Complete the magic hexagon by filling in the missing numbers. - Free Printable

Let's analyze the pattern in this circular arrangement to determine the missing number (marked with a "?").

We have a circle with alternating yellow circles and red squares:

- Yellow circles: contain numbers: 3, 9, 5, 8, 7, 7
- Red squares: contain numbers: 21, ?, 45, 40, 56, 48

The red squares are positioned between the yellow circles. Let's label them in order around the circle.

Starting from the top and going clockwise:

1. Yellow: 3
2. Red: 21
3. Yellow: 7
4. Red: 48
5. Yellow: 7
6. Red: 56
7. Yellow: 8
8. Red: 40
9. Yellow: 5
10. Red: 45
11. Yellow: 9
12. Red: ?

So the red squares are placed between two yellow circles. We can assume that each red square is derived from the two adjacent yellow circles.

Let’s examine each red square and see if there's a consistent rule relating it to its two neighboring yellow circles.

Step-by-step analysis:

#### 1. Red = 21 → Between Yellow 3 and 7
- 3 and 7 → 21
- 3 × 7 = 21 ✔

#### 2. Red = 48 → Between Yellow 7 and 7
- 7 × 7 = 49 ✘ (not 48)
- 7 + 7 = 14 ✘
- Maybe not multiplication?

Wait — maybe we need to look at the positions more carefully.

Let’s list all pairs:

| Red Square | Left Yellow | Right Yellow |
|------------|-------------|--------------|
| 21 | 3 | 7 |
| 48 | 7 | 7 |
| 56 | 7 | 8 |
| 40 | 8 | 5 |
| 45 | 5 | 9 |
| ? | 9 | 3 |

Now check operations:

1. 3 and 7 → 21 → 3 × 7 = 21 ✔
2. 7 and 7 → 48 → 7×7=49 ≠ 48 ✘
But 7×7 -1 = 48? That seems arbitrary. Try another idea.

Wait — what if the operation isn't always multiplication?

Try:

- 7 and 7 → 48 → 7×7 = 49 → close but not exact.
- 7×6 + 6 = 48? Not helpful.

Wait — perhaps it's not multiplication.

Try looking at other values:

3. 7 and 8 → 56 → 7 × 8 = 56 ✔
4. 8 and 5 → 40 → 8 × 5 = 40 ✔
5. 5 and 9 → 45 → 5 × 9 = 45 ✔
6. 9 and 3 → ? → 9 × 3 = 27

But earlier:
- 7 and 7 → 48 → but 7×7 = 49 ≠ 48 → contradiction?

Wait! Is the pair for 48 really 7 and 7?

Let’s double-check the positions.

From the diagram:

Going clockwise:

- Top: Yellow 3 → Red ? → Yellow 9
- Then: Yellow 9 → Red 45 → Yellow 5
- Then: Yellow 5 → Red 40 → Yellow 8
- Then: Yellow 8 → Red 56 → Yellow 7
- Then: Yellow 7 → Red 48 → Yellow 7
- Then: Yellow 7 → Red 21 → Yellow 3

Wait — now I see the issue. The sequence may be going clockwise or counterclockwise.

Let’s reconstruct the full circle properly.

Looking at the image layout:

Positions (clockwise):

1. Top: Yellow 3
2. Top-right: Red ?
3. Right: Yellow 9
4. Bottom-right: Red 45
5. Bottom: Yellow 5
6. Bottom-left: Red 40
7. Left: Yellow 8
8. Top-left: Red 56
9. Left-middle: Yellow 7
10. Middle-left: Red 48
11. Middle-bottom: Yellow 7
12. Top-left: Red 21
13. Back to Yellow 3

Wait — this is confusing. Let's assign positions clearly.

Let’s number the positions around the circle clockwise:

Start from the top:

- Position 1: Yellow circle with 3
- Position 2: Red square with ?
- Position 3: Yellow circle with 9
- Position 4: Red square with 45
- Position 5: Yellow circle with 5
- Position 6: Red square with 40
- Position 7: Yellow circle with 8
- Position 8: Red square with 56
- Position 9: Yellow circle with 7
- Position 10: Red square with 48
- Position 11: Yellow circle with 7
- Position 12: Red square with 21
- Back to position 1: Yellow 3

So the red squares are in positions: 2, 4, 6, 8, 10, 12

And they are between:

- Red 2 (?): between Yellow 3 and Yellow 9
- Red 4 (45): between Yellow 9 and Yellow 5
- Red 6 (40): between Yellow 5 and Yellow 8
- Red 8 (56): between Yellow 8 and Yellow 7
- Red 10 (48): between Yellow 7 and Yellow 7
- Red 12 (21): between Yellow 7 and Yellow 3

Now let’s write each red value and the two adjacent yellows:

1. ? → between 3 and 9 → 3 × 9 = 27 → maybe ?
2. 45 → between 9 and 5 → 9 × 5 = 45 ✔
3. 40 → between 5 and 8 → 5 × 8 = 40 ✔
4. 56 → between 8 and 7 → 8 × 7 = 56 ✔
5. 48 → between 7 and 7 → 7 × 7 = 49 ✘ → not 48
6. 21 → between 7 and 3 → 7 × 3 = 21 ✔

Only one inconsistency: 7 × 7 = 49, but red square is 48.

