things by induction, then here is an amazing fact:

Theorem. All horses are the same color.

Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color.

Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses.

Well, given any set of N+1 horses, if you exclude the last horse, you get a set of N horses. By the inductive step these N horses all have the same color. But by excluding the first horse in the pack of N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED.

Hmmn... clearly not all horses have the same color. So what's wrong with this proof by induction?

Presentation Suggestions:
This delightful puzzle is an excellent test of student understanding of proofs by induction.

The Math Behind the Fact:
Hint: what could be wrong? You showed the base case. And you showed the inductive step, right?

Well actually, the argument in the inductive step breaks down in going from n=1 to n=2, because the first 1 horse and the last 1 horse have no horses in common, and therefore may not all have the same color.

How to Cite this Page:
Su, Francis E., et al. "All Horses are the Same Color." Math Fun Facts. <>.

Keywords:    logic
Subjects:    other
Level:    Advanced
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
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!