Multiplication puzzle - Free Printable

Let’s solve each multiplication puzzle one by one. Each puzzle is a 3x3 grid with some numbers filled in, and the goal is to fill in the blanks so that:

- The product of the first row (left two boxes) equals the third box in that row.
- The product of the second row (left two boxes) equals the third box in that row.
- The product of the first column (top two boxes) equals the bottom box in that column.
- The product of the second column (top two boxes) equals the bottom box in that column.
- Also, the top-right number should equal the product of the top-left and top-middle? Wait — actually, looking at the structure, it seems like:

Actually, let me re-express the layout clearly.

Each puzzle has this shape:

```
[A]
[B][C][D]
[E][F][G]
[H][I][J]
```

But wait — looking at the first puzzle:

Top: 42
Then row: _ , 6 , 24
Then row: _ , _ , 49
Bottom row: 28, 42, 28

Wait — perhaps it's better to think of it as a 3x3 grid where:

Row 1: A B C → but only B and C are given sometimes? Actually, no.

Looking again — each puzzle is arranged like this:

There is a single number on top (above the 3x3 grid), then a 3x3 grid below it.

But actually, from the layout, it looks like:

For each puzzle, there is a 3x3 grid of boxes, and above the rightmost column, there’s a number; to the left of the bottom row, there’s a number? No.

Wait — let’s take the very first puzzle:

It shows:

42
□ 6 24
□ □ 49
28 42 28

So actually, the “42” on top is probably the product of the top row? But top row is □, 6, 24 — if 6 * something = 24, then that something is 4. So top-left is 4.

Then middle row: □, □, 49 — and bottom row is 28, 42, 28.

Also, columns: first column: top=4, middle=?, bottom=28 → so 4 * ? = 28 → ? = 7.

Second column: top=6, middle=?, bottom=42 → 6 * ? = 42 → ? = 7.

Third column: top=24, middle=49, bottom=28 — but 24 * ? = 49? That doesn’t work. Wait, maybe the bottom number is the product of the two above it in the column?

Check: third column: top=24, middle=49, bottom=28 — 24 * 49 is way bigger than 28. Doesn't make sense.

Alternative interpretation: Perhaps the number above the grid (like 42) is the product of the entire top row? But top row is three numbers: let’s say X, 6, 24. If X * 6 * 24 = 42? That would be tiny, impossible.

Wait — look at the bottom row: 28, 42, 28. And the rightmost column ends with 28. Maybe the bottom row is the product of the columns?

Let me try this: For each column, the bottom number is the product of the two numbers above it in that column.

First column: top = ?, middle = ?, bottom = 28 → so ? * ? = 28.

Second column: top = 6, middle = ?, bottom = 42 → 6 * ? = 42 → ? = 7.

Third column: top = 24, middle = 49, bottom = 28 — 24 * 49 ≠ 28. Not possible.

Another idea: Maybe the number on top (42) is the product of the first column? First column: top=?, middle=?, bottom=28 — too many unknowns.

Let’s look at the second puzzle for clarity:

Top: 9
Grid:
□ 3 3
□ □ 9
3 9 3

Bottom row: 3,9,3

If we assume that for each row, the first two numbers multiply to give the third:

Row 1: ? * 3 = 3 → ? = 1

Row 2: ? * ? = 9

Row 3: 3 * 9 = 27, but it says 3 — not matching.

Unless the bottom row is not part of the multiplication, but rather the product of the columns?

Column 1: top=1 (from above), middle=?, bottom=3 → 1 * ? = 3 → ? = 3

Column 2: top=3, middle=?, bottom=9 → 3 * ? = 9 → ? = 3

Column 3: top=3, middle=9, bottom=3 — 3*9=27≠3. Still not.

Wait — what if the bottom row is the product of the rows above? No.

Let’s read the title: "MULTIPLICATION PUZZLE"

Perhaps it's like a magic square but with multiplication.

Another approach: Look at the last number in each row and column.

In the first puzzle:

After filling top-left as 4 (since 4*6=24), then for the first column: 4 * x = 28 → x=7

Second column: 6 * y = 42 → y=7

Then middle row is 7,7,49 — and 7*7=49, perfect!

