Find A and B to make the sum equal on each side of the triangle.

Let's solve each question one by one, starting with Q.1, which is the first and most prominent problem in your image.

---

Q.1: Find out A and B so that the sum is equal on each side of this triangle.

We have a triangle with numbers at the vertices and midpoints of sides:

```
3
/ \
A B
/ \
2---6---1
```

So the three sides are:
- Left side: 3 → A → 2
- Right side: 3 → B → 1
- Base: 2 → 6 → 1

We need all three sides to have the same sum.

Let’s compute the base sum first:

> Base: 2 + 6 + 1 = 9

Now set up equations for the other two sides:

- Left side: 3 + A + 2 = A + 5
This must equal 9 → A + 5 = 9 → A = 4

- Right side: 3 + B + 1 = B + 4
This must equal 9 → B + 4 = 9 → B = 5

✔ So, A = 4, B = 5

Check:
- Left: 3 + 4 + 2 = 9 ✔
- Right: 3 + 5 + 1 = 9 ✔
- Base: 2 + 6 + 1 = 9 ✔

✔️ All sides sum to 9.

---

Q.2: Study the pattern and fill in the missing numbers.

We have four triangles:

```
△ 4P △ 3Q △ 2R △ ?
```

Each triangle has a number and a letter.

Let’s look at the pattern:

| Triangle | Number | Letter |
|---------|--------|--------|
| 1 | 4 | P |
| 2 | 3 | Q |
| 3 | 2 | R |
| 4 | ? | ? |

Notice:
- Numbers: 4, 3, 2 → decreasing by 1 → next should be 1
- Letters: P, Q, R → consecutive letters → next should be S

So the missing triangle should be: 1S

Answer: 1S

---

Q.3: Clouds with numbers

We have clouds:

```
{3} {6} {12} {?}
```

Sequence: 3, 6, 12, ?

Look at pattern:

- 3 × 2 = 6
- 6 × 2 = 12
- 12 × 2 = 24

So the next number is 24

Answer: 24

---

Q.4: Semi-circles with numbers

```
(80) (40) (20) (?)
```

Sequence: 80, 40, 20, ?

Pattern: Each number is half the previous:

- 80 ÷ 2 = 40
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10

Answer: 10

---

Q.5: Diamond shapes with numbers

```
6A7 7B7 8C7 ?
```

Each diamond has a number in the form: [digit][letter][7]

So:
- First: 6A7 → 6, A, 7
- Second: 7B7 → 7, B, 7
- Third: 8C7 → 8, C, 7
- Fourth: ? → ?, ?, 7

Look at the first digit: 6, 7, 8 → increasing by 1 → next: 9

Letter: A, B, C → consecutive letters → next: D

So the fourth diamond is: 9D7

Answer: 9D7

---

Q.6: Pyramid with numbers

```
30
20 ?
5 10 15
```

This looks like a pyramid where each number above is derived from the two below it.

Let’s label positions:

```
30
20 X
5 10 15
```

We know:
- 20 is above 5 and 10 → maybe 5 + 10 = 15? But we have 20 → not matching.
- Maybe multiplication or another operation?

Try:
- 5 + 10 = 15 → but above is 20 → not matching
- 5 × 10 = 50 → too big
- Maybe average? (5+10)/2 = 7.5 → no

Wait — what if it's sum of the two numbers below plus something?

But let's try looking at how 30 is formed from 20 and X.

Suppose the top number is sum of the two below it:

→ 20 + X = 30 → X = 10

But then check if 20 comes from 5 and 10:

→ 5 + 10 = 15 ≠ 20 → doesn't work.

Alternative idea: perhaps the top number is the sum of the bottom row?

Bottom: 5 + 10 + 15 = 30 → yes! That matches the top.

So total sum of bottom row = 30.

Now, the middle layer: 20 and X

Is there a relation?

Maybe the left middle number (20) is sum of 5 and 10 → 5 + 10 = 15 ≠ 20 → no

Wait — maybe each upper number is sum of two adjacent lower numbers?

Then:
- Left middle: 5 + 10 = 15 → but given as 20 → no
- Right middle: 10 + 15 = 25 → but we don’t know

But 20 is given, so maybe not.

Alternative idea: maybe the middle row numbers are sums of adjacent pairs, and the top is sum of middle row?

Try:
- Middle left: 5 + 10 = 15 → but it's 20 → no
- Unless it's double?

Wait — maybe the numbers are not directly sums.

Another idea: Look at the pattern of differences.

Or consider that the top number is 30, and the bottom row adds to 30.

