Scrambled Multiplication Table | ShowMe - Free Printable

Let's solve the Scrambled Multiplication Table step by step.

We are given a multiplication table where:

- The top row and leftmost column contain numbers (some missing), and
- The interior cells are products of the corresponding row and column headers.
- Some values are already filled in, and others are missing.

Our goal is to complete the table, meaning fill in all missing entries in both the headers and the interior.

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Step 1: Understand the structure

The table has:
- A top row with some known numbers: `7, 1, 10, 2, 4, 6, 8, ?, ?, ?`
- A leftmost column with some known numbers: `7, 10, ?, ?, ?, ?, ?, ?, ?`
- Interior cells that are products of the row header × column header.

We can use known products to deduce the missing headers.

Let’s label the rows and columns.

Let’s denote:

- Row 1: 7 → first row
- Row 2: 10 → second row
- Then we have more rows below.

Similarly, columns:
- Col 1: 7
- Col 2: 1
- Col 3: 10
- Col 4: 2
- Col 5: 4
- Col 6: 6
- Col 7: 8
- Then three more unknown columns: Col 8, 9, 10

So the table is 10×10, but only some values are given.

Let’s reconstruct it.

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Step 2: Use known values to deduce missing headers

Let’s look at the first row (row header = 7):

| × | 7 | 1 | 10 | 2 | 4 | 6 | 8 | ? | ? | ? |
|---|---|---|----|---|---|---|---|---|---|---|
| 7 | 49 | 7 | 70 | 14 | 28 | 42 | 56 | 21 | 35 | ? |

Wait — let's check this.

From the image:

First row:
- 7×7 = 49 ✔
- 7×1 = 7 ✔
- 7×10 = 70 ✔
- 7×2 = 14 ✔
- 7×4 = 28 ✔
- 7×6 = 42 ✔
- 7×8 = 56 ✔
- 7×? = 21 → So ? = 21 ÷ 7 = 3
- 7×? = 35 → ? = 35 ÷ 7 = 5
- 7×? = ? → last one missing

So the column headers must be:

- Col 1: 7
- Col 2: 1
- Col 3: 10
- Col 4: 2
- Col 5: 4
- Col 6: 6
- Col 7: 8
- Col 8: 3
- Col 9: 5
- Col 10: ?

Now, from the first row, last entry: 7 × ? = ?

But we don’t know what it is yet. Let's keep going.

So far, column headers are:

> [7, 1, 10, 2, 4, 6, 8, 3, 5, ?]

Now let’s go to second row: row header = 10

| × | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | ? |
|---|---|---|----|---|---|---|---|---|---|---|
| 10 | 70 | 10 | 100 | 20 | 40 | 60 | 80 | ? | 50 | 90 |

Check:
- 10×7 = 70 ✔
- 10×1 = 10 ✔
- 10×10 = 100 ✔
- 10×2 = 20 ✔
- 10×4 = 40 ✔
- 10×6 = 60 ✔
- 10×8 = 80 ✔
- 10×3 = 30 → so missing value in col 8 is 30
- 10×5 = 50 ✔
- 10×? = 90 → ? = 90 ÷ 10 = 9

So last column header is 9

Thus, full column headers are:

> [7, 1, 10, 2, 4, 6, 8, 3, 5, 9]

Now update the table:

| × | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
|---|---|---|----|---|---|---|---|---|---|---|
| 7 | 49 | 7 | 70 | 14 | 28 | 42 | 56 | 21 | 35 | 63 |
| 10 | 70 | 10 | 100 | 20 | 40 | 60 | 80 | 30 | 50 | 90 |
| ? | ? | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
| ? | ? | 6 | 60 | ? | 24 | 36 | 48 | 18 | 30 | 54 |
| ? | 14 | 2 | 20 | 4 | ? | 12 | 16 | 6 | 10 | 18 |
| ? | 56 | ? | 80 | 16 | 32 | 48 | ? | 24 | 40 | 72 |
| ? | 21 | 3 | 30 | ? | 12 | 18 | 24 | 9 | ? | 27 |
| ? | 28 | ? | 40 | 8 | 16 | 24 | 32 | ? | 20 | 36 |
| ? | 63 | 9 | ? | 18 | ? | 54 | ? | 27 | 45 | 81 |
| ? | ? | 5 | 50 | 10 | 20 | 30 | 40 | 15 | 25 | ? |

Now let’s determine the row headers.