Then bottom row: 28,42,28 — which are the products of the columns: col1: 4*7=28, col2:6*7=42, col3:24*49? 24*49 is 1176, not 28. Oh no.

But 24 and 49 are in the same column? Third column: top=24, middle=49, bottom=28 — that doesn't multiply.

Unless the bottom number is not the product of the column, but something else.

Notice that in the bottom row, the numbers are 28,42,28 — and 28 is also the first and last of the bottom row.

Also, the top number is 42, which matches the middle of the bottom row.

Perhaps the top number is the product of the middle column? Middle column: 6,7,42 — 6*7=42, yes! And the bottom middle is 42, which is consistent.

Similarly, for the first column: 4,7,28 — 4*7=28, and bottom is 28.

Third column: 24,49,28 — but 24*49 is not 28. However, 24 and 49 are both divisible by... 24=8*3, 49=7*7, no common factors.

Wait a minute — perhaps the bottom number in each column is the product of the two above it, but for the third column, it's different? That can't be.

Let's calculate 24 * 49 = 1176, and 1176 / 42 = 28? 42*28=1176, yes! 42*28=1176.

Oh! So perhaps the bottom number is not directly the product, but related to the top number.

Recall that the top number is 42, and for the third column, top=24, middle=49, bottom=28.

And 24 * 49 = 1176, and 1176 / 42 = 28. Yes! Because 42 * 28 = 1176.

Similarly, for the first column: top=4, middle=7, bottom=28, and 4*7=28, and 28 / 42 is not integer, but wait.

Perhaps the rule is: for each column, the product of the three numbers in the column equals the top number times the bottom number or something.

Let's formalize.

From the first puzzle:

We have:

Let me denote the grid as:

Row 1: A B C

Row 2: D E F

Row 3: G H I

With additional numbers: T on top (above C?), and perhaps L on left, R on right, but in the image, for first puzzle, T=42, and bottom row is G,H,I = 28,42,28.

From earlier, we deduced A=4 (because B*C=6*4=24? No, in row 1, it's A,B,C with C=24, B=6, so A*6=24 → A=4.

Then for column 1: A,D,G with G=28, A=4, so 4*D=28 → D=7.

Column 2: B,E,H with H=42, B=6, so 6*E=42 → E=7.

Then row 2: D,E,F =7,7,F and F=49, and 7*7=49, good.

Now column 3: C,F,I =24,49,I and I=28.

But 24*49=1176, and 1176 / 42 = 28, and 42 is T, the top number.

Similarly, for row 3: G,H,I=28,42,28, and 28*42=1176, and 1176 / 42 = 28, which is I, but not consistent.

Notice that T * I = 42 * 28 = 1176 = C * F = 24 * 49.

Similarly, for other columns.

In column 1: A*D*G =4*7*28=784, and T*G=42*28=1176, not equal.

Perhaps the product of the entire grid or something.

Another idea: perhaps the top number T is such that for each row, the product of the first two equals the third, and for each column, the product of the first two equals the third, and additionally, T is the product of the diagonal or something.

Let's check the second puzzle to verify.

Second puzzle:

T=9

Grid:

A B C = ? 3 3

D E F = ? ? 9

G H I = 3 9 3

Assume row 1: A*3=3 → A=1

Row 2: D*E=9

Row 3: 3*9=27, but I=3, not 27. Contradiction.

Unless the bottom row is not used for row multiplication, but for column multiplication.

Column 1: A,D,G =1,D,3 → 1*D=3 → D=3

Column 2: B,E,H =3,E,9 → 3*E=9 → E=3

Then row 2: D,E,F=3,3,9 and 3*3=9, good.

Column 3: C,F,I=3,9,3 — 3*9=27, but I=3, not 27.

But 3*9=27, and 27 / 9 = 3, and T=9, so again, C*F / T = I? 3*9/9=3, yes!

Similarly, in first puzzle, C*F/T =24*49/42=1176/42=28=I, yes.

Also, for row 3: G*H/I should be something, but in first puzzle G=28,H=42,I=28, 28*42/28=42, which is T.

In second puzzle, G=3,H=9,I=3, 3*9/3=9=T, yes!

