Mental Math

“You don’t have to be a mathematician to have a feel for numbers.” – John Forbes Nash, Jr.

Today’s puzzle is another original puzzle that I’ve named What Goes Where Puzzles.  It’s all about number sense.  For this puzzle, you need to  understand even and odd numbers as well as multiples (of 3 only).  I developed these puzzles to assist children in developing their number sense.

The puzzles are pretty straightforward.  You need to put the numbers 1 through 9 into the empty circles using the hints provided.  It’s important to use the numbers (1 through 9) once and only once.  Once you have filled in each circle with a number from 1 through 9, check by reviewing each hint to ensure your answers match.

Here’s today’s puzzle and good luck!


I love these puzzles, as they develop number sense while encouraging reading, understanding and following directions.  I wrote a book of 100 What Goes Where Puzzles available on Amazon.com – please take a look and pass this onto others. 

Good Luck and pass the puzzles onto others who may enjoy them!

“Eliminate all other factors, and the one that remains must be the truth.” – Sherlock Holmes

Inspector Galois was stumped. He had yet to solve his most recent case, the murder of the wealthy Oil Magnate Phil M. Mupp. There were five suspects in the murder, all of whom seemed to have an airtight alibi. The famous thoroughbred jockey, Horace Racer, was racing at Saratoga. The world famous physician, Karen Rejab, was at her health clinic in Africa. The popular DJ, Mike Refone, was hosting a 24 hour dance marathon in Los Angeles. The real estate mogul, Iva Lott, was closing on properties she recently purchased in Russia. The world famous explorer, Ben Therenbak, was mountain climbing in South America. All had numerous witnesses that eagerly corroborated their whereabouts.

While reviewing the evidence once again, there was a knock at the inspector’s office door. Without looking up, the inspector responded “Come in”. The door opened and entered Junior Inspector Abel. “It’s just the mail sir, only a single letter for you today.” said Abel as he handed the envelope to Galois. Galois took the letter and examined it curiously. It was addressed to Galois, Inspector of Police. Additionally, on the front of the envelope, at the bottom, was neatly printed “Correctly fill in each circle using only the numbers 1 through 5 – Good Luck”. The inspector curiously opened the envelope and removed a single sheet of paper. He unfolded the paper and examined it. Within seconds, he refolded it, looked at Junior Inspector Abel and said “The case is solved. I know the murderer”. The suspect was apprehended and within days the Inspector was proven to be correct. He did indeed know the murderer. What was on the sheet of paper? Look for yourself, it’s replicated below. Can you too solve the case and determine the murderer?

This is an original self-referential puzzle.  To solve the case, complete the puzzle by entering a number (1,2,3,4 or 5) in each circle that correctly matches the clue.  The number in the gray circle will be the murderer.  Remember the nature of self-referential puzzles, when one number is changed, it will likely effect other numbers.  Good Luck and please pass this puzzle and others onto others who may enjoy them!  Also, feel free to select the Facebook Like button below.

“Begin doing what you want to do now. We are not living in eternity. We have only this moment, sparkling like a star in our hand and melting like a snowflake.” – Francis Bacon

This is a new puzzle that is an extension of the Kuruko Puzzles that I created. Like Kurulko puzzles, they are similar to Sudoku puzzles.  I call them Snowflake puzzles because of their shape.  Snowflakes are symmetric shapes with six points.   The Snowflake puzzle follows this design.  It is comprised of 42 hexagons (six sided shapes).  There are white and yellow hexagons.  To complete the puzzle you must fill in each of the empty white hexagons.  There are two simple rules to follow:

1. Surrounding each yellow hexagon there are six white hexagons, these must contain the numbers 1 through 6 (each number once and only once)

2. The number in each yellow hexagon is the sum of the numbers in the three white hexagons that point to it (with a black triangle).

Here’s a snowflake puzzle for you to try –   Click here for a pdf file.

If you like the snowflake puzzle, take a look at the book of 100 Kuruko Puzzles that I wrote which is available on Amazon.com.  Good Luck and pass the puzzles onto others who may enjoy them!

“Do not let what you cannot do interfere with what you can do.” – John Wooden

Today’s puzzles are Hex Codes.  Each puzzle is a collection of adjacent hexagons that follow a pattern.  Each hexagon contains either a number/letter or is empty.  The objective is for you to fill in empty hexagons with the correct number/letter that follows the given pattern.  You must determine the pattern that each puzzle exhibits in order to correctly complete each puzzles.  The hexagons may vary in color which may be of importance (hint, hint, nudge, nudge).  Some of the puzzles may use addition or subtraction, while others may utilize some other connection (like adjacent hexagons or the color) between certain attributes.  That’s all for you to figure out. 

So all you need to do is to figure out the pattern in each puzzle and then use it fill in empty hexagons.  Best of luck!


As always, I hope you enjoyed these puzzles.  Please pass them onto others who may enjoy them!

“It is the brain, the little gray cells on which one must rely” – Hercule Poirot

Today’s puzzle is a great puzzle that everyone can enjoy.  They are called alphametics.  In each puzzle, there is a simple arithmetic problem.  However, instead of using numbers, letters are used.  Each letter stands for a single digit number (e.g. 0,1,2,3,4,5,6,7,8 or 9).   Within each puzzle, a number can be associated with only one letter, so two different letters can not equal the same number.  The meaning of the letters may change from puzzle to puzzle, so “N” may be “5” in the first puzzle and may be “3” in the second puzzle.  The objective is to assign each letter a number so the statement is correct.  It sounds straight-forward and is easy to understand, but may be very challenging.  Here’s an example:   

Make sure you understand the example. When you do, try the following puzzles. Good Luck!