We already have:
- Row 1: 7
- Row 2: 10

Look at Row 3: It starts with blank, but has values:

| ? | ? | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |

So, when multiplied by the column headers, gives:

- Column 2: 1 → so row header × 1 = 1 → row header = 1
- Column 3: 10 → row header × 10 = 10 → row header = 1 → consistent
- All entries in this row are just the column headers → so this row is 1

So Row 3: 1

Update:

| × | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
|---|---|---|----|---|---|---|---|---|---|---|
| 7 | 49 | 7 | 70 | 14 | 28 | 42 | 56 | 21 | 35 | 63 |
| 10 | 70 | 10 | 100 | 20 | 40 | 60 | 80 | 30 | 50 | 90 |
| 1 | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
| ? | ? | 6 | 60 | ? | 24 | 36 | 48 | 18 | 30 | 54 |
| ? | 14 | 2 | 20 | 4 | ? | 12 | 16 | 6 | 10 | 18 |
| ? | 56 | ? | 80 | 16 | 32 | 48 | ? | 24 | 40 | 72 |
| ? | 21 | 3 | 30 | ? | 12 | 18 | 24 | 9 | ? | 27 |
| ? | 28 | ? | 40 | 8 | 16 | 24 | 32 | ? | 20 | 36 |
| ? | 63 | 9 | ? | 18 | ? | 54 | ? | 27 | 45 | 81 |
| ? | ? | 5 | 50 | 10 | 20 | 30 | 40 | 15 | 25 | ? |

Now Row 4:
- Column 2: 6 → so row header × 1 = 6 → row header = 6
- Check: 6×10 = 60 ✔, 6×2=12 → but table says ? → wait, no: column 4 is 2 → 6×2 = 12 → but current value is blank → we can fill it
- But we have: 6×4 = 24 ✔, 6×6=36 ✔, 6×8=48 ✔, 6×3=18 ✔, 6×5=30 ✔, 6×9=54 ✔
- Only missing: 6×7 = ? → 42, 6×1=6 ✔, 6×10=60 ✔
- But column 1: 6×7 = 42 → so fill that
- Column 4: 6×2 = 12 → fill that

So Row 4: 6

Update:

| 6 | 42 | 6 | 60 | 12 | 24 | 36 | 48 | 18 | 30 | 54 |

Now Row 5:
- Column 1: 14 → row header × 7 = 14 → row header = 14 ÷ 7 = 2
- Check: 2×1 = 2 ✔, 2×10 = 20 ✔, 2×2 = 4 ✔, 2×4 = 8 → but table says ?, wait — no: column 5 is 4 → 2×4 = 8 → but currently blank
- Wait, column 5 is 4 → 2×4 = 8 → but in the table, it's blank → we can fill
- Also, 2×6 = 12 ✔, 2×8 = 16 ✔, 2×3 = 6 ✔, 2×5 = 10 ✔, 2×9 = 18 ✔
- Missing: 2×7 = 14 ✔ (already there), 2×1 = 2 ✔, 2×10 = 20 ✔, 2×4 = 8 → fill that

So Row 5: 2

Update:

| 2 | 14 | 2 | 20 | 8 | 8 | 12 | 16 | 6 | 10 | 18 |

Wait — but in column 5 (which is 4), we have 2×4 = 8 → yes.

But in the original table, the cell for row 5, col 5 is blank → now fill with 8.

Now Row 6:
- Column 1: 56 → row header × 7 = 56 → row header = 56 ÷ 7 = 8
- Check: 8×1 = 8 → but table shows ? in column 2 → should be 8
- 8×10 = 80 ✔, 8×2 = 16 ✔, 8×4 = 32 ✔, 8×6 = 48 ✔, 8×8 = 64 → but table says ? → wait, column 7 is 8 → 8×8 = 64 → missing
- 8×3 = 24 ✔, 8×5 = 40 ✔, 8×9 = 72 ✔
- 8×7 = 56 ✔, 8×1 = 8 → fill
- 8×8 = 64 → fill

