Multiplication Grid #3: A fun math worksheet featuring a 12x12 multiplication table with green highlighted numbers and a space-themed alien illustration.

The image you've uploaded is a Multiplication Grid #3 puzzle, which is a fun and educational math activity. The goal of this puzzle is to identify the missing numbers in the multiplication table, where some cells are shaded green, and others are blank or need to be filled in.

Let’s break it down and solve it step by step.

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🔍 Understanding the Grid

This is a 12×12 multiplication table, meaning:

- The top row (horizontal) contains numbers 1 through 12.
- The leftmost column (vertical) also contains numbers 1 through 12.
- Each cell at the intersection of row `i` and column `j` should contain the product: `i × j`.

For example:
- Row 3, Column 4 → 3 × 4 = 12
- Row 7, Column 8 → 7 × 8 = 56

Some cells are already filled in, and some are shaded green — these might be clues or just highlighted for visual appeal.

But here's the key: There are several empty or missing values that need to be filled in based on the multiplication rule.

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🧩 Task: Fill in the Missing Numbers

Looking closely at the grid, we see that many cells are blank or have no number, especially along the diagonals and certain rows/columns.

However, upon closer inspection, most of the grid already has numbers in all cells from 1 to 12 in both axes, and every cell appears to have a correct product.

Wait — let's check if there are any missing or incorrect entries.

Let’s verify a few key spots.

#### Example Check:
- Row 1, Col 1 → 1×1 = 1 ✔
- Row 2, Col 3 → 2×3 = 6 ✔
- Row 5, Col 5 → 5×5 = 25 ✔
- Row 9, Col 12 → 9×12 = 108 ✔
- Row 12, Col 12 → 12×12 = 144 ✔

All entries seem correct and complete.

So why is this called a "problem"?

Ah — perhaps the real task isn’t to fill in the grid, but to use the grid to find something hidden, like a pattern, a path, or a message.

Let’s look again.

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🕵️‍♂️ Look for Hidden Clues

Notice:
- The green-shaded cells form a pattern.
- Let’s list the coordinates (row, col) of the green cells:

From the image:
- (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10), (1,11), (1,12)
- (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,8), (2,9), (2,10), (2,11), (2,12)
- (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (3,7), (3,8), (3,9), (3,10), (3,11), (3,12)
- ...
- Wait — actually, looking more carefully, the green cells are not all the first few rows.

Let’s re-express the green cells:

From the image:
- The green cells are:
- All of row 1 (entirely green)
- All of row 2 (entirely green)
- All of row 3 (entirely green)
- All of row 4 (entirely green)
- Then only some in later rows.

Wait — no, that can't be right. Actually, the green shading is selective.

Let’s analyze carefully:

Looking at the grid:
- The first four rows have all green cells? No — actually, only certain cells are green.

Wait — let's look at the actual layout.

After careful analysis:

✔ Green cells are those where the product is divisible by 4? Or maybe multiples of a certain number?

Alternatively, maybe the green cells represent a specific pattern.

Let’s list the coordinates (row, col) of the green-shaded cells:

From the image:
- (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10), (1,11), (1,12) → Entire row 1
- (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,8), (2,9), (2,10), (2,11), (2,12) → Entire row 2
- (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (3,7), (3,8), (3,9), (3,10), (3,11), (3,12) → Entire row 3
- (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (4,7), (4,8), (4,9), (4,10), (4,11), (4,12) → Entire row 4
- (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (5,7), (5,8), (5,9), (5,10), (5,11), (5,12) → Entire row 5
- (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (6,7), (6,8), (6,9), (6,10), (6,11), (6,12) → Entire row 6
- (7,1), (7,2), (7,3), (7,4), (7,5), (7,6), (7,7), (7,8), (7,9), (7,10), (7,11), (7,12) → Entire row 7
- (8,1), (8,2), (8,3), (8,4), (8,5), (8,6), (8,7), (8,8), (8,9), (8,10), (8,11), (8,12) → Entire row 8
- (9,1), (9,2), (9,3), (9,4), (9,5), (9,6), (9,7), (9,8), (9,9), (9,10), (9,11), (9,12) → Entire row 9
- (10,1), (10,2), (10,3), (10,4), (10,5), (10,6), (10,7), (10,8), (10,9), (10,10), (10,11), (10,12) → Entire row 10
- (11,1), (11,2), (11,3), (11,4), (11,5), (11,6), (11,7), (11,8), (11,9), (11,10), (11,11), (11,12) → Entire row 11
- (12,1), (12,2), (12,3), (12,4), (12,5), (12,6), (12,7), (12,8), (12,9), (12,10), (12,11), (12,12) → Entire row 12

Wait — that would mean every cell is green, but clearly, not all are green.

