Worksheet with shape analogies for educational practice.

Let's solve this analogy puzzle step by step. The image presents a series of analogies in the form:

A : B :: C : ?

Each row shows a relationship between two shapes on the left, and then asks for the corresponding shape on the right that maintains the same relationship with the third shape.

We'll analyze each row one at a time.

---

Row 1:

```
△ : △□ :: □ : □□
```

Wait — actually, let's look carefully at the format. It seems like it's:

Shape A : Shape B :: Shape C : ?

But the notation might be ambiguous. Let’s interpret based on standard analogy patterns.

Looking closely:

- First row:
`△ : △□ :: □ : □□`
But actually, the symbols are:
- Left side: Triangle → Triangle with a square inside? Or maybe triangle next to square?
Wait — looking again at the image (based on your description), it appears the format is:

Each row has:
- First pair: Two shapes separated by a colon
- Then a double colon (indicating "as")
- Second pair: One shape, then a blank

So likely:
A : B :: C : ?

And we need to find the missing shape such that the relationship from A to B is the same as from C to ?

Let’s go row by row.

---

Row 1:

```
△ : △□ :: □ : ?
```

Wait — the second symbol is not clearly defined. Let me reconstruct based on common logic puzzles.

Actually, looking at your text:

> △ : △□ :: □ : □□

Wait — perhaps the second shape is a triangle with a square inside? Or is it a triangle followed by a square?

But given the layout, it seems more likely that:

- First column: a triangle
- Second column: a triangle with a square inside or next to?
But in many such puzzles, when you see △ : △□, it could mean a transformation.

Alternatively, perhaps it's:

- △ : △□ → meaning triangle becomes triangle with a square added?
But that doesn't make sense unless there's a pattern.

Wait — perhaps it's about adding a square?

But let’s re-express all rows properly.

From your input:

```
Analogies

△ : △□ :: □ : □□
□ : ■ :: ○ : ?
○ : ◎ :: △ : ?
□ : ■ :: ○ : ?
□ : ◻× :: ○ : ?
△ : ▽ :: ■ : ?
```

Wait — now I see. There are some symbols that are unclear. Let's clarify:

Based on standard notation:

- △ = triangle
- □ = square
- ■ = filled square (black)
- ○ = circle
- ◎ = circle with dot in center (bullseye)
- ◻× = square with cross (X) inside
- ▽ = downward-pointing triangle (inverted triangle)
- ■ = solid black square

Now let’s parse each row.

---

Row 1:

```
△ : △□ :: □ : □□
```

Wait — but △□ and □□ aren't standard. Are they composite shapes?

Possibility: Maybe the second shape in each pair is a combination.

But in standard analogies, it's usually a transformation.

Alternatively, maybe the colon separates two shapes, and the analogy is shape A transforms to shape B, so shape C should transform to ? in the same way.

But in Row 1:
- △ : △□ → triangle to triangle with square? That seems odd.
- But wait — perhaps it's not a composite, but a change in fill?

Wait — let’s check the actual symbols.

Looking at the image description, perhaps:

- In Row 1:
Left: triangle (empty) : triangle (with a square inside?)
Right: square (empty) : square (with a square inside?)

But that seems inconsistent.

Wait — another possibility: maybe the second shape is a filled version?

But no, the second shape in first pair is shown as △□ — which might mean triangle and square side by side?

That would be unusual.

Alternatively, perhaps it's a typo or misrendering.

Wait — let’s consider a different interpretation.

Maybe the colon is separating two shapes, and the analogy is:

A is to B as C is to D

So:

Row 1:
- △ is to △□ as □ is to □□

But what does △□ mean? Is it a triangle and a square? Then the relationship is “triangle paired with square”?

But then △ is alone, and △□ is paired — so maybe it's “add a square”.

So:
△ → △ + □ (add a square)
Then □ → □ + □ = □□

So answer: □□ (square with another square? or two squares?)

But that seems messy.

Wait — perhaps the second shape is a filled version.

Look at Row 2:

```
□ : ■ :: ○ : ?
```

Here, square → filled square (■). So filling the shape.

So the transformation is: outline → filled

Then: ○ → filled circle → ● (but not shown)

But the answer space is blank.

So likely: ○ → ● (filled circle)

But in the image, the answer might be a filled circle.

But let’s continue.

Row 3:
```
○ : ◎ :: △ : ?
```

○ → ◎ → circle with a dot in center (bullseye)

So transformation: add a dot in center.

Then △ → triangle with a dot in center? So a triangle with a central dot.

But we don’t have such symbol. Or maybe it’s a filled triangle?

No — ◎ is not filled, it's a circle with a dot.

So probably: add a dot in center.

So △ → △ with dot inside.

But if not available, maybe the answer is a filled triangle?

Wait — but that’s different.

Alternatively, maybe the dot is a smaller shape.

But let’s think.

Row 4:
```
□ : ■ :: ○ : ?
```

Same as Row 2? Wait — Row 2 and Row 4 are identical?

Wait — your list:

```
Row 1: △ : △□ :: □ : □□
Row 2: □ : ■ :: ○ : ?
Row 3: ○ : ◎ :: △ : ?
Row 4: □ : ■ :: ○ : ? ← same as Row 2
Row 5: □ : ◻× :: ○ : ?
Row 6: △ : ▽ :: ■ : ?
```

Wait — Row 2 and Row 4 are the same? That can't be.

Perhaps typo.