Hmm.

Wait — could it be a typo? Or is there another pattern?

Wait — what if the order is different?

Wait — let’s list the red squares and their neighbors again:

- Red 45: between 9 and 5 → 9×5=45 ✔
- Red 40: between 5 and 8 → 5×8=40 ✔
- Red 56: between 8 and 7 → 8×7=56 ✔
- Red 48: between 7 and 7 → 7×7=49 ✘ → should be 49?
- Red 21: between 7 and 3 → 7×3=21 ✔
- Red ?: between 3 and 9 → 3×9=27

But 7×7=49, but the value is 48.

Is there a mistake?

Alternatively, maybe the red square labeled 48 is between 7 and 7, but perhaps it's not multiplication?

Wait — maybe the pattern is sum of squares or something else?

Try:

- 7 and 7: 7+7=14, 7²=49, 7×7=49 → still 49

But value is 48.

Wait — unless the red squares are not all products.

Wait — what if the red squares are products of the two adjacent yellows, but one is an exception?

But that would break the pattern.

Wait — maybe I misread the positions.

Let me recheck the diagram.

Looking at the image:

- At top: Yellow 3
- To the right: Red ?
- Then Yellow 9
- Then Red 45
- Then Yellow 5
- Then Red 40
- Then Yellow 8
- Then Red 56
- Then Yellow 7
- Then Red 48
- Then Yellow 7
- Then Red 21
- Then back to Yellow 3

Yes.

So the red squares are:

- Between 3 and 9 → ?
- Between 9 and 5 → 45 → 9×5=45 ✔
- Between 5 and 8 → 40 → 5×8=40 ✔
- Between 8 and 7 → 56 → 8×7=56 ✔
- Between 7 and 7 → 48 → 7×7=49 ✘
- Between 7 and 3 → 21 → 7×3=21 ✔

All except one follow multiplication.

But 7×7=49, not 48.

Unless it's sum of squares?

7² + 7² = 49 + 49 = 98 → no.

Or average? 7+7=14 → no.

Wait — could it be that the red square between 7 and 7 is actually not 48?

Wait — maybe I misread the value.

Looking at the image:

- Red square between Yellow 7 and Yellow 7 → labeled "48"

But 7×7=49.

Could it be a typo?

Alternatively, maybe the pattern is (a + b) × something?

Wait — try another idea.

What if the red squares are the product of the two adjacent yellows, but only when they are different? But that doesn’t make sense.

Wait — perhaps the red square between 7 and 7 is not 48, but 49?

But it says 48.

Alternatively, maybe the pattern is a × b - 1?

For 7 and 7: 49 - 1 = 48 → yes!

Then check others:

- 3 and 9: 3×9 = 27 → 27 - 1 = 26? But we don't know yet.
- But 9 and 5: 45 → 45 - 1 = 44 ≠ 45 → no.

So that doesn’t work.

Wait — only 7 and 7 gives 48, which is 49 - 1.

But others are exact products.

So why is this one off?

Wait — maybe I have the wrong pair.

Wait — let’s double-check the sequence.

Perhaps the red square labeled 48 is not between 7 and 7.

Let’s go clockwise:

1. Yellow 3
2. Red ? → between 3 and 9
3. Yellow 9
4. Red 45 → between 9 and 5
5. Yellow 5
6. Red 40 → between 5 and 8
7. Yellow 8
8. Red 56 → between 8 and 7
9. Yellow 7
10. Red 48 → between 7 and 7
11. Yellow 7
12. Red 21 → between 7 and 3
13. Yellow 3

Yes, that’s correct.

So:

- 9×5 = 45 ✔
- 5×8 = 40 ✔
- 8×7 = 56 ✔
- 7×7 = 49 → but given 48 ✘
- 7×3 = 21 ✔
- 3×9 = 27 → so ? = 27

But why is 7×7 = 48 instead of 49?

Unless there’s a different pattern.

Wait — maybe the red squares are not products, but something else.

Let’s list all red values and their neighbors:

| Red | Left Yellow | Right Yellow | Product | Sum |
|-----|-------------|--------------|--------|-----|
| ? | 3 | 9 | 27 | 12 |
| 45 | 9 | 5 | 45 | 14 |
| 40 | 5 | 8 | 40 | 13 |
| 56 | 8 | 7 | 56 | 15 |
| 48 | 7 | 7 | 49 | 14 |
| 21 | 7 | 3 | 21 | 10 |

Now notice:

- 45 = 9×5
- 40 = 5×8
- 56 = 8×7
- 21 = 7×3
- ? = 3×9 = 27
- 48 vs 7×7 = 49 → off by 1

But 48 is very close to 49.

Is it possible that the red square between 7 and 7 is supposed to be 49, but printed as 48? Unlikely.

Wait — maybe the red squares are sums of squares?

7² + 7² = 49 + 49 = 98 → no.

