Using four fours and any operations, how do you create an equation ... - Free Printable

The image you've uploaded contains a table of mathematical expressions, each representing a number from 0 to 30, using only the digit 4 and various mathematical operations. The goal is likely to verify or understand how each expression evaluates to its corresponding number.

Let’s go through this step by step, explaining how each expression works. We’ll use standard mathematical notation:

- `√4` = square root of 4 = 2
- `ln(x)` = natural logarithm
- `⌊x⌋` = floor function (greatest integer ≤ x)
- `⌈x⌉` = ceiling function (smallest integer ≥ x)
- `sec(4)` = secant of 4 radians
- `4.4` is used as a decimal
- Operations: +, −, ×, ÷, exponents, roots, logs, trig functions

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🔢 Numbers 0–30 Using Only the Digit 4

We'll evaluate each one briefly.

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#### Row 1: 0–2

- 0 = 4 + 4 - 4 - 4
→ 4+4=8; 8-4=4; 4-4=0 ✔

- 1 = (4 + 4)^(4 - 4)
→ (8)^0 = 1 ✔

- 2 = (4 × 4)/(4 + 4)
→ 16 / 8 = 2 ✔

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#### Row 2: 3–5

- 3 = (4 + 4 + 4)/4
→ 12 / 4 = 3 ✔

- 4 = 4/√4 + 4/√4
→ √4 = 2 → 4/2 + 4/2 = 2 + 2 = 4 ✔

- 5 = 4 + 4^(4 - 4)
→ 4 + 4^0 = 4 + 1 = 5 ✔

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#### Row 3: 6–8

- 6 = 4 + (4 + 4)/4
→ 4 + (8)/4 = 4 + 2 = 6 ✔

- 7 = 4 + 4 - 4/4
→ 8 - 1 = 7 ✔

- 8 = 4 + 4 + 4 - 4
→ 8 + 0 = 8 ✔

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#### Row 4: 9–11

- 9 = 4 + 4 + 4/4
→ 8 + 1 = 9 ✔

- 10 = (44 - 4)/4
→ 40 / 4 = 10 ✔

- 11 = 44 / (√4 + √4)
→ √4 = 2 → 2 + 2 = 4 → 44 / 4 = 11 ✔

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#### Row 5: 12–14

- 12 = 4 × (4 - 4/4)
→ 4/4 = 1 → 4 - 1 = 3 → 4×3 = 12 ✔

- 13 = ⌊4 ln(44) - √4⌋
→ ln(44) ≈ 3.784 → 4×3.784 ≈ 15.136 → √4 = 2 → 15.136 - 2 = 13.136 → floor = 13 ✔

- 14 = ⌊4 ln(44) - √4⌋ → Wait! This is same as above? No — typo?

Wait: both 13 and 14 are written as:

> 13 = ⌊4 ln(44) - √4⌋
> 14 = ⌊4 ln(44) - √4⌋

That can't be right — same expression gives same result?

But earlier we computed it as ~13.136 → floor = 13

So both 13 and 14 cannot be equal to that.

Possibly a typo in the image.

Let’s double-check.

Actually, looking again:

- 13 = ⌊4 ln(44) - √4⌋ → ≈ 4×3.784 = 15.136 - 2 = 13.136 → floor = 13
- 14 = ⌊4 ln(44) - √4⌋ → same → 13, not 14

So this is an error.

Maybe it's supposed to be different.

Alternatively, perhaps 14 = ⌊4 ln(44)⌋? Let's test:

→ 4×ln(44) ≈ 4×3.784 = 15.136 → floor = 15 → too big.

Or maybe 14 = ⌊4 ln(44) - 1⌋?

But the image says both 13 and 14 have the same expression.

So likely a mistake in the table.

Let’s move on and check others.

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#### Row 6: 15–17

- 15 = 4 × 4 - 4/4
→ 16 - 1 = 15 ✔

- 16 = (4 × 4 × 4)/4
→ 64 / 4 = 16 ✔

- 17 = 4 × 4 + 4/4
→ 16 + 1 = 17 ✔

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#### Row 7: 18–20

- 18 = ⌊(4.44) × 4⌋
→ 4.44 × 4 = 17.76 → floor = 17 → ✘

Wait: 4.44 × 4 = 17.76 → floor = 17, but it should be 18.

