A Lot Of The High-level Sports Are

The Trouble With Integers Is That We Have Examined Only The Very Small Ones. Maybe All The Exciting Stuff Happens At Really Big Numbers, Ones We Can't Even Begin To Think About In Any Very Definite Way. Our Brains Have Evolved To Get Us Out Of The Rain, Find Where The Berries Are, And Keep Us From Getting Killed. Our Brains Did Not Evolve To Help Us Grasp Really Large Numbers Or To Look At Things In A Hundred Thousand Dimensions.

Incidentally, When We're Faced With A "prove Or Disprove," We're Usually Better Off Trying First To Disprove With A Counterexample, For Two Reasons: A Disproof Is Potentially Easier (we Need Just One Counterexample); And Nitpicking Arouses Our Creative Juices. Even If The Given Assertion Is True, Our Search For A Counterexample Often Leads To A Proof, As Soon As We See Why A Counterexample Is Impossible. Besides, It's Healthy To Be Skeptical.

A Lot Of The High-level Sports Are Really In Your Mind.

Some People Think That Mathematics Is A Serious Business That Must Always Be Cold And Dry; But We Think Mathematics Is Fun, And We Aren't Ashamed To Admit The Fact. Why Should A Strict Boundary Line Be Drawn Between Work And Play? Concrete Mathematics Is Full Of Appealing Patterns; The Manipulations Are Not Always Easy, But The Answers Can Be Astonishingly Attractive.

Ab=1/4((a+b)^2-(a-b)^2) Is An Amazing Identity, And Unfortunately, I Have To Remind My Current Students How To Prove It.