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

Let’s solve each question one by one, step by step.

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Q.1: Find out A and B so as to make sum equal on each side of this triangle.

We have a triangle with numbers at the corners: 3 (top), 2 (bottom left), 1 (bottom right).
On the sides, we have:
- Left side: 3 — A — 2 → sum = 3 + A + 2
- Right side: 3 — B — 1 → sum = 3 + B + 1
- Bottom side: 2 — 6 — 1 → sum = 2 + 6 + 1 = 9

So, all sides must add up to 9.

Left side: 3 + A + 2 = 9 → 5 + A = 9 → A = 4

Right side: 3 + B + 1 = 9 → 4 + B = 9 → B = 5

✔ Check:
Left: 3+4+2=9
Right: 3+5+1=9
Bottom: 2+6+1=9 → Perfect!

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

We see triangles with letters and numbers:

First: ▲ 4P → maybe “4” is number, “P” is letter? But next is ▼ 3Q → inverted triangle.

Wait — look at positions:

▲ 4P
▼ 3Q
▲ 2R
▼ ?

Pattern:
Triangle alternates direction: up, down, up, down → last should be ▼

Numbers: 4, 3, 2, ? → decreasing by 1 → next is 1

Letters: P, Q, R, ? → alphabet order → next is S

So answer: 1S

But written inside triangle: ▼ 1S

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

Look:
3 × 2 = 6
6 × 2 = 12
12 × 2 = 24

So next is 24

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Q.4: Shapes with numbers: 80, 40, 20, ?

Each time divided by 2:
80 ÷ 2 = 40
40 ÷ 2 = 20
20 ÷ 2 = 10

Answer: 10

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

Look at first digit: 6, 7, 8 → next is 9

Middle letter: A, B, C → next is D

Last digit: always 7 → stays 7

So next is 9D7

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Q.6: Pyramid of boxes

Top box: 30
Middle row: 20 and ?
Bottom row: 5, 10, 15

Notice: Each box above is sum of two below it?

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

Wait — maybe top is sum of middle? 20 + ? = 30 → ? = 10

Then check if middle boxes are sums of bottom?

Left middle: 20 → above 5 and 10 → 5 + 10 = 15 → not 20.

Alternative idea: Maybe each box is sum of the two directly under it?

Top: 30 → should be sum of 20 and ? → so ? = 10

Now, does 20 come from 5 and 10? 5 + 10 = 15 → no.

Wait — look at bottom row: 5, 10, 15 → these are multiples of 5.

Middle left: 20 → which is 5 + 15? Not adjacent.

Another idea: Perhaps the value in each box is the sum of the two boxes diagonally below?

Try:

Top box (30) = left middle (20) + right middle (?)

→ So ? = 10

Now, left middle (20) = bottom left (5) + bottom middle (10)? 5+10=15 → no.

Wait — what if it’s additive upward?

Bottom row: 5, 10, 15

If we add adjacent pairs:

5 + 10 = 15 → but that’s not 20.

10 + 15 = 25 → not matching.

Wait — let’s think differently.

What if the top number is the total sum?

5 + 10 + 15 = 30 → yes! Top is 30.

Then middle row: 20 and ?

If 20 + ? = 30 → ? = 10

And 20 could be 5 + 15? That’s not adjacent.

Alternatively, perhaps the middle left (20) is sum of bottom left and bottom right? 5 + 15 = 20 → YES!

Then middle right (?) should be sum of bottom middle and... wait, only three bottom boxes.

Actually, standard pyramid addition:

In many such puzzles, each upper box is sum of the two boxes directly beneath it.

So:

Box above 5 and 10 → should be 5 + 10 = 15 → but it’s labeled 20 → doesn’t match.

Unless… maybe it’s not direct sum.

Wait — look again:

Bottom: 5, 10, 15

Middle: 20, ?

Top: 30

What if:

20 = 5 + 15 (first and last of bottom)

Then ? = 10 + something? Only three numbers.

Another idea: Maybe the middle row numbers are sums of non-adjacent?

Or perhaps it’s a different rule.

Wait — try this:

Top = 30

Left middle = 20 → which is 5 * 4? No.

Notice: 5, 10, 15 → arithmetic sequence.

20 is 5*4, 10*2, etc.

Perhaps the missing number is 10, because 20 + 10 = 30, and 10 is also in the bottom.

But let’s test consistency.

Suppose the rule is: each box is sum of the two boxes immediately below it.

Then:

The box above 5 and 10 should be 15 → but it’s 20 → contradiction.

Unless the diagram is arranged differently.

Looking back at user input:

It says:

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

So visually, 20 is above 5 and 10? Or above 5 and the space between?

Standard interpretation: In a pyramid, each parent node covers two children.

So:

- 30 is above 20 and ?
- 20 is above 5 and 10
- ? is above 10 and 15

Ah! That makes sense.

So:

20 = 5 + 10 → 15? But it’s 20 → still not matching.

Wait — 5 + 10 = 15, but box says 20.