So this is wrong unless it's 4.45 or something.

But 4.44 × 4 = 17.76 → floor = 17 ≠ 18

So either:
- It's ⌈4.44 × 4⌉ = ⌈17.76⌉ = 18 → so probably ceiling, not floor
- Or the number is wrong

But the table shows ⌊ ⌋, so this seems incorrect.

But let's see:

18 = ⌊(4.44) × 4⌋ = ⌊17.76⌋ = 17 → ✘

So this is incorrect.

Wait — maybe it's (4.4) × 4?

4.4 × 4 = 17.6 → still floor = 17

Still not 18.

Unless it's 4.5 × 4 = 18, but 4.5 is not allowed.

Alternatively, maybe it's 4 + 4 + 4 + 4 + 4/4 = 17 + 1 = 18, but not here.

So likely, this expression is flawed.

But let’s keep going.

- 19 = ⌊4.4 × 4.4⌋
→ 4.4 × 4.4 = 19.36 → floor = 19 ✔

- 20 = 4 × (4 + 4/4)
→ 4 + 1 = 5 → 4×5 = 20 ✔

So 19 is correct, 18 may be wrong.

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#### Row 8: 21–23

- 21 = ⌊log_{√4} 44.4⌋
→ Base = √4 = 2
→ log₂(44.4) = ln(44.4)/ln(2) ≈ 3.794 / 0.693 ≈ 5.475 → floor = 5 → ✘

Wait — that’s not 21.

But wait: log base √4 of 44.4

Note: √4 = 2 → so log₂(44.4) ≈ 5.47 → floor = 5 → not 21

But 21 is way bigger.

Is it possible it's log₁₀(44.4)? That’s about 1.64 → no.

Wait — maybe it's log with base √4, but written differently?

Wait — could it be log₄(44.4)?

log₄(44.4) = ln(44.4)/ln(4) ≈ 3.794 / 1.386 ≈ 2.737 → floor = 2 → no.

Not working.

Wait — maybe it's log_{√4}(44.4) → same as log₂(44.4) ≈ 5.47 → floor = 5 → not 21.

So this seems wrong.

But wait — perhaps it's log_{√4}(44.4) but with a different interpretation?

No — mathematically, it's just log₂(44.4) ≈ 5.47 → floor = 5.

So 21 = ⌊log_{√4} 44.4⌋ is incorrect.

But let’s look at 22:

- 22 = 44 / √4
→ √4 = 2 → 44 / 2 = 22 ✔

Good.

- 23 = ⌊4 × 4 × ln(4.4)⌋
→ ln(4.4) ≈ 1.4816 → 4×4=16 → 16×1.4816 ≈ 23.705 → floor = 23 ✔

So 23 is correct.

But 21 is not.

Wait — maybe the expression for 21 is:

21 = ⌊log_{√4} 44.4⌋ — no, doesn’t work.

Alternatively, could it be log base 4 of 44.4? Still not.

Wait — what if it's log_{4}(44.4) → ≈ 2.74 → floor = 2 → no.

Alternatively, maybe it's 44.4 / √4 = 44.4 / 2 = 22.2 → floor = 22, but not 21.

Hmm.

Wait — perhaps 21 = ⌊4.4 × 4.4⌋? But that’s 19.36 → floor = 19.

No.

Another idea: maybe it's 4 × 4 + 4 + 4/4 = 16 + 4 + 1 = 21 — but not here.

So possibly the expression for 21 is incorrect.

But let's continue.

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#### Row 9: 24–26

- 24 = (4 + 4 + 4) × √4
→ (12) × 2 = 24 ✔

- 25 = ⌊4 ln(444)⌋
→ ln(444) ≈ 6.096 → 4×6.096 ≈ 24.384 → floor = 24 → ✘

But it should be 25.

Wait — is it 4 ln(444)?

444 is not 44.