So the rules are:

For each row i, the product of the first two cells equals the third cell.

For each column j, the product of the first two cells equals the third cell.

Additionally, the top number T satisfies: for the third column, C * F / T = I, but since I is already determined by the column rule, it must be consistent.

Actually, from the column rule, for column 3, C * F = I * K for some K, but in this case, it's C * F = T * I.

Similarly, for the bottom row, G * H = T * I? In first puzzle, 28*42=1176, T*I=42*28=1176, yes.

In second puzzle, G*H=3*9=27, T*I=9*3=27, yes.

So general rules:

1. For each row, left * middle = right

2. For each column, top * middle = bottom

3. Additionally, the top number T is such that T * (bottom-right) = (top-right) * (middle-right) , but since from column 3, top-right * middle-right = bottom-right * something, actually from rule 2, for column 3, C * F = I * ? no.

From rule 2 applied to column 3: C * F = I * X, but in the grid, the bottom of column 3 is I, so according to rule 2, the product of the first two in the column should equal the third, so C * F should equal I, but in first puzzle 24*49=1176 ≠ 28, so rule 2 does not apply to column 3 in the same way.

I think I made a mistake.

Let's redefine based on the puzzles.

From the first puzzle, after filling:

Row 1: 4, 6, 24 → 4*6=24

Row 2: 7, 7, 49 → 7*7=49

Row 3: 28, 42, 28 — but 28*42=1176 ≠ 28, so row 3 is not following the same rule.

However, for columns:

Col 1: 4,7,28 → 4*7=28

Col 2: 6,7,42 → 6*7=42

Col 3: 24,49,28 — 24*49=1176, not 28.

But 24*49 = 1176, and 1176 / 42 = 28, and 42 is the top number.

Also, 28*42 = 1176, same thing.

Moreover, the top number 42 is equal to the bottom-middle number.

In fact, in all puzzles, the top number is equal to the bottom-middle number of the grid.

In first puzzle: top=42, bottom-middle=42

Second puzzle: top=9, bottom-middle=9

Third puzzle: top=81, bottom-middle=9? Let's see third puzzle.

Third puzzle:

Top: 81

Grid:

2 □ 18

□ □ 9

18 9 2

Bottom row: 18,9,2

So bottom-middle is 9, but top is 81, not equal. 81 vs 9, not equal.

In third puzzle, bottom row is 18,9,2, so middle is 9, top is 81.

But 81 / 9 = 9, not sure.

Let's solve third puzzle.

Row 1: 2, B, 18 → 2*B=18 → B=9

Row 2: D, E, 9 → D*E=9

Row 3: 18,9,2 — 18*9=162 ≠2, so not following row rule.

Columns:

Col 1: 2,D,18 → 2*D=18 → D=9

Col 2: B,E,9 =9,E,9 → 9*E=9 → E=1

Then row 2: D,E,F=9,1,9 and 9*1=9, good.

Col 3: 18,9,2 — 18*9=162, not 2.

But 18*9=162, and 162 / 81 = 2, and top is 81, so C*F / T = I? 18*9/81=162/81=2=I, yes.

Also, bottom row: G,H,I=18,9,2, and G*H / T =18*9/81=162/81=2=I, same.

And T=81, while bottom-middle H=9, and 81 / 9 =9, not directly related.

But notice that in this case, T = 81, and H=9, and 9*9=81, but not always.

In first puzzle, T=42, H=42, so T=H.

In second puzzle, T=9, H=9, T=H.

In third puzzle, T=81, H=9, T≠H.

What's special about third puzzle? Bottom row is 18,9,2, and 18*2=36, not related.

Perhaps T is the product of the diagonals or something.

Another observation: in all cases, for the third column, C * F = T * I

In first puzzle: 24*49=1176, T*I=42*28=1176

Second puzzle: C=3, F=9, T=9, I=3, 3*9=27, 9*3=27

Third puzzle: C=18, F=9, T=81, I=2, 18*9=162, 81*2=162, yes!

Similarly, for the bottom row, G * H = T * I

First puzzle: 28*42=1176, 42*28=1176

Second: 3*9=27, 9*3=27

Third: 18*9=162, 81*2=162, yes.

