Number puzzle arranged in a star pattern with some cells filled and others blank.

Let's solve this puzzle step by step.

We are given a grid of numbers with some missing values. The goal is to determine the pattern or rule that governs the numbers and fill in the blanks. The final number in the bottom-right corner is 94, which may be a clue.

Here's the grid:

```
Row 1: 6 12 9 18 15 30
Row 2: ? 16 13 26 ? 46
Row 3: 11 22 ? 38 35 ?
Row 4: 12 ? 21 ? 29 ?
Row 5: 9 ? ? ? ? ?
Row 6: ? ? ? ? ? 94
```

Let’s denote the grid as `G[row][col]`, with rows from 1 to 6 and columns from 1 to 6.

---

Step 1: Look for patterns

#### Column-wise analysis

Let’s look at each column and see if we can find a pattern.

---

Column 1:
- Row 1: 6
- Row 2: ?
- Row 3: 11
- Row 4: 12
- Row 5: 9
- Row 6: ?

Values: 6, ?, 11, 12, 9, ?

Not obvious yet.

---

Column 2:
- 12, 16, 22, ?, ?, ?

Look at differences:
- 12 → 16 (+4)
- 16 → 22 (+6)

Maybe increasing by +4, +6, then +8, etc.? Let’s test:
- Next: 22 + 8 = 30 → Row 4: ?
- Then 30 + 10 = 40 → Row 5: ?
- Then 40 + 12 = 52 → Row 6: ?

But we don’t know if that holds.

Alternatively, maybe relate to row numbers?

Wait — perhaps there's a pattern across rows or diagonals?

---

Try looking at diagonals or arithmetic relationships

Another idea: Maybe the sum of certain cells relates to others?

Let’s look at Row 1:
6, 12, 9, 18, 15, 30

Notice:
- 6 × 2 = 12
- 9 × 2 = 18
- 15 × 2 = 30

So in Row 1, every odd-positioned number (1st, 3rd, 5th) multiplied by 2 gives the even-positioned one.

That is:
- Col1: 6 → Col2: 12
- Col3: 9 → Col4: 18
- Col5: 15 → Col6: 30

So even columns = 2 × odd columns in Row 1.

Now check Row 2:
- Col2: 16 → so Col1 should be 8? (since 8×2=16)
- Col4: 26 → so Col3 should be 13 → but it's already 13 ✔
- Col6: 46 → so Col5 should be 23

So:
- Col1 = 8
- Col5 = 23

So Row 2: 8, 16, 13, 26, 23, 46

This fits the same pattern: even columns = 2 × odd columns.

Let’s verify:
- 8×2=16 ✔
- 13×2=26 ✔
- 23×2=46 ✔

Great!

So Pattern: In each row, even-numbered columns = 2 × odd-numbered columns

So for any row:
- G[r][2] = 2 × G[r][1]
- G[r][4] = 2 × G[r][3]
- G[r][6] = 2 × G[r][5]

So we can use this to fill in missing values.

Let’s apply this to other rows.

---

Row 3:

Given: 11, 22, ?, 38, 35, ?

From pattern:
- Col2 = 2 × Col1 → 22 = 2×11 → ✔
- Col4 = 2 × Col3 → 38 = 2 × ? → So ? = 19
- Col6 = 2 × Col5 → ? = 2 × 35 = 70

So Row 3: 11, 22, 19, 38, 35, 70

---

Row 4:

Given: 12, ?, 21, ?, 29, ?

Apply pattern:
- Col2 = 2 × Col1 = 2×12 = 24
- Col4 = 2 × Col3 = 2×21 = 42
- Col6 = 2 × Col5 = 2×29 = 58

So Row 4: 12, 24, 21, 42, 29, 58

---

Row 5:

Given: 9, ?, ?, ?, ?, ?

We have only Col1 = 9

Then:
- Col2 = 2×9 = 18
- Col3 = ? → let's call it x → then Col4 = 2x
- Col5 = ? → call it y → then Col6 = 2y

So Row 5: 9, 18, x, 2x, y, 2y

We don't have enough info yet.

---

Row 6:

Only Col6 = 94

From pattern: Col6 = 2 × Col5 → so Col5 = 94 / 2 = 47

Then Col4 = 2 × Col3 → so if Col3 = z, then Col4 = 2z

Col2 = 2 × Col1 → if Col1 = a, then Col2 = 2a

So Row 6: a, 2a, z, 2z, 47, 94

Still unknowns.

---

Now, perhaps we can find column-wise patterns or sums?

Wait — another idea: Sum of all entries?

We know the total sum might be related to 94?

But 94 is just the last cell.

Wait — could the total sum of the entire grid be related?

But we don't know most values yet.

Alternatively, maybe each column has a pattern?

Let’s go back and list what we have now.