Unless it’s multiplied?

5 * 4 = 20? Arbitrary.

Another thought: Maybe the numbers represent something else.

Wait — let’s calculate differences.

From bottom to top:

If 20 is above 5 and 10, and 20 - (5+10) = 5 → extra 5.

Similarly, ? above 10 and 15 → if same logic, ? = 10 + 15 + 5 = 30? Then top would be 20 + 30 = 50 ≠ 30.

No.

Perhaps the top is sum of all bottom: 5+10+15=30 → correct.

Then middle row: 20 and ? must add to 30 → ? = 10

And 20 might be 5 + 15 (skipping middle), and 10 is just 10? Doesn't fit.

Wait — here's a better idea:

Maybe the value in each box is the sum of the numbers in the boxes directly below it, but in this case, the middle left box (20) is actually above the first and second bottom boxes? But 5+10=15≠20.

Unless there's a typo or I'm misreading.

Let me re-express:

Assume the structure is:

Level 1: [30]

Level 2: [20] [?]

Level 3: [5] [10] [15]

In standard pyramid addition puzzles, each number is the sum of the two numbers directly below it.

So:

[20] should be sum of [5] and [10] → 5+10=15 → but it's 20 → inconsistency.

Unless the puzzle uses a different rule.

Another possibility: The numbers are products? 5*4=20, but why 4?

Or perhaps it's cumulative.

Let’s try this:

What if the top number is the sum of the entire bottom row: 5+10+15=30 → good.

Then the middle row: 20 and ?

If 20 represents the sum of the first two bottom numbers: 5+10=15 → not 20.

5+15=20 → oh! 5 and 15 are the outer ones.

Then ? should be sum of the inner ones? But only one inner: 10.

Not working.

Perhaps ? is 10, and 20 is given, and 20 + 10 = 30, and we accept that 20 is not derived from below, but the puzzle only asks for the missing number based on top being sum of middle.

That seems forced.

Wait — let's look at the values:

Bottom: 5, 10, 15 → increasing by 5.

Middle: 20, ?

Top: 30

If we assume that the middle left (20) is 5 * 4, but 4 is not related.

Another idea: Perhaps each number is multiplied by its position.

I think I found it:

Notice that 5 + 15 = 20 (the left middle)

Then the right middle should be 10 + ? but only three numbers.

Perhaps the right middle is just 10, and it's placed there.

But then top is 20 + 10 = 30, which matches.

And 20 = 5 + 15, which are the first and last of bottom.

Then ? = 10, which is the middle of bottom.

So the rule is: the left middle box is sum of first and last bottom, right middle box is the middle bottom number.

Then top is sum of middle row.

So ? = 10

This works numerically.

To confirm:

Bottom: 5, 10, 15

Left middle: 5 + 15 = 20

Right middle: 10

Top: 20 + 10 = 30 → perfect.

So answer is 10

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Q.7: Table

Row 1: 3, 4, 5, 6

Row 2: 9, 15, 25, ?

Look at relationship between row1 and row2.

3 → 9 = 3^2

4 → 15? 4^2=16 ≠15

3*3=9

4*3.75=15? Not integer.

Differences:

From 3 to 9: +6

4 to 15: +11

5 to 25: +20

6 to ?: ?

Not clear.

Products:

3*3=9

4*3.75=15 — no.

Another idea: 9 = 3*3, 15=3*5, 25=5*5 — not consistent.

Look at squares:

3^2 = 9

4^2 = 16, but we have 15 → close.

5^2 = 25 → matches.

6^2 = 36 → so ? = 36?

But 4 gives 15, not 16.

Why 15 for 4?

Perhaps it's n*(n+1) or something.

For n=3: 3*3=9

n=4: 4*3.75 — no.

Another approach: differences between row2 numbers.

15 - 9 = 6

25 - 15 = 10

Next difference? 6, 10, then 14? (increasing by 4)

So 25 + 14 = 39

Then ? = 39

Check if fits.

Or ratios: 9/3=3, 15/4=3.75, 25/5=5 — not constant.

Notice: 9 = 3^2

15 = 4^2 - 1 = 16-1

25 = 5^2

So for 6: 6^2 = 36, or 6^2 -1 =35? Pattern not clear.

For odd positions: 3->9=3^2, 5->25=5^2

Even positions: 4->15=4^2 -1, so 6->6^2 -1=35

That could be.

So ? = 35

Let me verify:

Position 1 (odd): 3^2 = 9

Position 2 (even): 4^2 -1 = 16-1=15

Position 3 (odd): 5^2 = 25

Position 4 (even): 6^2 -1 = 36-1=35

Yes, consistent.

So answer is 35

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Now, compiling all answers:

Q1: A=4, B=5

Q2: 1S

Q3: 24

Q4: 10

Q5: 9D7

Q6: 10

Q7: 35

Final Answer:

Q1: A = 4, B = 5
Q2: 1S
Q3: 24
Q4: 10
Q5: 9D7
Q6: 10
Q7: 35