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 edges:

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

Wait — actually, looking closely, it seems like the triangle has:

- Top vertex: 3
- Bottom-left vertex: 2
- Bottom-right vertex: 1
- The middle edge between 2 and 1 has a number: 6
- On the left side (from 3 to 2): there’s a number A
- On the right side (from 3 to 1): there’s a number B

So the three sides are:

1. Left side: 3 → A → 2
2. Right side: 3 → B → 1
3. Base: 2 → 6 → 1

We need the sum of each side to be equal.

Let’s compute the base sum:

> Base: 2 + 6 + 1 = 9

Now set up equations for the other two sides:

- Left side: 3 + A + 2 = 5 + A
- Right side: 3 + B + 1 = 4 + B

Set both equal to 9:

- 5 + A = 9 → A = 4
- 4 + B = 9 → B = 5

✔ So:
- A = 4
- B = 5

Let’s verify:
- 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:

1. ▲ with 4P
2. ▲ with 3Q
3. ▲ with 2R
4. ▲ with ?

Looks like a sequence: 4P, 3Q, 2R, ?

We see:
- Numbers decreasing: 4, 3, 2 → next should be 1
- Letters increasing: P, Q, R → next is S

So the pattern is:
- Number: 4, 3, 2, 1
- Letter: P, Q, R, S

So the answer is: 1S

✔ Answer: 1S

---

Q.3: Clouds with numbers: 3, 6, 12, ?

Sequence: 3, 6, 12, ?

Look at the pattern:
- 3 × 2 = 6
- 6 × 2 = 12
- 12 × 2 = 24

So next number is 24

✔ Answer: 24

---

Q.4: Semi-circles with numbers: 80, 40, 20, ?

Sequence: 80, 40, 20, ?

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

✔ Answer: 10

---

Q.5: Diamonds with numbers: 6A7, 7B7, 8C7, ?

Each diamond has format: X _ Y, where X increases from 6 to 7 to 8, and last digit is 7.

So:
- First: 6A7 → A is unknown
- Second: 7B7 → B is unknown
- Third: 8C7 → C is unknown
- Fourth: ? → probably 9D7?

But we need to find the pattern in the middle letters.

Let’s assume the middle letter corresponds to some logic.

Let’s look at the numbers:
- First: 6 A 7
- Second: 7 B 7
- Third: 8 C 7

Notice:
- First digit increases: 6 → 7 → 8 → likely 9
- Last digit is always 7
- Middle letter: A, B, C → suggests alphabetical order

So next should be 9D7

But wait — could there be more to it?

Alternatively, perhaps the middle letter represents a digit?

But A, B, C are letters, not digits.

Wait — maybe the middle letter is related to the sum or difference?

Try:
- 6 + 7 = 13 → but A is 1st letter?
- 7 + 7 = 14 → B is 2nd?
- 8 + 7 = 15 → C is 3rd?

Not matching.

Alternatively, maybe the middle letter corresponds to the first digit?

But 6 → A, 7 → B, 8 → C → yes! That makes sense.

So:
- 6 → A (6th letter? No: A=1, B=2, C=3...)

Wait — A is 1st letter, but first number is 6.

But:
- 6 → A
- 7 → B
- 8 → C

That would mean:
- 6 → A (1st letter)
- 7 → B (2nd)
- 8 → C (3rd)

No — that doesn't match.

But if it's number = position of letter?

- A = 1 → but first number is 6 → no.

Wait — what if the letter is the same as the first digit?

6 → A? But A is not 6.

But A = 1, B = 2, C = 3...

But here:
- First: 6A7 → A is 1st letter
- Second: 7B7 → B is 2nd
- Third: 8C7 → C is 3rd

So the letter is increasing alphabetically, while the first digit increases by 1.

So:
- 6A7 → A
- 7B7 → B
- 8C7 → C
- Next: 9D7 → D

So the missing one is 9D7

✔ Answer: 9D7

---

Q.6: Pyramid-like structure

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

It looks like a pyramid where each number is derived from the two below.

Bottom row: 5, 10, 15

Second row: 20 and ?