Updated Grid:

| | C1 | C2 | C3 | C4 | C5 | C6 |
|-------|----|----|----|----|----|----|
| R1 | 6 | 12 | 9 | 18 | 15 | 30 |
| R2 | 8 | 16 | 13 | 26 | 23 | 46 |
| R3 | 11 | 22 | 19 | 38 | 35 | 70 |
| R4 | 12 | 24 | 21 | 42 | 29 | 58 |
| R5 | 9 | 18 | x | 2x | y | 2y |
| R6 | a | 2a | z | 2z | 47 | 94 |

Now, let’s look at column sums or progressions.

Try Column 1:
R1: 6
R2: 8
R3: 11
R4: 12
R5: 9
R6: a

Sequence: 6, 8, 11, 12, 9, a

No clear arithmetic progression.

But maybe not needed.

Wait — another idea: Are the numbers in each column following a pattern?

Let’s try Column 6:
R1: 30
R2: 46
R3: 70
R4: 58
R5: 2y
R6: 94

Sequence: 30, 46, 70, 58, 2y, 94

Differences:
- 46 - 30 = 16
- 70 - 46 = 24
- 58 - 70 = -12
- 2y - 58 = ?
- 94 - 2y = ?

Not helpful.

Wait — maybe row sums?

Try Row 1 sum: 6+12+9+18+15+30 = 90
Row 2: 8+16+13+26+23+46 = 132
Row 3: 11+22+19+38+35+70 = 195
Row 4: 12+24+21+42+29+58 = 196
Row 5: 9+18+x+2x+y+2y = 27 + 3x + 3y = 3(9 + x + y)
Row 6: a + 2a + z + 2z + 47 + 94 = 3a + 3z + 141 = 3(a+z+47)

Not obviously helpful.

Wait — maybe the last column has a pattern?

Column 6:
- R1: 30
- R2: 46
- R3: 70
- R4: 58
- R5: 2y
- R6: 94

Hmm.

But notice: R6 is 94 — maybe it's the sum of something?

Wait — perhaps the entire grid is built so that the sum of all cells equals 94? But that seems too small.

Total of known values:
- R1: 90
- R2: 132
- R3: 195
- R4: 196
- R5: 27 + 3x + 3y
- R6: 3a + 3z + 141

Already R1 to R4 sum to: 90+132=222; +195=417; +196=613

Even without R5 and R6, we’re at 613 — and 94 is less than that — so no.

So 94 is not the total.

Wait — perhaps the sum of the last column is 94?

But R6 is 94, and others are already larger: 30+46+70+58 = 204 > 94 → no.

So not that.

Alternative idea: Is there a diagonal pattern?

Let’s look at main diagonal (top-left to bottom-right):

- R1C1: 6
- R2C2: 16
- R3C3: 19
- R4C4: 42
- R5C5: y
- R6C6: 94

Sequence: 6, 16, 19, 42, y, 94

Differences:
- 16-6=10
- 19-16=3
- 42-19=23
- y-42=?
- 94-y=?

Not helpful.

Other diagonal?

Try anti-diagonal: R1C6=30, R2C5=23, R3C4=38, R4C3=21, R5C2=18, R6C1=a

Sequence: 30, 23, 38, 21, 18, a

No pattern.

Back to the row pattern — we have a strong pattern: in each row, even columns are double the odd ones.

So far we’ve filled:
- R2: 8,16,13,26,23,46
- R3: 11,22,19,38,35,70
- R4: 12,24,21,42,29,58

Now, maybe column-wise, we can find more.

Let’s look at Column 1:
R1: 6
R2: 8
R3: 11
R4: 12
R5: 9
R6: a

Sequence: 6, 8, 11, 12, 9, a

What could a be?

Look at differences:
- 6→8: +2
- 8→11: +3
- 11→12: +1
- 12→9: -3
- 9→a: ?

No clear pattern.

But wait — maybe Column 3?

Column 3:
R1: 9
R2: 13
R3: 19
R4: 21
R5: x
R6: z

Sequence: 9, 13, 19, 21, x, z

Differences:
- 13-9=4
- 19-13=6
- 21-19=2
- x-21=?
- z-x=?

Not obvious.

But look at Column 5:
R1: 15
R2: 23
R3: 35
R4: 29
R5: y
R6: 47

Sequence: 15, 23, 35, 29, y, 47

Differences:
- 23-15=8
- 35-23=12
- 29-35=-6
- y-29=?
- 47-y=?

Not clear.

Wait — maybe each column has alternating increase/decrease?

Another idea: Perhaps the sum of each pair of adjacent rows has a pattern?

But maybe not.

Wait — here’s a new idea: Maybe the entire grid satisfies a linear relationship?

Or perhaps each column follows an arithmetic sequence?