I hope you enjoyed the puzzles.  Please pass the puzzles onto others who may enjoy them!

“Look at the stars and not down at your feet. Try to make sense of what you see, and wonder what makes the universe exist. Be curious.” – Stephen Hawking

Here’s a new type of puzzle that I call Number Sense Puzzles.  They are geared for younger puzzlers to help them improve their number sense.  There will be 8 statements each corresponding to a number.  You have to use the numbers 0 through 9 once and only once to fill in the correct number for each statement.  You will need to use and develop your deductive problem solving skills in order to correctly place the numbers.  Since there are only 8 answers and 10 numbers (0 through 9), some of the answers will require two digits. 

I created an interactive puzzle that will automatically check your answer and provide feedback. I also created a pdf file that can be used in classrooms or with pencils for those so inclined. The goal is to have fun and challenge yourself.

I hope you enjoy these puzzles.  If you find these interesting, Click Here for a selection of more Number Sense puzzles that I’ve created.  I will be adding to the selection over time. Good Luck and pass the puzzles onto others who may enjoy them!

“Even if you fall on your face, you are still moving forward.” – Victor Kiam

Here’s a new original puzzle that is another great challenge for everyone. It emphasizes combining even numbers (2,4,6 and 8) with addition, subtraction, multiplication and division using the rules of order of operations.

The rules are simple:

1. Use only the numbers 2, 4, 6 and 8
2. Use each number once and only once
3. Use any of + – x / to combine the numbers
4. Use parenthesis () as desired
5. Following rules 1-4, make the numbers 0-9

For example, “11” can be made as “(8 + 6) / 2 + 4″. In this example, each number (2,4,6,8) was used once and only once as required. There are other ways to make “11” which would also work.

I put together a little interactive program for this puzzle. You can enter your answers for each number and then select check. Your answers will be scored and a response provided (e.g. “got it!” or “need to use 2,4,6 and 8″). Give it a try and let me know what you think.  Click here for a pdf file.

I hope you enjoy this puzzle.  Good uck and pass it onto others who may enjoy them!

In order to use the interactive puzzle, you need to have adobe flash enabled.

“You must be the change you wish to see in the world.” – Mahatma Ghandi

Here’s another new number sense puzzle. It’s pretty easy to explain and hopefully to understand. You’re given a start number and an end number. You have to “morph” the start number into the end number by using the provided operations and entering numbers. There are two rules.

1. digits can only be used once (at most)
2. order of operations is always left to right

Here’s an example:

You have to fill in the two “boxes” with numbers (1-digit or 2-digit numbers) so that the result equals 1. Remember, that digits can not be used more than once, so right away you know that “4” and “1” can not be in either of the two entered numbers. Below is the solution:

“4 x 2″ is 8 and then “8 – 7″ is 1.  That’s all there is to it.  Here are a few more puzzles to try:

Good luck with the puzzles and pass them along to others who may enjoy them!

“There is no greatness where there is not simplicity, goodness, and truth.”
― Leo Tolstoy

Dividing by 5 mentally can be accomplished very easily. The best part is that there’s no division required. That’s right, there’s no division required to mentally divide a number by 5!  That sounds almost too good to be true.

All you need to do is multiply the number by 2 and then shift the decimal point left 1 place. That’s right, just two steps, multiplying by 2 and then shifting the decimal point of the product. Let me put them in bullet format:

 step 1: multiply the number by 2

 step 2: shift the decimal point left 1 place
 

So, if you want to divide 120 by 5, just do the following:

So 120 ÷ 5 = 24.  That’s pretty easy, isn’t it?  Well, it’s easy if you can quickly multiply numbers by 2 mentally.  This is referred to as “doubling” a number and it’s the basis of a lot of mental arithmetic techniques.  So in order to mentally divide by 5, first master doubling numbers mentally.  Let’s look at another example using a decimal number instead of an integer.   To divide 3246.4 by 5, just do the following: 

So 3246.4 ÷ 5 = 649.28.  Again, that’s pretty easy, isn’t it?  You may be asking “Why does it work?”.  Well, 5 is equal to 10 ÷ 2.  So when dividing by 5, it’s equivalent to dividing by “10 ÷ 2″.  This in turn is equivalent to multiplying by 2 and then dividing by 10.  In our base 10 (decimal) number system, dividing by 10 can be accomplished by shifting the decimal point left 1 place.  There we have it, instead of dividing by 5, we can just double and shift.

Give it a try on some other problems.  Start with integers that are easy to double (like 111 or 234) to get the idea.  Once you’re comfortable with those problems try a bit more challenging problems (like 756 or 1867) which are more difficult to double. 

Remember, mastering doubling numbers mentally is the key.  Practice and master that first and then dividing by 5 mentally will come easily.  Check out my Mental Math Anti-Calculator where you can practice your doubling of numbers and dividing by 5 interactively.  Access will be free to everyone until the end of March.  After that, contact me and I can set up an account for you.  Good luck and remember to practice doubling.