Top: 30

Let’s see how 20 is formed from 5 and 10.

Possibility: 5 + 10 = 15 → but it’s 20 → too small.

5 × 10 = 50 → too big.

Maybe: 5 + 10 + 5 = 20? Not clear.

Wait — perhaps it's sum of two numbers below, but not adjacent?

But 5 and 10 are adjacent.

Wait — maybe the rule is: each upper number is the sum of the two numbers directly below it?

So:
- 20 is above 5 and 10 → 5 + 10 = 15 ≠ 20 → no.

But 5 + 10 = 15, not 20.

What if it's 5 × 2 + 10 = 20? Or 10 × 2 = 20?

Wait — 10 is in the middle.

Wait — maybe the top number 30 is sum of 20 and ?

So: 20 + ? = 30 → ? = 10

Then the missing number is 10

Now check if that fits with bottom row.

So second row: 20 and 10

Now, 20 is above 5 and 10 → 5 + 10 = 15 ≠ 20 → still not matching.

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

Standard pyramid: each number is sum of two below.

But here:

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

So:
- 20 should come from 5 and 10 → 5 + 10 = 15 → not 20
- ? should come from 10 and 15 → 10 + 15 = 25
- Then top: 20 + 25 = 45 ≠ 30 → no

Alternative idea: Maybe the top number is sum of all bottom numbers?

5 + 10 + 15 = 30 → yes!

So total sum is 30.

Now, 20 is given as one of the middle numbers.

So the other middle number must be such that 20 + ? = 30 → ? = 10

So missing number is 10

But now, how are 20 and 10 derived from bottom?

Maybe:
- 20 = 5 + 10 + 5? Not consistent.

Wait — perhaps the two middle numbers are sums of adjacent pairs?

Try:
- Left middle: 5 + 10 = 15 → but it's 20 → no
- Right middle: 10 + 15 = 25 → but we want 10? No.

Another idea: Maybe it's weighted sum?

Or perhaps it's not a standard pyramid, but the top number is sum of bottom, and the middle numbers are partial sums?

But 20 is already there.

Wait — suppose the left middle is sum of left two: 5 + 10 = 15 → but it's 20 → no.

Unless it's 5 × 4 = 20, but why?

Wait — let’s try another approach.

Suppose:
- The top number is 30
- It is made from 20 and ? → so ? = 10

Now, can we justify how 20 comes from 5 and 10?

Maybe: 5 × 4 = 20 → but 10 is also there.

Wait — maybe the middle numbers are averages or something?

Alternatively, think of it as:

- 5 and 10 → 20
- 10 and 15 → ?

If 5 and 10 give 20, then maybe 10 and 15 give 30? But then 20 + 30 = 50 ≠ 30

No.

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

But 5 + 10 + 15 = 30 → matches top.

Now, the left middle is 20, which is greater than 5+10=15.

Wait — unless it's 5 × 4 = 20, but 4 from where?

Alternatively, maybe the middle numbers are not derived from the bottom, but the top is sum of the two middle numbers.

So:
- 20 + ? = 30 → ? = 10

And we don’t care about derivation from bottom.

But the bottom row might be used to derive the middle ones.

Wait — look again:

Bottom: 5, 10, 15

Middle: 20, ?

Top: 30

Perhaps:
- 20 = 5 + 10 + 5? Not logical.

Wait — maybe it's multiplication?

5 × 10 = 50 → no.

Wait — what if the middle numbers are sums of adjacent pairs, but scaled?

Try:
- 5 + 10 = 15 → but 20 is there → no

Wait — unless the 20 is 5 × 4, but 4 isn’t there.

Another idea: Perhaps the top number is the sum of the two middle numbers, and each middle number is the sum of the two numbers below it.

So:
- Left middle: 5 + 10 = 15 → but it's 20 → contradiction

Unless the numbers are not aligned.

Wait — maybe the alignment is:

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

So:
- 20 is above 5 and 10
- ? is above 10 and 15

Then:
- 20 = 5 + 10 + 5? No

Wait — what if the top number is sum of all three bottom numbers: 5 + 10 + 15 = 30 → correct.