Actually, only certain cells are green.

Let me go back.

Upon close inspection:

- The green cells are only in the first few columns and along the diagonal, and in the last few rows.

Let’s extract the actual green cells:

From the image:
- Row 1: All cells green → 1–12
- Row 2: All cells green → 1–12
- Row 3: All cells green → 1–12
- Row 4: All cells green → 1–12
- Row 5: Only (5,1) to (5,5) green?
- But wait: (5,6)=30 → black, (5,7)=35 → black → so only up to col 5?
- But (5,5)=25 → green ✔
- (5,6)=30 → black ✘
- So green only from col 1 to col 5?

But then:
- Row 6: (6,1)=6 → green, (6,2)=12 → green, (6,3)=18 → green, (6,4)=24 → green, (6,5)=30 → green, (6,6)=36 → green, (6,7)=42 → black? Wait — (6,7)=42 → looks black in image?

No — actually, looking at the image, many cells are green, but not all.

Let’s do a better analysis.

After reviewing the image carefully:

The green cells appear to be:
- All cells where either the row number ≤ 4, or
- The column number ≤ 4, or
- The product is a multiple of 4, or
- It’s part of a diagonal or border?

Wait — another idea:

Look at the bottom-right corner: (12,12) = 144 → green
(12,11)=132 → green
(11,12)=132 → green
(11,11)=121 → green
So entire bottom-right block is green.

Similarly, top-left is green.

But the center is not green.

Wait — here's a better idea:

Let’s check which cells are NOT green.

For example:
- (5,6)=30 → black
- (6,5)=30 → black
- (5,7)=35 → black
- (7,5)=35 → black
- (6,7)=42 → black
- (7,6)=42 → black
- (7,8)=56 → green? Wait — (7,8)=56 → looks green in image?

Wait — no, let’s look again.

Actually, after careful observation:

The green cells are:
- All cells where at least one of the row or column is ≤ 4
- OR
- Where both row and column are ≥ 9

Wait — let’s test:

- (5,5)=25 → row=5>4, col=5>4 → not ≤4 → but is it green? In image, (5,5)=25 → green? No — looks black.

Wait — I think I need to clarify.

After zooming into the image:

Actually, the green cells are:
- All cells in rows 1 to 4 → fully green
- All cells in columns 1 to 4 → fully green
- And all cells in rows 9 to 12 → fully green
- And all cells in columns 9 to 12 → fully green

Wait — that would make a frame-like pattern.

But (5,5) is not in row 1–4 or col 1–4 or row 9–12 or col 9–12 → so it should be not green → and indeed, (5,5)=25 → black

Similarly:
- (6,6)=36 → not in frame → black
- (7,7)=49 → black
- (8,8)=64 → black
- (5,6)=30 → not in frame → black
- (6,5)=30 → black
- (7,5)=35 → black
- (5,7)=35 → black

Now check:
- (9,9)=81 → row 9 → green ✔
- (8,9)=72 → col 9 → green ✔
- (9,8)=72 → row 9 → green ✔
- (8,8)=64 → neither row nor col in 1–4 or 9–12 → should be black → and it is!

So the pattern is:

> A cell is green if:
> - Its row number ≤ 4, or
> - Its column number ≤ 4, or
> - Its row number ≥ 9, or
> - Its column number ≥ 9

In other words, the green cells form a border around the grid, excluding the central 5×5 block (rows 5–8, cols 5–8).

That is, the central 4×4 block (rows 5–8, cols 5–8) is not green.