Also, for the first two columns, the column rule holds: top * middle = bottom

And for the first two rows, left * middle = right

For the third row and third column, they are determined by the above, but with the relation involving T.

Specifically, once the first two rows and first two columns are filled using the multiplication rules, then the bottom-right cell I can be found from C * F = T * I, or from G * H = T * I, and it should be consistent.

In practice, for solving, we can use the row and column rules for the first two rows and first two columns to fill most cells, then use the top number to find the missing ones if needed, but in the puzzles, usually the bottom-right is given or can be calculated.

In the given puzzles, all cells except some are given, so we can solve step by step.

Let's solve each puzzle systematically.

Start with Puzzle 1:

Top: 42

Grid:

Row 1: A, 6, 24 → A * 6 = 24 → A = 4

Row 2: D, E, 49 → D * E = 49

Row 3: 28, 42, 28

Column 1: A, D, 28 → 4 * D = 28 → D = 7

Then from row 2: 7 * E = 49 → E = 7

Column 2: 6, E, 42 → 6 * 7 = 42, good.

Column 3: 24, 49, 28 — check if 24 * 49 = 42 * 28? 24*49=1176, 42*28=1176, yes.

Bottom row: 28,42,28 — 28*42=1176, 42*28=1176, good.

So filled grid:

4 6 24

7 7 49

28 42 28

Puzzle 2:

Top: 9

Grid:

Row 1: A, 3, 3 → A*3=3 → A=1

Row 2: D, E, 9 → D*E=9

Row 3: 3, 9, 3

Column 1: A,D,3 → 1*D=3 → D=3

Then row 2: 3*E=9 → E=3

Column 2: 3,E,9 → 3*3=9, good.

Column 3: 3,9,3 — 3*9=27, T*I=9*3=27, good.

Filled:

1 3 3

3 3 9

3 9 3

Puzzle 3:

Top: 81

Grid:

Row 1: 2, B, 18 → 2*B=18 → B=9

Row 2: D, E, 9 → D*E=9

Row 3: 18, 9, 2

Column 1: 2,D,18 → 2*D=18 → D=9

Then row 2: 9*E=9 → E=1

Column 2: B,E,9 =9,1,9 → 9*1=9, good.

Column 3: 18,9,2 — 18*9=162, T*I=81*2=162, good.

Filled:

2 9 18

9 1 9

18 9 2

Puzzle 4:

Top: 20

Grid:

Row 1: 8, B, 80 → 8*B=80 → B=10

Row 2: D, E, 12 → D*E=12

Row 3: 16, 60, 48

Column 1: 8,D,16 → 8*D=16 → D=2

Then row 2: 2*E=12 → E=6

Column 2: B,E,60 =10,6,60 → 10*6=60, good.

Column 3: 80,12,48 — 80*12=960, T*I=20*48=960, good.

Filled:

8 10 80

2 6 12

16 60 48

Puzzle 5:

Top: 20

Grid:

Row 1: A, B, 6 → A*B=6

Row 2: D, 8, 80 → D*8=80 → D=10

Row 3: 30, 16, 24

Column 1: A,D,30 =A,10,30 → A*10=30 → A=3

Then row 1: 3*B=6 → B=2

Column 2: B,8,16 =2,8,16 → 2*8=16, good.

Column 3: 6,80,24 — 6*80=480, T*I=20*24=480, good.

Filled:

3 2 6

10 8 80

30 16 24

Puzzle 6:

Top: 8

Grid:

Row 1: 2, B, 2 → 2*B=2 → B=1

Row 2: D, E, 40 → D*E=40

Row 3: 16, 5, 10

Column 1: 2,D,16 → 2*D=16 → D=8

Then row 2: 8*E=40 → E=5

Column 2: B,E,5 =1,5,5 → 1*5=5, good.

Column 3: 2,40,10 — 2*40=80, T*I=8*10=80, good.

Filled:

2 1 2

8 5 40

16 5 10

Puzzle 7:

Top: 35

Grid:

Row 1: A, 7, 42 → A*7=42 → A=6

Row 2: D, E, 45 → D*E=45

Row 3: 30, 63, 54

Column 1: A,D,30 =6,D,30 → 6*D=30 → D=5

Then row 2: 5*E=45 → E=9

Column 2: 7,E,63 =7,9,63 → 7*9=63, good.

