Christmas Tree Adding Puzzle 1 - A fun math worksheet where each number is the sum of the two numbers below it.

Let's solve the Christmas Tree Adding Puzzle 1 step by step.

🔍 Rule:

> Each number is the total of the two numbers below it.

This means that each hexagon above is the sum of the two hexagons directly beneath it. We work from bottom to top, filling in missing numbers.

We'll solve each tree one at a time.

---

## 🌲 Tree 1 (Top Left)

Given:
```
?
? ?
9 ? 11
3 6 2 9
```

Start from the bottom row and work upward.

Step 1: Fill in the second row from bottom

- The hexagon above 3 and 6 → 3 + 6 = 9 ✔ (already given)
- Above 6 and 2 → 6 + 2 = 8
- Above 2 and 9 → 2 + 9 = 11 ✔ (already given)

So now we have:
```
?
? ?
9 8 11
3 6 2 9
```

Step 2: Third row up

- Above 9 and 8 → 9 + 8 = 17
- Above 8 and 11 → 8 + 11 = 19

Now:
```
?
17 19
9 8 11
3 6 2 9
```

Step 3: Top

- Above 17 and 19 → 17 + 19 = 36

✔ Final Tree 1:
```
36
17 19
9 8 11
3 6 2 9
```

---

## 🌲 Tree 2 (Top Right)

Given:
```
?
? ?
? 12 ?
6 4 8 5
```

Step 1: Second row from bottom

- Above 6 and 4 → 6 + 4 = 10
- Above 4 and 8 → 4 + 8 = 12 ✔ (given)
- Above 8 and 5 → 8 + 5 = 13

So:
```
?
? ?
10 12 13
6 4 8 5
```

Step 2: Third row up

- Above 10 and 12 → 10 + 12 = 22
- Above 12 and 13 → 12 + 13 = 25

Now:
```
?
22 25
10 12 13
6 4 8 5
```

Step 3: Top

- Above 22 and 25 → 22 + 25 = 47

✔ Final Tree 2:
```
47
22 25
10 12 13
6 4 8 5
```

---

## 🌲 Tree 3 (Bottom Left)

Given:
```
?
? ?
21 ? 7
? 10 ? 4
? 7 4 ? ?
```

Wait — this tree has five bottom hexagons, so it’s a bit different.

Let’s label positions clearly.

We’ll write it as:

```
?
? ?
21 ? 7
? 10 ? 4
? 7 4 ? ?
```

We know:
- Bottom row: 5 hexagons
- Row above: 4 hexagons
- Then 3, then 2, then 1

We are told:
- In row 4 (from bottom): 10 and 4 are given
- In row 3: 21 and 7 are given
- In row 2: 7 and 4 are given

Let’s fill in step by step.

Step 1: Fill row 4 (just above bottom)

Each number is sum of two below it.

Let’s denote bottom row: A B C D E
Row above: F G H I
Then: J K L
Then: M N
Then: O

We know:
- F = A + B → F = ?
- G = B + C → G = 10
- H = C + D → H = ?
- I = D + E → I = 4

Also:
- J = F + G = 21
- K = G + H = ?
- L = H + I = 7

And:
- M = J + K = ?
- N = K + L = ?

Finally:
- O = M + N = ?

Let’s use known values.

We know:
- G = B + C = 10
- I = D + E = 4
- J = F + G = 21 → F + 10 = 21 → F = 11
- L = H + I = 7 → H + 4 = 7 → H = 3

Now:
- F = A + B = 11
- G = B + C = 10
- H = C + D = 3
- I = D + E = 4

We also know:
- Bottom row has: A, B, C, D, E
- Middle row: F=11, G=10, H=3, I=4

Now let’s find A, B, C, D, E.

From:
1. A + B = 11
2. B + C = 10
3. C + D = 3
4. D + E = 4

We can solve these equations.

From (3): C + D = 3 → C = 3 - D
From (2): B + C = 10 → B + (3 - D) = 10 → B = 7 + D
From (1): A + B = 11 → A + (7 + D) = 11 → A = 4 - D
From (4): D + E = 4 → E = 4 - D

Now all variables in terms of D.

Try possible integer values for D (must be such that all numbers are positive integers).

Try D = 1:
- C = 3 - 1 = 2
- B = 7 + 1 = 8
- A = 4 - 1 = 3
- E = 4 - 1 = 3

Check:
- A + B = 3 + 8 = 11 ✔
- B + C = 8 + 2 = 10 ✔
- C + D = 2 + 1 = 3 ✔
- D + E = 1 + 3 = 4 ✔

Perfect!

So bottom row: A=3, B=8, C=2, D=1, E=3

Now we can build up.

Row 4: F=11, G=10, H=3, I=4

Row 3: J = F+G = 11+10 = 21 ✔ (given), K = G+H = 10+3 = 13, L = H+I = 3+4 = 7 ✔

Row 2: M = J+K = 21+13 = 34, N = K+L = 13+7 = 20

Top: O = M+N = 34+20 = 54

✔ Final Tree 3:
```
54
34 20
21 13 7
11 10 3 4
3 8 2 1 3
```

---

## 🌲 Tree 4 (Bottom Right)

Given:
```
?
? ?
14 11 ?
9 ? ? 3
```

Let’s label:

Row 1 (top): O
Row 2: M, N
Row 3: J, K, L
Row 4: F, G, H, I
Row 5: A, B, C, D

Given:
- A = 9
- D = 3
- F = 9 + B → but not yet known
- G = B + C
- H = C + D = C + 3
- I = D = 3? Wait — no, I is above D, but there’s no fifth hexagon on right?

Wait — let’s count.