Let’s test Column 1:
6, 8, 11, 12, 9, a

Not arithmetic.

But look at Column 2:
R1: 12
R2: 16
R3: 22
R4: 24
R5: 18
R6: 2a

Sequence: 12, 16, 22, 24, 18, 2a

Differences:
- +4, +6, +2, -6, ?

Not helpful.

Wait — maybe look at the difference between consecutive rows in a column?

For example, Column 3:
- R1: 9
- R2: 13 → +4
- R3: 19 → +6
- R4: 21 → +2
- R5: x → ?
- R6: z → ?

+4, +6, +2 — maybe next is +4 or +6?

But then R5 would be 21+? → 25 or 27?

But we need more.

Wait — let’s consider Row 5 and Row 6.

We have:
- Row 5: 9, 18, x, 2x, y, 2y
- Row 6: a, 2a, z, 2z, 47, 94

And we know R6C5 = 47, R6C6 = 94 → consistent with pattern.

Now, perhaps there is a pattern in the values across rows for each position.

Let’s look at Column 5 again:
- R1: 15
- R2: 23
- R3: 35
- R4: 29
- R5: y
- R6: 47

List: 15, 23, 35, 29, y, 47

Let’s see if there’s a pattern.

Note: 15, 23, 35, 29, ?, 47

Check differences:
- 23-15=8
- 35-23=12
- 29-35=-6
- y-29=?
- 47-y=?

Now, 8, 12, -6 — maybe next is +6? Then y = 29+6=35 → then 47-35=12 → possible?

But then sequence: 15,23,35,29,35,47

But R3C5 is already 35, R5C5=35 — possible.

Then R5C5 = 35 → so R5C6 = 70

But R5C5 = y = 35 → R5C6 = 70

Then R5: 9, 18, x, 2x, 35, 70

Now, can we find x?

Look at Column 3:
R1: 9
R2: 13
R3: 19
R4: 21
R5: x
R6: z

Suppose x = 25? Then R5C4 = 50

But we need more.

Wait — look at Column 4:
R1: 18
R2: 26
R3: 38
R4: 42
R5: 2x
R6: 2z

So: 18, 26, 38, 42, 2x, 2z

Differences:
- 26-18=8
- 38-26=12
- 42-38=4
- 2x-42=?
- 2z-2x=?

8, 12, 4 — maybe next is 8? Then 2x = 42+8=50 → x=25

Then next: 2z = 50 + ? — if pattern continues, maybe +12? → 62 → z=31

Let’s try that.

So assume:
- R5C3 = x = 25 → R5C4 = 50
- R6C3 = z = 31 → R6C4 = 62

Then R5: 9, 18, 25, 50, 35, 70

R6: a, 2a, 31, 62, 47, 94

Now, check Column 1:
R1: 6
R2: 8
R3: 11
R4: 12
R5: 9
R6: a

Sequence: 6,8,11,12,9,a

Now, look at Column 5:
R1:15, R2:23, R3:35, R4:29, R5:35, R6:47

Now differences:
- 15→23: +8
- 23→35: +12
- 35→29: -6
- 29→35: +6
- 35→47: +12

Oh! Pattern: +8, +12, -6, +6, +12

Wait — not symmetric.

But look: +8, +12, -6, +6, +12 — maybe next would be -6?

But we’re done.

Wait — perhaps alternating operations?

But not clear.

But let’s see if Column 1 has a pattern.

6,8,11,12,9,a

Let’s see if it relates to other columns.

Wait — maybe Row 6 must satisfy the pattern, and we can find a.

We have R6: a, 2a, 31, 62, 47, 94

We also have R5: 9,18,25,50,35,70

Now, look at Column 1:
6,8,11,12,9,a

Let’s see if there’s a pattern in the differences between rows.

R1 to R2: +2
R2 to R3: +3
R3 to R4: +1
R4 to R5: -3
R5 to R6: a-9

Could be alternating: +2,+3,+1,-3, then maybe +2?

Then a = 9+2 = 11

Then R6C1 = 11 → R6C2 = 22

But R3C1 is already 11 — okay.

So a = 11

Then R6: 11, 22, 31, 62, 47, 94

Now check if this makes sense.

Now Column 1: 6,8,11,12,9,11

Sequence: 6,8,11,12,9,11

Still not perfect, but possible.

Now Column 2:
R1:12
R2:16
R3:22
R4:24
R5:18
R6:22

Sequence: 12,16,22,24,18,22

Differences: +4,+6,+2,-6,+4

Possible.

Now Column 3:
9,13,19,21,25,31

Differences: +4,+6,+2,+4,+6 → nice! Alternating +4,+6,+2,+4,+6

Wait:
- 9→13: +4
- 13→19: +6
- 19→21: +2
- 21→25: +4
- 25→31: +6

So pattern: +4, +6, +2, +4, +6 — not regular.

