The "four fours" problem demonstrating how to form numbers 0 through 20 using exactly four 4s and various mathematical operations.

The image presents "The Four Fours" problem, a classic mathematical puzzle where the goal is to express every integer from 0 to 20 (and beyond) using exactly four 4s and a set of allowed mathematical operations.

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🔍 What is the "Four Fours" Problem?

You must use four 4s and any combination of mathematical operations (like addition, subtraction, multiplication, division, exponents, factorials, square roots, decimals, etc.) to create expressions that evaluate to each number from 0 to 20. Each expression must use exactly four 4s, no more, no less.

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✔ Rules (implied by the examples):

- Use exactly four 4s.
- You can use:
- Basic operations: +, −, ×, ÷
- Exponents: $ 4^4 $
- Factorials: $ 4! = 24 $
- Square roots: $ \sqrt{4} = 2 $
- Decimals: e.g., $ .4 $ or $ 4.4 $
- Concatenation: e.g., $ 44 $ (using two 4s)
- Parentheses for grouping
- The goal is to generate numbers from 0 to 20 (as shown).

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🧮 Let's go through the solutions in the image and verify/understand them:

We'll examine each expression and confirm it uses four 4s and evaluates correctly.

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#### 0
$$
(4 + 4) - (4 + 4) = 8 - 8 = 0
$$
✔ Uses four 4s.

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#### 1
$$
(4 + 4)/(4 + 4) = 8 / 8 = 1
$$
✔ Correct.

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#### 2
$$
(4/4) + (4/4) = 1 + 1 = 2
$$
✔ Correct.

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#### 3
$$
4 - (4^{4-4}) = 4 - (4^0) = 4 - 1 = 3
$$
Note: $ 4^{4-4} = 4^0 = 1 $
✔ Correct.

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#### 4
$$
4 + ((4 - 4) \times 4) = 4 + (0 \times 4) = 4 + 0 = 4
$$
✔ Correct.

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#### 5
$$
4 + (4^{4-4}) = 4 + 1 = 5
$$
✔ Correct.

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#### 6
$$
4 + ((4 + 4)/4) = 4 + (8/4) = 4 + 2 = 6
$$
✔ Correct.

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#### 7
$$
(4 + 4) - (4/4) = 8 - 1 = 7
$$
✔ Correct.

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#### 8
$$
(4 + 4) + (4 - 4) = 8 + 0 = 8
$$
✔ Correct.

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#### 9
$$
(4 + 4) + (4/4) = 8 + 1 = 9
$$
✔ Correct.

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#### 10
$$
(44 - 4)/4 = 40 / 4 = 10
$$
✔ Uses concatenation (44), which is allowed.

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#### 11
$$
44 / (\sqrt{4} \times \sqrt{4}) = 44 / (2 \times 2) = 44 / 4 = 11
$$
✔ Correct.

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#### 12
$$
4 \times (4 - (4/4)) = 4 \times (4 - 1) = 4 \times 3 = 12
$$
✔ Correct.

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#### 13
$$
(44 / 4) + \sqrt{4} = 11 + 2 = 13
$$
✔ Correct.

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#### 14
$$
4 + 4 + 4 + \sqrt{4} = 4 + 4 + 4 + 2 = 14
$$
✔ Correct.

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#### 15
$$
(44 / 4) + 4 = 11 + 4 = 15
$$
✔ Correct.

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#### 16
$$
(4^{4/4}) \times 4 = (4^1) \times 4 = 4 \times 4 = 16
$$
Note: $ 4/4 = 1 $, so $ 4^1 = 4 $
✔ Correct.

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#### 17
$$
(4 \times 4) + (4/4) = 16 + 1 = 17
$$
✔ Correct.

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#### 18
$$
(4 \times 4) + 4 - \sqrt{4} = 16 + 4 - 2 = 18
$$
✔ Correct.

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#### 19
$$
4! - 4 - (4/4) = 24 - 4 - 1 = 19
$$
✔ $ 4! = 24 $, used correctly.

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#### 20
$$
(4 \times 4) + \sqrt{4} + \sqrt{4} = 16 + 2 + 2 = 20
$$
✔ Correct.

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✔ Summary

All expressions from 0 to 20 are correctly formed using exactly four 4s and standard mathematical operations.

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🎯 Why is this problem interesting?

- It encourages creative thinking and familiarity with mathematical notation.
- It demonstrates how simple digits can generate complex results.
- It’s a fun way to practice order of operations, exponents, factorials, and more.

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💡 Can we go beyond 20?

Yes! With more advanced operations like decimal points ($ .4 $), repeating decimals ($ .\overline{4} $), or even tetration, you can extend the sequence much further.

For example:
- $ 4! + 4 + 4 + 4 = 24 + 4 + 4 + 4 = 36 $
- $ 44 - 4 - 4 = 36 $
- $ 4! + 4! - 4 - 4 = 24 + 24 - 4 - 4 = 40 $

But the challenge remains: use only four 4s.

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✔ Final Answer:

The image successfully solves the "Four Fours" problem for integers from 0 to 20, using exactly four 4s per expression. All equations are correct and follow standard rules of the puzzle.

✔ Problem solved!