Four Fours

I stumbled upon this little brain teaser a few months ago. Since then, I've kept a sheet of paper near my computer that I've added to when something strikes me, and I think I've finally completed it. Here's the puzzle:

1. Start with four fours: 4 4 4 4


2. Add mathematical operators between and around these four fours so that you end up with equations that equal the numbers 1 through 20.

Hint: You'll have to use six different operators, as well as parentheses, to accomplish this.

Here are my answers. Some of these (the first one, is a good example) have more than one possible answer. I went with the one that involves the fewest HTML codes. (And, yes, some of the parentheses are redundant because of the order of operations, but I think they're also easier to read.)

See if you can figure it out, and then click the link below to see my answers.

1. (4 + 4) ÷ (4 + 4)


2. 4 × (4 ÷ (4 + 4))


3. (4 + 4 + 4) ÷ 4


4. (√4 + √4) × (4 ÷ 4)


5. ((4 × 4) + 4) ÷ 4


6. √4 + 4 × (4 ÷ 4)


7. (4 + 4) – (4 ÷ 4)


8. 4 + 4 + 4 – 4


9. 4 + 4 + (4 ÷ 4)


10. 4 + 4 + 4 – √4


11. (4! ÷ √4) – (4 ÷ 4)


12. (√4 + √4 + √4) × √4


13. (4! ÷ √4) + (4 ÷ 4)


14. 4 + 4 + 4 + √4


15. (4 × 4) – (4 ÷ 4)


16. 4 + 4 + 4 + 4


17. (4 × 4) + (4 ÷ 4)


18. 4! – (√4 + √4 + √4)


19. 4! – 4 – (4 ÷ 4)


20. (4 × 4) + (√4 × √4)