Or (a + b)²? (7+7)² = 196 → no.

Another idea: maybe the red squares represent a × b, but the values are swapped?

Wait — look at the red square between 7 and 7: 48

But 7×7 = 49.

Wait — could it be (a + b) × c? No clear c.

Wait — what if the red squares are a × b, but the 48 is actually for a different pair?

Wait — perhaps the red square labeled 48 is between Yellow 7 and Yellow 7, but maybe it's not multiplication.

Wait — let’s try to see if there’s a pattern in the red values:

List red values: ?, 45, 40, 56, 48, 21

But we know:

- 45 = 9×5
- 40 = 5×8
- 56 = 8×7
- 48 = ? → but 7×7 = 49
- 21 = 7×3

Wait — 7×7 = 49, but it's 48 — difference of 1.

But others are exact.

Unless it's a typo, and it should be 49.

But then ? = 3×9 = 27.

But let’s suppose it’s not a typo.

Alternative idea: maybe the red squares are a × b, but only if a ≠ b, otherwise something else.

But 7 and 7 are equal, so maybe different rule?

But 48 is close to 49.

Wait — what if the red square between 7 and 7 is 7×7 - 1 = 48, and the others are exact?

But then why only this one?

That seems arbitrary.

Wait — perhaps the pattern is a × b, and the 48 is correct, but the pair is not 7 and 7?

Wait — is it possible that the red square labeled 48 is between Yellow 7 and Yellow 7, but maybe the value is not from multiplication?

Wait — let’s try to see if there’s a different pattern.

Look at the red values:

- 21: between 7 and 3 → 7×3 = 21
- 45: between 9 and 5 → 9×5 = 45
- 40: between 5 and 8 → 5×8 = 40
- 56: between 8 and 7 → 8×7 = 56
- 48: between 7 and 7 → 7×7 = 49 → but given 48
- ?: between 3 and 9 → 3×9 = 27

So five out of six are products.

Only one discrepancy.

Maybe it's a typo in the image, and 48 should be 49.

But let’s suppose it’s not.

Another possibility: maybe the red squares are a × b, but the yellow circles are not the immediate neighbors?

No, they are.

Wait — could it be that the red squares are a × b, but the numbers are read in a different order?

Wait — what if the red square between 7 and 7 is actually between Yellow 7 and Yellow 7, but the value is 48 because of a different rule?

Wait — let’s calculate the differences:

- 7×7 = 49 → 48 → difference of 1
- All others match exactly.

So perhaps the pattern is: product of the two adjacent yellow numbers

Then:

- ? = 3 × 9 = 27

And the 48 might be a typo, or perhaps there’s a different explanation.

But let’s consider: is there any other way?

Wait — what if the red squares are sum of the two yellows times something?

No.

Wait — look at the values:

- 21 = 3×7
- 45 = 5×9
- 40 = 5×8
- 56 = 7×8
- 48 = 6×8 or 6×8=48, but we have 7 and 7
- 21 = 3×7

Wait — 48 = 6×8, but we have 7 and 7.

Not matching.

Wait — perhaps the red square labeled 48 is not between 7 and 7, but between 8 and 6? But there’s no 6.

No.

Wait — maybe the red square labeled 48 is between Yellow 7 and Yellow 7, but the value is 48 because it’s (7+1)×6 = 48? No.

Another idea: maybe the red squares are a × b, but the yellow circles are not the ones directly adjacent?

No.

Wait — let’s try to see if the red squares are a × b, and accept that 7×7=49, but it’s written as 48, so maybe it’s a typo.

In that case, the missing one is 3×9 = 27.

But let’s see if there’s a different pattern.

Wait — look at the sequence of red values:

- ?
- 45
- 40
- 56
- 48
- 21

And the yellow pairs:

- 3,9 → ?
- 9,5 → 45
- 5,8 → 40
- 8,7 → 56
- 7,7 → 48
- 7,3 → 21

Now, notice that 45, 40, 56, 48, 21 are all products, except possibly 48.

But 7×7=49.

Unless it’s not multiplication.

Wait — what if the red squares are a × b, but the yellow numbers are not the ones on the sides, but something else?

No.

Wait — perhaps the red squares are a × b, and the only anomaly is 48, but maybe it’s meant to be 49.

Given that all others fit perfectly, it’s likely that the missing value is 3×9 = 27.

Moreover, 3×9 = 27, and it fits the pattern.

So even though 7×7=49 but shown as 48, perhaps it’s a typo, or perhaps there’s a different rule.

But let’s consider: is there a possibility that the red squares are a × b, and the 48 is correct because of a different reason?

Wait — what if the red square between 7 and 7 is not 48, but the value is 48 because of a different calculation?

Wait — perhaps the pattern is a × b, but only when a and b are different, and when a=b, it’s a² - 1?

But that’s arbitrary.

Alternatively, maybe the red squares are a × b, and the image has a typo.

Given that five out of six follow the multiplication rule, it’s reasonable to conclude that the missing value is 3 × 9 = 27.

Therefore, the answer is:

27

✔ Final Answer: 27