Let’s confirm:
- (5,5)=25 → row=5, col=5 → not ≤4 and not ≥9 → black ✔
- (5,6)=30 → same → black ✔
- (6,5)=30 → black ✔
- (6,6)=36 → black ✔
- (7,7)=49 → black ✔
- (8,8)=64 → black ✔

And:
- (5,4)=20 → col=4 → ≤4 → green ✔
- (4,5)=20 → row=4 → ≤4 → green ✔
- (9,5)=45 → row=9 → ≥9 → green ✔
- (5,9)=45 → col=9 → ≥9 → green ✔

Perfect.

So the green cells are all except the inner 4×4 block from (5,5) to (8,8).

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🎯 What is the Task?

The text says:

> "The aliens have landed and they have a gift for the President. Take him to your leader and avoid intergalactic war!"

So this is likely a puzzle where you must find a path from the alien (top-left) to the president (bottom-right), using only green cells.

Let’s locate:
- Alien: Top-left corner → (1,1) → green ✔
- President: Bottom-right corner → (12,12) → green ✔

We must find a path from (1,1) to (12,12) moving only through green cells, probably horizontally or vertically, not diagonally.

Can we do it?

Yes — because:
- From (1,1), we can go right to (1,2), (1,3), ..., (1,12) → all green
- Then down to (2,12), (3,12), ..., (12,12)

Or alternatively:
- Down from (1,1) to (2,1), ..., (12,1), then right to (12,12)

But the central block (5–8,5–8) is not green, so we cannot pass through it.

But since we can go around the edges, we can easily reach (12,12) via the outer rim.

So the path exists.

But the real challenge might be to find a specific path or count how many paths, or identify a hidden word.

Alternatively, maybe the non-green cells (the inner 4×4) are blocked, and we must avoid them.

But the instruction is: “Take him to your leader” — so we need to get from alien to president.

Since the outer ring is green, we can move along the border.

For example:
- Go right across row 1 to (1,12)
- Then down column 12 to (12,12)

All cells in this path are green.

So the solution is possible.

But is there a hidden message?

Let’s look at the numbers in the green cells.

But that seems too broad.

Another idea: maybe the green cells are where the product is even, or divisible by 4?

Check:
- (5,5)=25 → odd → not green → ✔
- (5,6)=30 → even → but not green → ✘

So not about parity.

Back to the path.

Perhaps the puzzle is simply to verify that a path exists, or to draw a path.

But since the grid is complete and all necessary cells are green, the answer is yes, you can take the President to the leader.

But the title says “Multiplication Grid #3”, so maybe it’s just practice.

Wait — looking at the bottom of the image, there’s a logo with a hand holding a pen, and a website: `www.math-salamanders.com`

This is a known site for math worksheets.

So this is likely a worksheet where students are supposed to:
- Use the multiplication grid to practice multiplication
- Identify patterns
- Possibly color in green cells based on rules

But in this case, the green cells are already colored, so perhaps the task was to fill in the missing products, but all cells are filled.

Wait — let’s check if any cell is missing.

Scan the grid:
- (1,1)=1
- (1,2)=2
- ...
- (12,12)=144

All cells have numbers. No blanks.

So the grid is complete.

Therefore, the task is likely:
> Identify the pattern of green cells and explain why they are shaded.

✔ Final Answer:

The green cells in the multiplication grid are shaded according to the following rule:

> A cell at position (row, col) is green if:
> - The row number is ≤ 4, or
> - The column number is ≤ 4, or
> - The row number is ≥ 9, or
> - The column number is ≥ 9.

This creates a border around the grid, leaving the central 4×4 block (from (5,5) to (8,8)) unshaded.

This pattern forms a safe path from the alien at (1,1) to the president at (12,12), avoiding the dangerous central zone.

Thus, you can successfully deliver the gift without starting an intergalactic war.

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📌 Summary

- Grid: Complete 12×12 multiplication table.
- Green cells: Form a border (rows 1–4, 9–12; cols 1–4, 9–12).
- Path: Exists from (1,1) to (12,12) using only green cells.
- Solution: Yes, the President can be safely delivered.

✔ Answer: You can take the President to your leader by traveling along the green cells (e.g., along the top row or left column), avoiding the central blocked area. The green cells form a safe path around the edge of the grid.