Then the two middle numbers are partial sums?

But 20 is larger than 5+10=15.

Wait — unless it's 5 × 4 = 20, but why?

Wait — maybe the middle number is 5 × (10/2) = 25? No.

Wait — let’s look at the numbers: 5, 10, 15 → arithmetic sequence.

Sum = 30

Now, 20 is given.

If 20 + ? = 30 → ? = 10

So the missing number is 10

Even if we don't know the derivation, since top = sum of middle numbers, and 20 + ? = 30 → ? = 10

So likely the intended answer is 10

✔ Answer: 10

---

Q.7: Table

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

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

Find relation between rows.

Look at:
- 3 → 9 → 3×3 = 9
- 4 → 15 → 4×3.75 = 15? Not nice.
- 5 → 25 → 5×5 = 25

Wait — 3→9 = 3²
5→25 = 5²
But 4→15? 4² = 16, not 15

Hmm.

Wait — maybe it's (n+1)² - 1?

For n=3: (3+1)² -1 = 16 -1 = 15 → no, we have 9

Wait — maybe different pattern.

Try:
- 3 → 9 = 3 × 3
- 4 → 15 = 4 × 3.75 → no
- 5 → 25 = 5 × 5
- 6 → ? = 6 × ?

But 3×3=9, 5×5=25 → squares

But 4→15? 4×3.75 → no

Wait — maybe not multiplication.

Try differences:

From 3 to 9: +6
4 to 15: +11
5 to 25: +20

Now differences: +6, +11, +20

Differences of differences: 11-6=5, 20-11=9 → not constant

Wait — maybe it's n² + something?

- For n=3: 9 = 3² → yes
- n=4: 15 → 4² = 16 → 16 -1 = 15
- n=5: 25 = 5² → yes
- n=6: 6² = 36 → maybe 36?

But pattern: 3², 4² -1, 5², 6²?

That would be: 9, 15, 25, 36

But 4² -1 = 15 → yes

So:
- n=3: 3² = 9
- n=4: 4² -1 = 15
- n=5: 5² = 25
- n=6: 6² = 36

But alternating? 3², 4²-1, 5², 6² → not consistent

Wait — maybe it's (n+1)² - 1?

For n=3: (4)² -1 = 16-1=15 → no

Wait — look at the second row: 9, 15, 25, ?

Try to see if it's related to multiplication:

- 3×3 = 9
- 4×3.75 = 15 → no
- 5×5 = 25
- 6×? = ?

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

Wait — maybe the second row is (first row) × (something)

But 3→9 = ×3
4→15 = ×3.75
5→25 = ×5
6→? = ×?

Not consistent.

Another idea: Look at the products:

Wait — maybe the second row is (first row)² - something?

- 3² = 9 → matches
- 4² = 16 → but 15 = 16 -1
- 5² = 25 → matches
- 6² = 36 → so maybe 36?

But why does 4 give 15 instead of 16?

Wait — maybe it's (n+1)² - 1 for n=4?

n=4: (4+1)² -1 = 25-1=24 → no

Wait — maybe it's n × (n+1)?

3×4 = 12 → no

Wait — let’s try:

- 3 → 9 = 3×3
- 4 → 15 = 3×5 → no
- 5 → 25 = 5×5
- 6 → ? = 6×6 = 36?

But 4→15 — not 16.

Wait — maybe it's (n^2) when n odd, (n^2 -1) when even?

- n=3 (odd): 3² = 9 ✔
- n=4 (even): 4² -1 = 15 ✔
- n=5 (odd): 5² = 25 ✔
- n=6 (even): 6² -1 = 36 -1 = 35

So answer: 35

Check: 6 → 35

Yes — pattern:
- Odd n: n²
- Even n: n² -1

So:
- 3 (odd) → 9
- 4 (even) → 15
- 5 (odd) → 25
- 6 (even) → 35

✔ Answer: 35

---

✔ Final Answers:

| Question | Answer |
|--------|--------|
| 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 explanations in a simpler form or visual diagrams!