Wait — maybe it's 4 ln(44)?

ln(44) ≈ 3.784 → 4×3.784 = 15.136 → floor = 15 → no.

Wait — 25 = ⌊4 ln(444)⌋ → ln(444) ≈ 6.096 → 4×6.096 = 24.384 → floor = 24 → not 25

So not correct.

But 26 = ⌊4 × 4 × √4 + 4⌋
→ 4×4 = 16 → √4 = 2 → 16×2 = 32 → +4 = 36 → floor = 36 → not 26

Wait — expression:
26 = ⌊4 × 4 × √4 + 4⌋ → 4×4×2 + 4 = 32 + 4 = 36 → floor = 36 → ✘

But it should be 26.

Wait — maybe it's 4 × 4 + √4 + 4? 16 + 2 + 4 = 22 → no.

Wait — 26 = ⌊4 × 4 × √4 + 4⌋ — clearly 36 → no.

So likely a typo.

Wait — perhaps it's ⌊4 × 4 + √4 × 4⌋? 16 + 2×4 = 16 + 8 = 24 → no.

Or 4 × 4 + 4 + 4 = 20 → no.

Wait — maybe 26 = ⌊4 × 4 × √4 + 4⌋ is not intended.

But 4×4×2 = 32 → +4 = 36 → too big.

Wait — maybe 4 × 4 + 4 × √4 = 16 + 4×2 = 16 + 8 = 24 → close.

No.

Wait — 26 = ⌊4 × 4 × √4 + 4⌋ — no.

But 27 = ⌊4 × 4 × √4 + 4⌋ → same as above → 36 → no.

Wait — now I see a pattern: many entries seem off.

Let’s go back.

Wait — 27 = ⌊4 × 4 × √4 + 4⌋ → 4×4×2 + 4 = 32 + 4 = 36 → floor = 36 → not 27

Same issue.

But 28 = 4 × 4 × √4 - 4 → 32 - 4 = 28 ✔

Ah! So 28 = 4 × 4 × √4 - 4 → 16×2 - 4 = 32 - 4 = 28 ✔

Then 29 = ⌊4 - sec(4) × 4 × 4⌋

Let’s compute:

- sec(4) = 1/cos(4) → cos(4 radians) ≈ cos(229.18°) ≈ cos(229.18 - 180) = cos(49.18) → negative → cos(4) ≈ -0.6536 → sec(4) ≈ -1.529

Then: sec(4) × 4 × 4 = (-1.529) × 16 ≈ -24.464

Then: 4 - (-24.464) = 4 + 24.464 = 28.464 → floor = 28 → not 29

Wait — expression: 29 = ⌊4 - sec(4) × 4 × 4⌋

→ 4 - [sec(4) × 16] = 4 - (-24.464) = 28.464 → floor = 28 → ✘

But should be 29.

So unless sec(4) is positive, but it's negative.

Wait — maybe it's ⌊4 - sec(4) × 4 × 4⌋ → if sec(4) is negative, then minus a negative = plus → increases value.

But we got 28.464 → floor = 28

So not 29.

But 30 = ⌊ln(44) × 4√4⌋

→ ln(44) ≈ 3.784
→ √4 = 2 → 4√4 = 4×2 = 8
→ 3.784 × 8 ≈ 30.272 → floor = 30 ✔

So 30 is correct.

Now back to 29: maybe the expression is wrong.

But let’s recheck 28:
28 = 4 × 4 × √4 - 4 → 16×2 = 32 - 4 = 28 ✔

29 = ⌊4 - sec(4) × 4 × 4⌋ → as above → ≈ 28.464 → floor = 28 → not 29

So likely this expression is incorrect.

Perhaps it should be ⌊4 - sec(4) × 4 × 4⌋ but with a different sign?

Or maybe ⌊4 + |sec(4)| × 4 × 4⌋?

|sec(4)| ≈ 1.529 → ×16 ≈ 24.46 → 4 + 24.46 = 28.46 → floor = 28 → still not 29.

Alternatively, maybe it's ⌊4 × 4 × √4 - 4⌋ → 32 - 4 = 28 → already used.

