The Four 4s Challenge is a mathematical puzzle where the objective is to use exactly four 4s and any mathematical operators to create expressions that equal the numbers from 1 to 100. Here is a detailed answer with examples for each number:

1. 1 = 4/4 + 4 - 4 Explanation: Divide one 4 by another 4, add 4, and subtract 4.

2. 2 = 4/4 + 4/4 Explanation: Divide one 4 by another 4 and add it to the result of dividing another 4 by 4.

3. 3 = (4 + 4 + 4)/4 Explanation: Add three 4s together and divide the sum by 4.

4. 4 = 4 + 4 - 4 + 4 Explanation: Add two 4s together, subtract 4, and add another 4.

5. 5 = 4 + 4/4 Explanation: Add two 4s together and divide one of them by 4.

6. 6 = 4 + 4/4 + 4/4 Explanation: Add two 4s together, divide one of them by 4, and add the result to the other 4 divided by 4.

7. 7 = 4 + 4 - 4/4 Explanation: Add two 4s together and subtract the result of dividing a 4 by 4.

8. 8 = 4 + 4 + 4 - 4 Explanation: Add three 4s together and subtract 4.

9. 9 = 4 + 4 + 4/4 Explanation: Add three 4s together and divide one of them by 4.

10. 10 = 4 + 4 - 4/4 Explanation: Add two 4s together and subtract the result of dividing a 4 by 4.

11. 11 = 4 + 4 + 4/4 + 4/4 Explanation: Add three 4s together, divide two of them by 4, and add the results.

12. 12 = 4 + 4 + 4 Explanation: Add three 4s together.

13. 13 = 4 + 4 + 4/4 + 4 Explanation: Add three 4s together, divide one of them by 4, and add another 4.

14. 14 = 4 + 4 + 4 - 4/4 Explanation: Add three 4s together and subtract the result of dividing a 4 by 4.

15. 15 = 4 + 4 + 4 + 4/4 Explanation: Add three 4s together and divide one of them by 4, then add another 4.

16. 16 = 4 * 4 Explanation: Multiply two 4s together.

17. 17 = 4 * 4 + 4/4 Explanation: Multiply two 4s together and add the result of dividing a 4 by 4.

18. 18 = 4 * 4 + 4/4 + 4/4 Explanation: Multiply two 4s together, add the result of dividing a 4 by 4, and add another 4 divided by 4.

19. 19 = 4 * 4 + 4 - 4/4 Explanation: Multiply two 4s together, add another 4, and subtract the result of dividing a 4 by 4.

20. 20 = 4 * 4 + 4 + 4/4 Explanation: Multiply two 4s together, add another 4, and divide one of them by 4.

21. 21 = 4 * 4 + 4 + 4/4 + 4/4 Explanation: Multiply two 4s together, add another 4, divide two of them by 4, and add the results.

22. 22 = 4 * 4 + 4 + 4 - 4/4 Explanation: Multiply two 4s together, add two 4s, and subtract the result of dividing a 4 by 4.

23. 23 = 4 * 4 + 4 + 4 + 4/4 Explanation: Multiply two 4s together, add three 4s, and divide one of them by 4.

24. 24 = 4! - 4 + 4 + 4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial, which is 432*1), then add three 4s.

25. 25 = 4! - 4 + 4 + √4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add two 4s and the square root of 4.

26. 26 = 4! - 4 + 4 + 4/4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add two 4s and divide one of them by 4.

27. 27 = 4! - 4 + 4 + 4/4 + 4/4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add two 4s and divide two of them by 4.

28. 28 = 4! - 4 + 4 + 4 - 4/4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add three 4s and subtract the result of dividing a 4 by 4.

29. 29 = 4! - 4 + 4 + 4 + 4/4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add four 4s and divide one of them by 4.

30. 30 = 4! - 4 + 4 + 4 + 4 Explanation: Using the factorial operator (!), subtract 4 from 4! (4 factorial), then add four 4s.

This is just a small sample of the numbers from 1 to 100, but the pattern continues. By using various mathematical operators and the four 4s, it is possible to create expressions that equal any number from 1 to 100.