The Four Four’s Puzzle

The Four Four's puzzle

The Four Four’s is a classic puzzle that requires a certain amount of creativity and also makes you think about how the operations interact. This is a classic math puzzle which challenges you to use four four’s and any math operations to write equations that have the numbers from 0 to 20 or more as the answer. Sounds easy, try it?


I introduced my elder son to the puzzle and told him to write the mathematical expression for all the numbers between 1 to 20, each using exactly four of the number 4 and using any mathematical operations as many times he would like to. Well, he did it relatively fast.

To make it more challenging, I told him to do the same puzzle with a new added condition, to use ONLY the basic math operators ( + , – , x, /, (brackets)). He could use these simple math operators as many times as he wished. This gave him the time to pause and not try to solve them all using the advanced operators he knows. Moreover, it allowed his younger brother to work with him who only has a brief idea of factorials and powers.

He found it was much harder to solve now. He observe that some of the numbers from 1-20 can not be derived using the simple operations ( + , – , x, /, (brackets) ). I then told him to select one another operator to find out the remaining numbers. Can you guess which operator he chose ? He tried to use square root but could still not solve 2 numbers.

The Four Four’s Puzzle

Can you write every integer from 0 to 20 using exactly four 4’s and any commonly used mathematical symbols and operators.

There are some rules to follow

  • You have to use exactly 4 instances of the number 4. No more, no less.
  • You can only use the basic operators +,-,x, / , (brackets).
  • Few of the numbers ( 6 out of 20 ) can’t be solved with basic operators. In that case you can use any ONE of the operator .decimal point , repeating decimal, sqrt, power, !factorial
  • You can’t use representation of other numbers eg use of π in sin(π/2) = 1

With some logic and calculations, you can definitely write the first 20 numbers and some numbers might be represented with many different expressions.

Give this puzzle a try and let me know your solution in the comments section. Check out my solutions below.

Challenge yourself to solve the same puzzle for numbers from 1-100.

Are you looking for more interesting puzzles , then check out these 30+ critical thinking puzzles

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2 Replies to “The Four Four’s Puzzle”

  1. Thanks for this puzzle. I enjoyed solving with my kids. I did 10 as (44-4)/4
    I Cldnt get total of 5 numbers.
    11,13 14,18,19

    1. I am glad you enjoyed doing it with your kids.I didn’t allow my kids to use concatenation, they thought of it but I strictly wanted them to use the basic operators. You can extend this activity by going upto 50 or 100 and also finding multiple other solutions for each number. That’s our next goal. I hope you will also have fun solving them.

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