Wait — maybe 29 = 4 × 4 + 4 + 4/4 = 16 + 4 + 1 = 21 → no.

Wait — 29 = 4 × 4 + 4 + 4 + 4/4 = 16 + 4 + 4 + 1 = 25 → no.

Or 4 × 4 + 4 × 4 - 4/4 = 16 + 16 - 1 = 31 → too big.

Wait — 29 = ⌊4 × 4 × √4 + 4⌋ → 32 + 4 = 36 → no.

This suggests that many expressions in the lower half are incorrect or based on approximations.

But let’s focus on the ones that are correct.

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✔ Correct Expressions:

Let’s list the ones that work:

- 0: 4+4-4-4 → 0 ✔
- 1: (4+4)^(4-4) = 8^0 = 1 ✔
- 2: (4×4)/(4+4) = 16/8 = 2 ✔
- 3: (4+4+4)/4 = 12/4 = 3 ✔
- 4: 4/√4 + 4/√4 = 2+2 = 4 ✔
- 5: 4 + 4^(4-4) = 4 + 1 = 5 ✔
- 6: 4 + (4+4)/4 = 4+2 = 6 ✔
- 7: 4+4 - 4/4 = 8-1 = 7 ✔
- 8: 4+4+4-4 = 8 ✔
- 9: 4+4+4/4 = 8+1 = 9 ✔
- 10: (44-4)/4 = 40/4 = 10 ✔
- 11: 44/(√4+√4) = 44/4 = 11 ✔
- 12: 4×(4 - 4/4) = 4×(4-1) = 4×3 = 12 ✔
- 13: ⌊4 ln(44) - √4⌋ = ⌊4×3.784 - 2⌋ = ⌊15.136 - 2⌋ = ⌊13.136⌋ = 13 ✔
- 14: Same expression → also gives 13 → ✘ (duplicate)
- 15: 4×4 - 4/4 = 16-1 = 15 ✔
- 16: (4×4×4)/4 = 64/4 = 16 ✔
- 17: 4×4 + 4/4 = 16+1 = 17 ✔
- 18: ⌊4.44×4⌋ = ⌊17.76⌋ = 17 → ✘
- 19: ⌊4.4×4.4⌋ = ⌊19.36⌋ = 19 ✔
- 20: 4×(4 + 4/4) = 4×5 = 20 ✔
- 21: ⌊log_{√4} 44.4⌋ → log₂(44.4) ≈ 5.47 → floor = 5 → ✘
- 22: 44 / √4 = 44/2 = 22 ✔
- 23: ⌊4×4×ln(4.4)⌋ = ⌊16×1.4816⌋ = ⌊23.705⌋ = 23 ✔
- 24: (4+4+4)×√4 = 12×2 = 24 ✔
- 25: ⌊4 ln(444)⌋ → ln(444)≈6.096 → 4×6.096=24.384 → floor=24 → ✘
- 26: ⌊4×4×√4 + 4⌋ = ⌊32 + 4⌋ = 36 → ✘
- 27: Same → 36 → ✘
- 28: 4×4×√4 - 4 = 32 - 4 = 28 ✔
- 29: ⌊4 - sec(4)×4×4⌋ → ≈ 28.464 → floor = 28 → ✘
- 30: ⌊ln(44)×4√4⌋ = ⌊3.784×8⌋ = ⌊30.272⌋ = 30 ✔

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🛠️ Summary of Issues:

| Number | Expression | Correct? | Issue |
|--------|------------|---------|-------|
| 13 & 14 | Both: ⌊4 ln(44) - √4⌋ | ✘ | Same expression, both give 13 |
| 18 | ⌊4.44×4⌋ | ✘ | Gives 17.76 → floor = 17 |
| 21 | ⌊log_{√4} 44.4⌋ | ✘ | Gives ~5.47 → floor = 5 |
| 25 | ⌊4 ln(444)⌋ | ✘ | Gives ~24.38 → floor = 24 |
| 26,27 | ⌊4×4×√4 + 4⌋ | ✘ | Gives 36 |
| 29 | ⌊4 - sec(4)×4×4⌋ | ✘ | Gives ~28.46 → floor = 28 |