So Row 6: 8

Update:

| 8 | 56 | 8 | 80 | 16 | 32 | 48 | 64 | 24 | 40 | 72 |

Now Row 7:
- Column 1: 21 → row header × 7 = 21 → row header = 3
- Check: 3×1 = 3 ✔, 3×10 = 30 ✔, 3×2 = 6 → but column 4 is 2 → 3×2 = 6 → but table says ? → fill
- 3×4 = 12 ✔, 3×6 = 18 ✔, 3×8 = 24 ✔, 3×3 = 9 ✔, 3×5 = 15 → but table says ? → fill
- 3×9 = 27 ✔
- 3×7 = 21 ✔, 3×1 = 3 ✔, 3×10 = 30 ✔
- 3×2 = 6 → fill
- 3×5 = 15 → fill

So Row 7: 3

Update:

| 3 | 21 | 3 | 30 | 6 | 12 | 18 | 24 | 9 | 15 | 27 |

Now Row 8:
- Column 1: 28 → row header × 7 = 28 → row header = 4
- Check: 4×1 = 4 → but table says ? in column 2 → should be 4
- 4×10 = 40 ✔, 4×2 = 8 ✔, 4×4 = 16 ✔, 4×6 = 24 ✔, 4×8 = 32 ✔, 4×3 = 12 → but table says ? → fill
- 4×5 = 20 ✔, 4×9 = 36 ✔
- 4×1 = 4 → fill
- 4×3 = 12 → fill

So Row 8: 4

Update:

| 4 | 28 | 4 | 40 | 8 | 16 | 24 | 32 | 12 | 20 | 36 |

Now Row 9:
- Column 1: 63 → row header × 7 = 63 → row header = 9
- Check: 9×1 = 9 ✔, 9×10 = 90 → but table says ? → fill
- 9×2 = 18 ✔, 9×4 = 36 → but table says ? → fill
- 9×6 = 54 ✔, 9×8 = 72 → but table says ? → fill
- 9×3 = 27 ✔, 9×5 = 45 ✔, 9×9 = 81 ✔
- 9×10 = 90 → fill
- 9×4 = 36 → fill
- 9×8 = 72 → fill

So Row 9: 9

Update:

| 9 | 63 | 9 | 90 | 18 | 36 | 54 | 72 | 27 | 45 | 81 |

Now Row 10:
- Column 1: ? → row header × 7 = ?
- Column 2: 5 → row header × 1 = 5 → row header = 5
- Check: 5×10 = 50 ✔, 5×2 = 10 ✔, 5×4 = 20 ✔, 5×6 = 30 ✔, 5×8 = 40 ✔, 5×3 = 15 ✔, 5×5 = 25 ✔, 5×9 = 45 → but table says ? → fill
- 5×7 = 35 → so column 1 = 35
- 5×1 = 5 ✔, 5×10 = 50 ✔, etc.
- 5×9 = 45 → fill

So Row 10: 5

Update:

| 5 | 35 | 5 | 50 | 10 | 20 | 30 | 40 | 15 | 25 | 45 |

Now we can complete the entire table.

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Final Table:

| × | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
|---|---|---|----|---|---|---|---|---|---|---|
| 7 | 49 | 7 | 70 | 14 | 28 | 42 | 56 | 21 | 35 | 63 |
| 10 | 70 | 10 | 100 | 20 | 40 | 60 | 80 | 30 | 50 | 90 |
| 1 | 7 | 1 | 10 | 2 | 4 | 6 | 8 | 3 | 5 | 9 |
| 6 | 42 | 6 | 60 | 12 | 24 | 36 | 48 | 18 | 30 | 54 |
| 2 | 14 | 2 | 20 | 8 | 8 | 12 | 16 | 6 | 10 | 18 |
| 8 | 56 | 8 | 80 | 16 | 32 | 48 | 64 | 24 | 40 | 72 |
| 3 | 21 | 3 | 30 | 6 | 12 | 18 | 24 | 9 | 15 | 27 |
| 4 | 28 | 4 | 40 | 8 | 16 | 24 | 32 | 12 | 20 | 36 |
| 9 | 63 | 9 | 90 | 18 | 36 | 54 | 72 | 27 | 45 | 81 |
| 5 | 35 | 5 | 50 | 10 | 20 | 30 | 40 | 15 | 25 | 45 |

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Summary of Missing Headers:

- Column headers: [7, 1, 10, 2, 4, 6, 8, 3, 5, 9]
- Row headers: [7, 10, 1, 6, 2, 8, 3, 4, 9, 5]

All missing values have been filled using multiplication.

✔ Table is now complete!

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