Four Fours Puzzle Challenge

Four Fours Puzzle Challenge

The 'Four Fours Puzzle' is a mathematical recreation that challenges you to find an expression for every integer from 0 to some maximum, using only four fours and a variety of mathematical operations. The allowed operations are addition (+), subtraction (-), multiplication (*), division (/), square root (√), factorial (!), exponents, decimal point (.), and concatenation (e.g., 44). You can also use parentheses to group operations. Your goal is to achieve the target number using exactly four '4's.

Use exactly four '4's and basic operations (+, -, *, /) to achieve the following numbers:

1. 0 =  

2. 1 =  

3. 2 =  

4. 4 =  

5. 8 =  

Now, you can use factorials (!) and square roots (√) in addition to the basic operations.

6. Find an expression for 3 using four fours. Show your work.

7. Find an expression for 5 using four fours. Show your work.

8. Find an expression for 7 using four fours. Show your work.

For these challenges, you may use decimal points (.), concatenation (e.g., 44), and exponents, along with all previously mentioned operations.

9. 10 =  

10. 15 =  

11. 17 =  

12. 20 =  

13. Explain why some numbers might be harder to achieve than others in the Four Fours puzzle.

14. Can you find an expression for 13 using four fours? If so, write it below. If not, explain why you think it's difficult.