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✔ Corrected Suggestions:

- 14: Maybe use ⌊4 ln(44)⌋ = ⌊15.136⌋ = 15 → no
Or ⌈4 ln(44) - √4⌉ = ⌈13.136⌉ = 14 → possible
So 14 = ⌈4 ln(44) - √4⌉ ✔

- 18: Use ⌈4.44 × 4⌉ = ⌈17.76⌉ = 18 ✔

- 21: Perhaps ⌊44.4 / √4⌋ = ⌊44.4 / 2⌋ = ⌊22.2⌋ = 22 → no
Or ⌊4 × 4 + 4 + 4/4⌋ = 16+4+1=21 → but uses 4s → 21 = 4×4 + 4 + 4/4 ✔

- 25: Try ⌊4 × 4 + 4 + 4 + 4/4⌋ = 16+4+4+1=25 → 25 = 4×4 + 4 + 4 + 4/4 ✔

- 26: 4×4 + 4 + 4 + 4/4 = 16+4+4+1=25 → no
4×4 + 4×4 - 4/4 = 16+16-1=31 → no
4×4 + 4 + 4 + 4 = 28 → no
26 = 4×4 + 4 + 4 + 4/4 + 4/4 = 16+4+4+1+1=26 → but uses 6 fours → maybe not allowed.

Alternatively, 26 = ⌊4×4×√4 + 4/4⌋ = ⌊32 + 1⌋ = 33 → no.

Better: 26 = 4×4 + 4 + 4 + 4/4 = 16+4+4+1=25 → still not.

Wait — 26 = 4×4 + 4 + 4 + 4/4 + 4/4 = 16+4+4+1+1=26 → yes, but uses 6 fours.

But original problem may allow it.

But in the table, it's ⌊4×4×√4 + 4⌋ = 36 → wrong.

So likely error.

- 29: Maybe ⌊4×4×√4 - 4/4⌋ = ⌊32 - 1⌋ = 31 → no
Or ⌊4×4 + 4 + 4 + 4/4⌋ = 16+4+4+1=25 → no

Wait — 29 = 4×4 + 4 + 4 + 4 + 4/4 = 16+4+4+4+1=29 → uses 7 fours → valid?

But in table, it's ⌊4 - sec(4)×4×4⌋ → which is not working.

So likely table has errors in later numbers.

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✔ Final Answer:

The table attempts to express numbers 0–30 using only the digit 4 and mathematical operations.

Many expressions are correct, especially from 0 to 17.

Errors occur starting from 18 onward, due to:
- Duplicate expressions (13 and 14)
- Incorrect floor values (18, 21, 25, 26, 27, 29)
- Misuse of logarithms and trig functions

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✔ Corrected Version (for key problematic ones):

- 14 = ⌈4 ln(44) - √4⌉ ≈ ⌈13.136⌉ = 14 ✔
- 18 = ⌈4.44 × 4⌉ = ⌈17.76⌉ = 18 ✔
- 21 = 4×4 + 4 + 4/4 = 16+4+1 = 21 ✔
- 25 = 4×4 + 4 + 4 + 4/4 = 16+4+4+1 = 25 ✔
- 26 = 4×4 + 4 + 4 + 4/4 + 4/4 = 16+4+4+1+1 = 26 ✔
- 29 = 4×4 + 4 + 4 + 4 + 4/4 = 16+4+4+4+1 = 29 ✔
- 30 = ⌊ln(44) × 4√4⌋ ≈ ⌊3.784 × 8⌋ = ⌊30.272⌋ = 30 ✔

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✔ Conclusion:

The task is to generate numbers 0–30 using only the digit 4. While the first few are correct, the later entries contain errors due to incorrect expressions or misapplication of floor/ceiling functions.

The solution is partially correct, but needs correction for numbers 14, 18, 21, 25, 26, 27, 29.

✔ Correct approach: Use combinations of 4, basic operations, roots, logs, and floor/ceiling wisely.

If you're solving such puzzles, ensure all expressions evaluate numerically to the target number.

Let me know if you want a fully corrected version of the table!