Actually, bottom row has 4 hexagons: A, B, C, D
Above: 3 hexagons: F, G, H
Then: 2: J, K
Then: 1: O

But wait — the diagram shows:
```
?
? ?
14 11 ?
9 ? ? 3
```

So bottom row: 9, ?, ?, 3 → four hexagons

Row above: 14, 11, ? → three hexagons

Then: ?, ? → two

Then: ? → one

So:

Let’s define:
- Bottom: A=9, B=?, C=?, D=3
- Row 4: F = A+B = 9+B
- G = B+C
- H = C+D = C+3
- Row 3: J = F+G = 14
- K = G+H = 11
- Row 2: M = J+K = ?
- Row 1: O = M

We know:
- J = F + G = (9+B) + (B+C) = 9 + 2B + C = 14 → Equation (1)
- K = G + H = (B+C) + (C+3) = B + 2C + 3 = 11 → Equation (2)

Solve:

(1): 9 + 2B + C = 14 → 2B + C = 5
(2): B + 2C + 3 = 11 → B + 2C = 8

Now solve system:
From (1): C = 5 - 2B
Plug into (2):
B + 2(5 - 2B) = 8
B + 10 - 4B = 8
-3B = -2 → B = 2/3 ✘ Not integer

Wait — something wrong?

But numbers should be whole numbers.

Wait — maybe I misread.

Look again:

Tree 4:
```
?
? ?
14 11 ?
9 ? ? 3
```

Is it possible that the third row has three numbers: 14, 11, and another?

Yes.

But the second row from bottom is:
- First: 9
- Second: ?
- Third: ?
- Fourth: 3

So bottom row: 9, x, y, 3

Row above:
- First: 9 + x
- Second: x + y
- Third: y + 3

Row above that:
- First: (9+x) + (x+y) = 9 + 2x + y = 14
- Second: (x+y) + (y+3) = x + 2y + 3 = 11
- Third: (y+3) + ? → but only three hexagons in row 3, so third is just y+3? No.

Wait — actually, the structure is like a pyramid:

Each hexagon supports two below it.

So in row 3 (from bottom), the leftmost is sum of bottom-left and middle-left.

So:

Let’s define:

Bottom row: A=9, B=?, C=?, D=3

Row 4 (above):
- F = A+B = 9+B
- G = B+C
- H = C+D = C+3

Row 3 (middle):
- J = F+G = (9+B)+(B+C) = 9 + 2B + C = 14 → Eq (1)
- K = G+H = (B+C)+(C+3) = B + 2C + 3 = 11 → Eq (2)
- L = H + ??? — but only three hexagons in row 3? Wait, no.

Wait — the diagram shows:
- Bottom: 4 hexagons
- Above: 3 hexagons
- Above: 2 hexagons
- Top: 1

So row 3 (from bottom) has 3 hexagons: J, K, L

But J = F+G
K = G+H
L = H + ??? — but H is the last in row 4, so nothing to the right? So L cannot exist unless there’s a fourth in row 4.

Wait — contradiction.

Actually, in a standard triangular pattern with 4 hexagons at base, the next row has 3, then 2, then 1.

So yes, row 3 (from bottom) has 3 hexagons:
- First: sum of first two of bottom → A+B
- Second: sum of second and third → B+C
- Third: sum of third and fourth → C+D

So row 4 (from bottom) has:
- F = A+B
- G = B+C
- H = C+D

Then row 3 (from bottom) has:
- J = F+G = (A+B)+(B+C) = A + 2B + C
- K = G+H = (B+C)+(C+D) = B + 2C + D
- L = H + ??? — but no fourth hexagon, so L is just H? No.

Wait — no, the row above (row 3) has two hexagons: J and K

Because:

- Row 4: 3 hexagons → F, G, H
- Row 3: 2 hexagons → J = F+G, K = G+H
- Row 2: 1 hexagon → M = J+K

But in the diagram, row 3 (from bottom) has three numbers: 14, 11, ?

Wait — look again:

The image shows:

```
?
? ?
14 11 ?
9 ? ? 3
```

So the third row (from bottom) has three numbers: 14, 11, and a blank.

That suggests four hexagons in the row above bottom.

But bottom has four, so row above must have three.

But here, three numbers are shown in the row above bottom.

So:

- Row 4 (bottom): 9, ?, ?, 3 → 4 hexagons
- Row 3: 14, 11, ? → 3 hexagons
- Row 2: ?, ? → 2 hexagons
- Row 1: ? → 1

So:
- Row 3: first = 9 + second → 9 + x = 14 → x = 5 → so B = 5
- Second in row 3: x + y = 11 → 5 + y = 11 → y = 6 → C = 6
- Third in row 3: y + 3 = 6 + 3 = 9 → so blank = 9

So row 3: 14, 11, 9

Now row 2:
- First: 14 + 11 = 25
- Second: 11 + 9 = 20

Top: 25 + 20 = 45

✔ Final Tree 4:
```
45
25 20
14 11 9
9 5 6 3
```

Wait — check:
- 9 + 5 = 14 ✔
- 5 + 6 = 11 ✔
- 6 + 3 = 9 ✔
- 14 + 11 = 25 ✔
- 11 + 9 = 20 ✔
- 25 + 20 = 45 ✔

Perfect!

---

✔ Final Answers:

#### Tree 1:
```
36
17 19
9 8 11
3 6 2 9
```

#### Tree 2:
```
47
22 25
10 12 13
6 4 8 5
```

#### Tree 3:
```
54
34 20
21 13 7
11 10 3 4
3 8 2 1 3
```

#### Tree 4:
```
45
25 20
14 11 9
9 5 6 3
```

---

✔ Summary of Top Numbers:

- Tree 1: 36
- Tree 2: 47
- Tree 3: 54
- Tree 4: 45

You can fill these in the blank spaces accordingly.

Let me know if you'd like a printable version or visual explanation!