counterexample

“Learning never exhausts the mind.” – Leonardo Di Vinci

This weeks twister is a real thinking puzzle that’s really pretty easy to understand.  There will be a series of statements and all you have to do is say whether each statement is true or false.  Well, there’s just a little more to it, if you say the statement is false, then you have to give a counter example that shows why the statement is false.  

Here’s an example to show you how it works.  The statement is

All numbers are greater than 5

This is obviously FALSE, and a counter example to this statement is 3, because it’s a number that’s NOT greater than 5.  So the number 3 disproves the statement.  The number 3 is not the only counter example to this statement.  Other counter examples are 2, 0, -13 or even 4.1267.

Below is a five statement counter example game. Within this game four of the five statements are FALSE, so only one is TRUE (so it doesn’t have a counter example).

1. No state in the United States begins and ends with the same letter.

2. All triangles have three sides.

3. All even numbers end in 2, 4, 6 or 8.

4. If a number is not negative then it is positive.

5. All four-sided shapes are squares.

These statements are from varied subjects (spelling, geometry and arithmetic).  So, if you want to make your own counter example statements, any topic will work (not just math).  Regardless of the subject matter, make it fun.  Challenge yourself and others to think beyond your/their immediate response.  “I don’t know” is not allowed – thinking is encouraged.  Originality and accomplishment should be rewarded. 

I love these puzzles because they develop thinking while encouraging reading and understanding.  Good Luck and pass the puzzles onto others who may enjoy them!  Also, click “Like” below if you like it!

“Three things cannot be long hidden: the sun, the moon, and the truth.” – Budda

A counterexample is an example that shows a statement is false. Counterexamples are very important as they can be used to disprove a proposition. For example, given the proposition: “All prime numbers are odd”, 2 serves as a counterexample that disproves the proposition. 2 is a prime number, but it is not odd. It is important to realize that although counterexamples can be used to disprove propositions, examples can not be used to prove propositions.

The Counterexample Game is really straight forward. Given a statement, say whether it’s TRUE or FALSE. If it’s FALSE, then give a counterexample that shows that the proposition is FALSE. What’s great about the Counterexample Game is that it gets you thinking. It’s a great quick game that can be used as a starter for a group (in a classroom environment) or rapid-fire with individuals. The other great feature is that the subject does not have to be math. It can be based on nearly all subjects. Make a statement and if it’s false, just ask for a counterexample.

Below is a five statement counterexample game. Within this game four of the five statements are FALSE, so only one is TRUE (so it doesn’t have a counterexample).

1. All of the state names in the United States contain the letter “e”.

2. All rectangles are squares.

3. All multiples of 3 are odd.

4. All multiples of 2 are even.

5. Connecting three unique dots will always make a triangle.

These statements are from varied subjects (spelling, geometry and arithmetic).  You can gear the statement(s) to the subject matter being reviewed or studied.  Regardless of the subject matter, make it fun.  Challenge the students to think beyond their immediate response.  “I don’t know” is not allowed – encourage thinking.  Reward originality and accomplishment. 

Good luck and have fun.  Please comment if you  like these.  I plan on generating more of these to be used in classrooms.