Now, the left middle number is 20, and it's above 5 and 10.

Could it be that 20 = 5 × 4? 10 × 2? Not clear.

Wait — perhaps each number is the sum of the two numbers below it, but shifted?

Try this:

Assume the structure is:

```
30
20 X
5 10 15
```

Suppose:
- 20 = 5 + 10 + ? → 5 + 10 = 15 → difference 5
- X = 10 + 15 = 25 → then 20 + 25 = 45 ≠ 30 → no

Alternatively, suppose the top number is the sum of the two middle numbers:

→ 20 + X = 30 → X = 10

Now, can we verify if 20 and 10 are related to the bottom?

Try:
- 20 = 5 + 10 + 5? No
- 20 = 5 × 4 → no

Wait — perhaps each middle number is the sum of the two numbers below it?

But:
- 20 = 5 + 10 = 15 → no
- 10 = 10 + 15 = 25 → no

No.

Wait — what if the middle numbers are not sums, but the top number is sum of bottom row, and the middle numbers are averages or something else?

But 5 + 10 + 15 = 30 → matches top.

Now, perhaps the middle numbers are sums of adjacent bottom numbers, but scaled?

Try:
- Left middle: 5 + 10 = 15 → but it's 20 → off by 5
- Right middle: 10 + 15 = 25 → but we don't know

But if top = 20 + X = 30 → X = 10

But then right middle is 10, but 10 + 15 = 25 → not matching.

Alternatively, maybe the middle numbers are multiplied?

Wait — think differently.

What if the top number is sum of the two middle numbers, and each middle number is sum of two adjacent bottom numbers?

Then:
- Left middle = 5 + 10 = 15 → but it's 20 → no
- Right middle = 10 + 15 = 25 → unknown

But 15 + 25 = 40 ≠ 30

No.

Wait — what if the top number is the sum of the bottom row, and the middle numbers are partial sums?

For example:
- 20 = 5 + 10 + 5? No
- Or maybe 20 = 5 × 4?

Another idea: The middle numbers are the sum of the two numbers below them, but the top is sum of both middle numbers.

So:
- Top = 20 + X = 30 → X = 10

Now, is 20 = 5 + 10 = 15? No → still doesn't work.

Wait — unless the bottom numbers are used differently.

Let me try this: suppose the middle number is the sum of the two numbers below it, but only if they are adjacent.

But 5 and 10 → sum 15, but value is 20 → not matching.

Wait — what if it's 5 × 4 = 20? But why 4?

Or 10 × 2 = 20? Still arbitrary.

Wait — look at the positions.

Maybe the left middle number (20) is 5 × 4, but 4 isn't visible.

Alternatively, notice:

- Bottom row: 5, 10, 15 → arithmetic sequence: +5 each
- Middle left: 20 → maybe 5 + 10 + 5 = 20? Where does extra 5 come from?

Wait — perhaps the middle number is the sum of the two numbers below it, plus the one before or after?

Not helpful.

Wait — here's a better idea:

Perhaps the middle numbers are the sum of the two numbers below them, but the top is the sum of the two middle numbers.

So:
- Top = 20 + X = 30 → X = 10

Now, check if 20 can be explained from 5 and 10.

But 5 + 10 = 15 ≠ 20

Unless it's 5 × 4 = 20, but no clue.

Wait — what if the numbers are not based on addition, but on multiplication or another rule?

Let’s reverse-engineer.

Suppose:
- The top number is the sum of the bottom row: 5 + 10 + 15 = 30 → ✔
- Now, the middle numbers might be partial sums:
- Left middle: 5 + 10 = 15 → but it's 20 → no
- Right middle: 10 + 15 = 25 → but if X = 10, that doesn't match

Wait — unless the middle numbers are not derived from direct addition, but from another logic.

Wait — maybe the top number is 30, and the middle numbers are 20 and X, and the bottom row is 5, 10, 15.

Now, suppose the left middle number (20) is sum of 5 and 10 and 5 again? No.

Wait — could it be that each number in the upper level is the sum of the two numbers below it, but the top is sum of the two middle numbers?

Then:
- Top = 20 + X = 30 → X = 10
- Left middle = 5 + 10 = 15 → but it's 20 → contradiction

No.

Wait — what if the middle numbers are the sum of the two numbers below them, but the top is the sum of the entire bottom row?

Then:
- 5 + 10 + 15 = 30 → ✔
- Left middle: 5 + 10 = 15 → but it's 20 → ✘
- Right middle: 10 + 15 = 25 → but X = 10 → ✘

Still doesn't work.

