Magic Triangle puzzles requiring placement of numbers 1 to 6 without repetition to achieve equal sums on each side of the triangle.

Let’s solve each magic triangle one by one. Remember: we must use numbers 1 to 6, no repeats, and each side of the triangle must add up to the given sum.

---

Problem 1: Sum = 9

Given:
- Top corner: 3
- Bottom left: 1
- Bottom right: 2

We need to fill in the three middle circles (one on each side).

Let’s label positions for clarity:

```
A (top) = 3
/ \
B C
/ \
D=1 — E — F=2
```

Sides:
- Left side: A + B + D = 3 + B + 1 → must equal 9 → so B = 5
- Right side: A + C + F = 3 + C + 2 → must equal 9 → so C = 4
- Bottom side: D + E + F = 1 + E + 2 → must equal 9 → so E = 6

Check used numbers: 3, 1, 2, 5, 4, 6 → all from 1–6, no repeats ✔

✔ Solution for #1:
Left middle = 5, Right middle = 4, Bottom middle = 6

---

Problem 2: Sum = 10

Given:
- Bottom left: 3
- Bottom middle: 2
- Right middle: 4

Label:

```
A
/ \
B C=4
/ \
D=3 — E=2 — F
```

Sides:
- Bottom: D + E + F = 3 + 2 + F = 10 → F = 5
- Right: A + C + F = A + 4 + 5 = 10 → A = 1
- Left: A + B + D = 1 + B + 3 = 10 → B = 6

Used numbers: 3, 2, 4, 5, 1, 6 → all unique, 1–6 ✔

✔ Solution for #2:
Top = 1, Left middle = 6, Bottom right = 5

---

Problem 3: Sum = 11

Given:
- Left middle: 5
- Bottom left: 4
- Bottom middle: 1

Label:

```
A
/ \
B=5 C
/ \
D=4 — E=1 — F
```

Sides:
- Bottom: D + E + F = 4 + 1 + F = 11 → F = 6
- Left: A + B + D = A + 5 + 4 = 11 → A = 2
- Right: A + C + F = 2 + C + 6 = 11 → C = 3

Used: 5, 4, 1, 6, 2, 3 → all 1–6, no repeats ✔

✔ Solution for #3:
Top = 2, Right middle = 3, Bottom right = 6

---

Problem 4: Sum = 12

Given:
- Left middle: 1
- Right middle: 2
- Bottom middle: 3

Label:

```
A
/ \
B=1 C=2
/ \
D — E=3 — F
```

Sides:
- Left: A + B + D = A + 1 + D = 12 → A + D = 11
- Right: A + C + F = A + 2 + F = 12 → A + F = 10
- Bottom: D + E + F = D + 3 + F = 12 → D + F = 9

Now we have:
(1) A + D = 11
(2) A + F = 10
(3) D + F = 9

Subtract (2) from (1):
(A + D) - (A + F) = 11 - 10 → D - F = 1 → D = F + 1

Plug into (3):
(F + 1) + F = 9 → 2F + 1 = 9 → 2F = 8 → F = 4
Then D = 5
From (2): A + 4 = 10 → A = 6

Used: 1, 2, 3, 4, 5, 6 → perfect!

✔ Solution for #4:
Top = 6, Bottom left = 5, Bottom right = 4

---

Final Answer:

1) Left middle: 5, Right middle: 4, Bottom middle: 6
2) Top: 1, Left middle: 6, Bottom right: 5
3) Top: 2, Right middle: 3, Bottom right: 6
4) Top: 6, Bottom left: 5, Bottom right: 4