Column 3: 42,45,54 — 42*45=1890, T*I=35*54=1890, good.

Filled:

6 7 42

5 9 45

30 63 54

Puzzle 8:

Top: 14

Grid:

Row 1: A, B, 18 → A*B=18

Row 2: 7, E, 56 → 7*E=56 → E=8

Row 3: 63, 16, 72

Column 1: A,7,63 → A*7=63 → A=9

Then row 1: 9*B=18 → B=2

Column 2: B,E,16 =2,8,16 → 2*8=16, good.

Column 3: 18,56,72 — 18*56=1008, T*I=14*72=1008, good.

Filled:

9 2 18

7 8 56

63 16 72

Puzzle 9:

Top: 24

Grid:

Row 1: A, B, 28 → A*B=28

Row 2: D, 4, 24 → D*4=24 → D=6

Row 3: 42, 16, 28

Column 1: A,D,42 =A,6,42 → A*6=42 → A=7

Then row 1: 7*B=28 → B=4

Column 2: B,4,16 =4,4,16 → 4*4=16, good.

Column 3: 28,24,28 — 28*24=672, T*I=24*28=672, good.

Filled:

7 4 28

6 4 24

42 16 28

Puzzle 10:

Top: 14

Grid:

Row 1: A, B, 16 → A*B=16

Row 2: 7, E, 63 → 7*E=63 → E=9

Row 3: 56, 18, 72

Column 1: A,7,56 → A*7=56 → A=8

Then row 1: 8*B=16 → B=2

Column 2: B,E,18 =2,9,18 → 2*9=18, good.

Column 3: 16,63,72 — 16*63=1008, T*I=14*72=1008, good.

Filled:

8 2 16

7 9 63

56 18 72

Puzzle 11:

Top: 36

Grid:

Row 1: A, 9, 27 → A*9=27 → A=3

Row 2: D, E, 28 → D*E=28

Row 3: 12, 63, 21

Column 1: A,D,12 =3,D,12 → 3*D=12 → D=4

Then row 2: 4*E=28 → E=7

Column 2: 9,E,63 =9,7,63 → 9*7=63, good.

Column 3: 27,28,21 — 27*28=756, T*I=36*21=756, good.

Filled:

3 9 27

4 7 28

12 63 21

Puzzle 12:

Top: 4

Grid:

Row 1: A, B, 18 → A*B=18

Row 2: D, 4, 8 → D*4=8 → D=2

Row 3: 18, 8, 36

Column 1: A,D,18 =A,2,18 → A*2=18 → A=9

Then row 1: 9*B=18 → B=2

Column 2: B,4,8 =2,4,8 → 2*4=8, good.

Column 3: 18,8,36 — 18*8=144, T*I=4*36=144, good.

Filled:

9 2 18

2 4 8

18 8 36

Puzzle 13:

Top: 48

Grid:

Row 1: A, 6, 18 → A*6=18 → A=3

Row 2: D, E, 16 → D*E=16

Row 3: 24, 12, 6

Column 1: A,D,24 =3,D,24 → 3*D=24 → D=8

Then row 2: 8*E=16 → E=2

Column 2: 6,E,12 =6,2,12 → 6*2=12, good.

Column 3: 18,16,6 — 18*16=288, T*I=48*6=288, good.

Filled:

3 6 18

8 2 16

24 12 6

Puzzle 14:

Top: 36

Grid:

Row 1: A, B, 36 → A*B=36

Row 2: D, 5, 45 → D*5=45 → D=9

Row 3: 81, 20, 45

Column 1: A,D,81 =A,9,81 → A*9=81 → A=9

Then row 1: 9*B=36 → B=4

Column 2: B,5,20 =4,5,20 → 4*5=20, good.

Column 3: 36,45,45 — 36*45=1620, T*I=36*45=1620, good.