Wait — maybe the middle numbers are not derived from the bottom, but from some other rule?

But 20 is given.

Another idea: perhaps the numbers are increasing by 5, and 20 is just placed there, and we need to find X such that 20 + X = 30 → X = 10

Even if the derivation is unclear, maybe the only logical way is that top = sum of middle, so:

→ 20 + X = 30 → X = 10

And the bottom row adds to 30, which matches.

So perhaps the middle numbers are not derived from the bottom, but the top is derived from the middle.

So answer: X = 10

Let’s go with that.

Answer: 10

---

Q.7: Table with numbers

```
3 4 5 6
9 15 25 ?
```

First row: 3, 4, 5, 6
Second row: 9, 15, 25, ?

Look at columns:

- Col 1: 3 → 9 → 9 = 3 × 3
- Col 2: 4 → 15 → 15 = 4 × 3.75 → not nice
- Col 3: 5 → 25 → 25 = 5 × 5

Hmm.

Wait:
- 3 → 9 = 3²
- 5 → 25 = 5²
- 4 → 15 → not 4² = 16
- 6 → ? → 6² = 36

But 4 → 15 ≠ 16

Wait — maybe not squares.

Look at ratios:

- 9 / 3 = 3
- 15 / 4 = 3.75
- 25 / 5 = 5
- ? / 6 = ?

Not constant.

Wait — look at differences:

- 9 = 3 × 3
- 15 = 3 × 5
- 25 = 5 × 5
- ? = ?

Wait — see:

- 3 → 9 = 3 × 3
- 4 → 15 = 3 × 5 → 3 and 5 are primes?
- 5 → 25 = 5 × 5
- 6 → ? = ?

Wait — maybe second row is: (first row) × (next prime?)?

Not clear.

Another idea: look at products.

Wait — 9 = 3 × 3
15 = 3 × 5
25 = 5 × 5
? = ?

Wait — 3, 3; 3, 5; 5, 5 → maybe the multipliers are increasing?

Wait — 3×3=9, 3×5=15, 5×5=25, then next could be 5×7=35?

But what about the first number?

Wait — the second row seems to be:

- 9 = 3 × 3
- 15 = 3 × 5
- 25 = 5 × 5
- ? = 5 × 7 = 35?

But 6 → 35?

But 6 is in first row.

Wait — perhaps the second row is n × m, where m is the next odd number?

Wait — 3 → 9 = 3×3
4 → 15 = 3×5 → not 4×something

Wait — maybe it's (first row) × (first row + 1)?

- 3×4 = 12 ≠ 9
- 4×5 = 20 ≠ 15

No.

Wait — look at the second row values:

9, 15, 25, ?

Now:
- 9 = 3²
- 15 = 3×5
- 25 = 5²

Now, 3, 4, 5, 6 → first row

So:
- For 3: 3² = 9
- For 4: ? → 15
- For 5: 5² = 25
- For 6: ?

But 4 → 15 → not 4² = 16

But 15 = 3×5 → maybe it's (n-1) × (n+1)?

For n=4: (4-1)(4+1) = 3×5 = 15 → ✔

For n=5: (5-1)(5+1) = 4×6 = 24 ≠ 25 → no

But 25 = 5²

Wait — for n=3: (3-1)(3+1) = 2×4 = 8 ≠ 9

No.

Wait — for n=3: 3² = 9
n=4: 3×5 = 15
n=5: 5² = 25
n=6: ?

Wait — 3, 3; 3,5; 5,5 → maybe alternating?

But 3→9=3²
4→15=3×5
5→25=5²
6→?

Now, 3 and 5 are primes.

Wait — perhaps the second row is:

- For odd n: n²
- For even n: (n-1) × (n+1)

Check:

- n=3 (odd): 3² = 9 ✔
- n=4 (even): (4-1)(4+1) = 3×5 = 15 ✔
- n=5 (odd): 5² = 25 ✔
- n=6 (even): (6-1)(6+1) = 5×7 = 35

So answer: 35

Yes!

So the pattern is:
- If n is odd → n²
- If n is even → (n-1)(n+1) = n² - 1

Check:
- n=3: 9 = 3²
- n=4: 15 = 16 - 1 = 15 ✔
- n=5: 25 = 5² ✔
- n=6: 36 - 1 = 35 ✔

So missing number is 35

Answer: 35

---

## ✔ Final Answers:

Q.1: A = 4, B = 5
Q.2: 1S
Q.3: 24
Q.4: 10
Q.5: 9D7
Q.6: 10
Q.7: 35

Let me know if you'd like diagrams or further explanation!