Math Puzzle

“Solitary trees, if they grow at all, grow strong” – Winston Churchill

You awake one day to find you’ve been shipwrecked on an island – The Island of Wishy Washy. The Island of Wishy-Washy had three types of natives. The Wishy, the Washy and the Wishy-Washy. Only natives could tell each of them apart. The Wishy always tell the truth and the Washy always lie. The Wishy-Washy sometimes tell the truth and sometimes lie. It all depended what they heard immediately before. If what they heard immediately before was the truth, then they would lie. If what they heard immediately before was a lie, then they would tell the truth.

So, if a Wishy answered immediately before a Wishy-Washy, then the Wishy-Washy would lie because a Wishy always tells the truth. If a Washy answered immediately before a Wishy-Washy, then the Wishy-Washy would tell the truth because a Washy always lies. If you don’t know what the statement was before a Wishy-Washy speaks, it could be either a true or false, depending on what the Wishy-Washy heard last (not you).

A curious fact about the natives of the Island of Wishy-Washy is that they never are seen with one of their own kind. That is, a Wishy is never with another Wishy. If a Wishy is walking with another native it is most certainly either a Washy or a Wishy-Washy.

Puzzle 1 – After awaking and realizing you’re on the Island of Wishy-Washy, you are immediately approached by three natives

• – the first says, “Hi, I’m a Wishy”
• – the second says, “Hi, I’m a Washy”
• – the third says “Hi, I’m a Wishy-Washy”

Can you determine which is which?

Puzzle 2 – The next day, you’re taking a walk on the beach, and again three natives approach. This time, you ask “Are you a Wishy-Washy?” to all three.

• – the first replies, “Yes, I’m a Wishy-Washy.”
• – immediately the second points to the first and says, “That’s a lie!”
• – to this, the third points to the second and says “No, that’s a lie!”

Can you determine which is which?

Puzzle 3 – The next day just two natives approach you. The first says, “I’m a Wishy-Washy” and the second says the same. Can you tell what they really are and which is which?

The Island of Wishy-Washy is an original logic puzzle. There are many further, more challenging puzzles concerning the Island of Wishy-Washy that will be introduced throughout the next year. Please enter a comment with any solutions in order share with others. Best of luck with solving these puzzles.

This puzzle is a reprinted puzzle from my original BLOG (on Facebook).  I will be introducing new and more Wishy-Washy puzzles in the future.

“What people don’t realize is that professionals are sensational because of fundamentals” – Barry Larkin

Number sense if a fundamental skill.  I like to refer to it as “Number Familiarity“. How familiar are are you with numbers.  For example, take the number 64.  How familiar are you with the number 64?  

• – Do you know it’s a perfect square?    
  – Do you know it’s a power of 2?    
  – Do you know it’s a perfect cube?  
  – Do you know it’s the product of two perfect squares?
  

64 equals all of these – it’s 8²,   2⁶,   4³  and   (2²)(4²). 

I think that simply stated number puzzles are a great way to increase number sense.  It doesn’t matter if you’re seven or eighty-seven, concise number puzzles can be beneficial.  Here’s a few concise number puzzles to try:

1. Can you think of two numbers that when multiplied make 8 and when added make 6?
2. Can you think of two numbers that when multiplied make 32 and when added make 18?
3. Can you think of two numbers that when multiplied make 48 and when added make 14?

These puzzles should not be solved algebraically (e.g. via a system of equations).  Instead, these puzzles should stimulate thinking and making number connections.  They should also generate confidence.  There are very few things that generate confidence better than success.  Once these puzzles are mastered, change it up a bit.  Keep the puzzles generally the same, but ask for something different.  Instead of asking for the numbers, ask for the difference of the numbers, or ask for the smaller or larger number.  This adds another level of challenge to the problem.  Here are a few additional number puzzles to try:

1. What is the difference of two numbers that when multiplied make 18 and when added make 9?
2. What is the larger number of two numbers that when multiplied make 30 and when added make 13?
3. What is the smaller number of two numbers that when multiplied make 48 and when added make 19?

These puzzles are similar to the first set of puzzles, but differ as they are more challenging because the two numbers must be further manipulated to obtain the correct answer.  Continuing this theme of making the puzzles more challenging, here are a few more number puzzles to try:

1. Can you think of two numbers that when multiplied make 16 and adding one number to twice the other makes 8?
2. Can you think of two numbers that when multiplied make 24 and adding one number to twice the other makes 14?
3. Can you think of two numbers that when multiplied make 36 and adding one number to three times the other makes 21?

These puzzles will help build fundamental number sense.  They can be easily stated, understood and quickly solved.  They serve as great 1-minute problems and can assist in building confidence which will lead to even better problem solving skills.  Give them a try and share them if you like them.  Have fun and good luck!

“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.