But maybe it's two interleaved sequences?

Odd rows: R1:9, R3:19, R5:25 → 9,19,25 → +10, +6

Even rows: R2:13, R4:21, R6:31 → 13,21,31 → +8, +10

Not helpful.

But let’s accept the current values and see if they fit.

Now, let’s write the full grid:

```
Row 1: 6 12 9 18 15 30
Row 2: 8 16 13 26 23 46
Row 3: 11 22 19 38 35 70
Row 4: 12 24 21 42 29 58
Row 5: 9 18 25 50 35 70
Row 6: 11 22 31 62 47 94
```

Now, check if all even columns are double odd columns:

- Row 5: 9,18 → 2×9=18 ✔; 25,50 → 2×25=50 ✔; 35,70 → 2×35=70 ✔
- Row 6: 11,22 → 2×11=22 ✔; 31,62 → 2×31=62 ✔; 47,94 → 2×47=94 ✔

Perfect.

Now, is there any contradiction?

Check Column 5:
15, 23, 35, 29, 35, 47

No issue.

But notice: Row 5 and Row 6 both have 35 in C5, and 70 in C6 — okay.

Now, is there a reason to believe our assumption is correct?

We assumed:
- x = 25 (from Column 4 pattern: +8,+12,+4,+8 → then +12?)
- Then z = 31
- a = 11

But how did we get x=25?

From Column 4:
18, 26, 38, 42, 50, 62

Differences: +8, +12, +4, +8, +12

Yes! That’s a pattern: +8, +12, +4, +8, +12

So:
- R1→R2: +8
- R2→R3: +12
- R3→R4: +4
- R4→R5: +8 → so 42+8=50 → so R5C4=50 → R5C3=25
- R5→R6: +12 → 50+12=62 → R6C4=62 → R6C3=31

Perfect!

Similarly, Column 3:
9, 13, 19, 21, 25, 31

Differences: +4, +6, +2, +4, +6

So pattern: +4, +6, +2, +4, +6

So R5C3 = 21+4=25, R6C3=25+6=31 — matches.

Now Column 1:
6, 8, 11, 12, 9, 11

Differences: +2, +3, +1, -3, +2

Not as clean, but maybe acceptable.

Now Column 2:
12, 16, 22, 24, 18, 22

Differences: +4, +6, +2, -6, +4

Symmetric: +4, +6, +2, -6, +4

Interesting.

Now Column 5:
15, 23, 35, 29, 35, 47

Differences: +8, +12, -6, +6, +12

Not symmetric, but okay.

Now, Column 6:
30, 46, 70, 58, 70, 94

Differences: +16, +24, -12, +12, +24

Pattern: +16, +24, -12, +12, +24

So R5C6 = 58 + 12 = 70 → yes

R6C6 = 70 + 24 = 94 → yes

Perfect!

So the pattern in Column 6 is: +16, +24, -12, +12, +24

Why? Not sure, but it works.

So our solution is consistent.

Now, we can fill in the blanks.

Original grid had blanks:

```
6 12 9 18 15 30
? 16 13 26 ? 46
11 22 ? 38 35 ?
12 ? 21 ? 29 ?
9 ? ? ? ? ?
? ? ? ? ? 94
```

Now fill in:

- Row 2: C1 = 8, C5 = 23
- Row 3: C3 = 19, C6 = 70
- Row 4: C2 = 24, C4 = 42, C6 = 58
- Row 5: C2 = 18, C3 = 25, C4 = 50, C5 = 35, C6 = 70
- Row 6: C1 = 11, C2 = 22, C3 = 31, C4 = 62, C5 = 47

So final grid:

```
6 12 9 18 15 30
8 16 13 26 23 46
11 22 19 38 35 70
12 24 21 42 29 58
9 18 25 50 35 70
11 22 31 62 47 94
```

All blanks filled.

Final Answer:

The missing values are:

- Row 2, Col1: 8
- Row 2, Col5: 23
- Row 3, Col3: 19
- Row 3, Col6: 70
- Row 4, Col2: 24
- Row 4, Col4: 42
- Row 4, Col6: 58
- Row 5, Col2: 18
- Row 5, Col3: 25
- Row 5, Col4: 50
- Row 5, Col5: 35
- Row 5, Col6: 70
- Row 6, Col1: 11
- Row 6, Col2: 22
- Row 6, Col3: 31
- Row 6, Col4: 62
- Row 6, Col5: 47

And the key pattern is:
> In each row, the value in an even column is twice the value in the preceding odd column.

Additionally, some columns show additive patterns (e.g., Column 4: +8, +12, +4, +8, +12), but the row doubling rule is the primary one.

✔ All constraints satisfied.