Filled:

9 4 36

9 5 45

81 20 45

Puzzle 15:

Top: 6

Grid:

Row 1: A, B, 16 → A*B=16

Row 2: D, 2, 6 → D*2=6 → D=3

Row 3: 24, 4, 16

Column 1: A,D,24 =A,3,24 → A*3=24 → A=8

Then row 1: 8*B=16 → B=2

Column 2: B,2,4 =2,2,4 → 2*2=4, good.

Column 3: 16,6,16 — 16*6=96, T*I=6*16=96, good.

Filled:

8 2 16

3 2 6

24 4 16

Now, to present the answers, since the user didn't specify which puzzle, but likely wants all solved, but the instruction is to solve the problem, and since it's a set, perhaps list all filled grids.

But the final answer should be concise. Since it's a puzzle sheet, maybe just state that all puzzles are solved as per the rules, but I think for the response, since it's homework help, perhaps provide the completed grids.

But the format says "Final Answer" should be the correct answer, so perhaps for each puzzle, the missing numbers.

Since the user might expect the filled grids, but to save space, I can list the missing values or just confirm.

Given that, and since the assistant is to provide the final answer after reasoning, I'll summarize.

But in the context, perhaps the student needs to fill in the blanks, so for each puzzle, the missing numbers are as calculated.

To box the answer, since it's multiple, perhaps state that the puzzles are solved correctly with the values as above.

But let's see the response format.

I think for the Final Answer, since it's a set, I can say:

All multiplication puzzles have been solved by ensuring that for each row, the product of the first two numbers equals the third; for each column, the product of the first two numbers equals the third; and the top number times the bottom-right number equals the product of the top-right and middle-right numbers (which is consistent with the column rule when considering the top number).

The completed grids are as follows:

[Then list them, but it might be long]

Since the user may have specific puzzles in mind, but the image has 15 puzzles, and I've solved all, perhaps in the final answer, I can provide one example or just state it's done.

But to follow instructions, I'll provide the final answer as the solution to the puzzles, implying they are filled correctly.

Perhaps the "final answer" is not applicable since it's multiple, but I think for the purpose, I'll box the key insight or something.

Another thought: perhaps the user wants the missing numbers for a particular puzzle, but since not specified, and the image is given, likely all.

In the response, after reasoning, I can say the puzzles are solved, and for verification, here are the filled grids.

But to keep it neat, I'll write the final answer as the completion of the puzzles.

Since the assistant is to provide the final correct answer, and it's a puzzle, perhaps there is no single answer, but I think for this context, the final answer is that all blanks are filled as per the multiplication rules described.

To be precise, I'll provide the filled grid for the first puzzle as an example, but that might not be sufficient.

Looking back at the user's request: "Solve the problem accurately." and "provide the final correct answer".

Perhaps in such cases, the final answer is the set of solutions.

But to adhere to the format, I'll state:

The multiplication puzzles are solved by filling in the missing numbers such that each row's first two numbers multiply to the third, each column's first two numbers multiply to the third, and the top number times the bottom-right number equals the product of the top-right and middle-right numbers. All puzzles are consistent with these rules.

For instance, in the first puzzle, the missing numbers are 4, 7, 7 for the respective blanks.

But to give a concrete final answer, since the problem might be to complete the puzzles, and there are 15, perhaps list the missing values for each.

That could be messy.

Notice that in each puzzle, there are typically 3-4 blanks, and I've filled them.

Perhaps for the final answer, I can say the puzzles are completed successfully.

I think for the sake of this, I'll provide the filled grid for the first puzzle as a representative, but that's not accurate.

Another idea: perhaps the "problem" is to understand how to solve it, but the user said "solve the problem", meaning complete the puzzles.

Given the constraints, I'll box the solution for the first puzzle as an example, but indicate that all are solved similarly.

But let's see the response format requirement: "Final Answer: [Provide only the final correct answer]"

So perhaps for this type of problem, the final answer is the completed set, but since it's text, I'll describe.

I recall that in some contexts, for such puzzles, the answer is the filled grid, but here there are many.

Perhaps the user expects the missing numbers listed.

To save time, I'll assume that the final answer is the verification that the puzzles are solvable with the rules, and provide one example.

But to be thorough, I'll include in the final answer the filled grids for all, but that might be too long.

Since the assistant is to help with homework, and the student can see the reasoning