Wait — maybe Row 4 is:
```
□ : ◻× :: ○ : ?
```

Yes — you wrote:
> □ : ◻× :: ○ : ?

And Row 5 is:
> □ : ◻× :: ○ : ?

Wait — no:

You wrote:
```
□ : ■ :: ○ : ?
○ : ◎ :: △ : ?
□ : ■ :: ○ : ?
□ : ◻× :: ○ : ?
△ : ▽ :: ■ : ?
```

Wait — Row 2 and Row 4 are both:
```
□ : ■ :: ○ : ?
```

That’s duplicate.

But Row 3 is:
```
○ : ◎ :: △ : ?
```

And Row 5:
```
□ : ◻× :: ○ : ?
```

So likely:

- Row 1: △ : △□ :: □ : □□
- Row 2: □ : ■ :: ○ : ?
- Row 3: ○ : ◎ :: △ : ?
- Row 4: □ : ■ :: ○ : ? ← duplicate?
- Row 5: □ : ◻× :: ○ : ?
- Row 6: △ : ▽ :: ■ : ?

Wait — perhaps Row 4 is meant to be different.

But in your message:

> □ : ■ :: ○ : ?
> ○ : ◎ :: △ : ?
> □ : ■ :: ○ : ?
> □ : ◻× :: ○ : ?
> △ : ▽ :: ■ : ?

So Row 2 and Row 4 are identical? That’s odd.

Unless it’s a typo.

But let’s assume it’s intentional.

Wait — maybe the fourth row is:

```
□ : ◻× :: ○ : ?
```

Yes — you wrote:

> □ : ◻× :: ○ : ?

So let’s number them properly:

1. △ : △□ :: □ : □□
2. □ : ■ :: ○ : ?
3. ○ : ◎ :: △ : ?
4. □ : ◻× :: ○ : ?
5. △ : ▽ :: ■ : ?

Wait — you listed five rows after "Analogies", but six lines.

Wait — count:

1. △ : △□ :: □ : □□
2. □ : ■ :: ○ : ?
3. ○ : ◎ :: △ : ?
4. □ : ■ :: ○ : ?
5. □ : ◻× :: ○ : ?
6. △ : ▽ :: ■ : ?

Ah — yes, six rows.

But Row 2 and Row 4 are both "□ : ■ :: ○ : ?", so possibly a duplication error.

But maybe it's intentional — perhaps testing consistency.

But let’s proceed.

---

Let’s reinterpret each row carefully.

---

Row 1:

```
△ : △□ :: □ : □□
```

This is ambiguous.

If △□ means triangle and square together, then:

△ → △ + □ (add a square)

Then □ → □ + □ = □□

So answer: two squares? Or square with another square?

But □□ might mean two squares side by side.

So the transformation is: add a square

But why only in the first case?

Alternatively, perhaps it’s a triangle with a square inside?

But △□ is not standard.

Another idea: perhaps it’s rotation or reflection?

But unlikely.

Wait — perhaps the second shape is a filled version?

But △□ doesn’t look like filled.

Alternatively, maybe it’s a composite shape.

But let’s skip and come back.

---

Row 2:

```
□ : ■ :: ○ : ?
```

Square → filled square (■)

So transformation: outline to filled

Therefore, circle → filled circle → ●

So answer: filled circle (●)

---

Row 3:

```
○ : ◎ :: △ : ?
```

Circle → circle with a dot in center (◎)

So transformation: add a dot in the center

Therefore, triangle → triangle with a dot in center

Answer: △ with a dot in center (like ◎ but triangle)

But since no such symbol, maybe it's represented as a triangle with a small circle inside.

But in multiple-choice style, it might be a filled triangle? No — that’s different.

So likely: triangle with a central dot.

But let’s see if other interpretations fit.

---

Row 4:

```
□ : ■ :: ○ : ?
```

Same as Row 2!

So again: outline → filled → circle → filled circle → ●

Answer: ●

---

Row 5:

```
□ : ◻× :: ○ : ?
```

Square → square with an X inside (cross)

So transformation: add an X inside

Therefore, circle → circle with an X inside

Answer: ○ with × inside → like a target or X inside circle

Symbol: ○× or ◯× (circle with cross)

---

Row 6:

```
△ : ▽ :: ■ : ?
```

Triangle → inverted triangle (▽)

So transformation: flip upside down

Therefore, filled square (■) → flipped filled square?

But a square flipped upside down looks the same.

So answer: ■ (same shape)

Because squares are symmetric under rotation.

So flipping a square gives the same appearance.

Thus, answer: ■

---

Now go back to Row 1.

Row 1:

```
△ : △□ :: □ : □□
```

We’re stuck here.

But notice: in the original, it might be:

△ : △ with something :: □ : □ with something

But △□ might be a typo.

Alternatively, perhaps it’s:

△ : △ with a square inside :: □ : □ with a square inside?

But that would be strange.

Another idea: perhaps the second shape is a combination: triangle and square.

So △ → △ and □

Then □ → □ and □ → two squares

So answer: □□ (two squares side by side)

But then the relationship is “add a square”

But why would triangle get a square added, but square gets another square added?

Inconsistent.

Alternatively, maybe it’s filling?

But △□ isn’t filled.

Wait — perhaps the second shape is a filled triangle?

But it’s written as △□ — not ■.

Unless □ is a typo.

Another possibility: perhaps it’s color change or pattern.

Wait — look at Row 5: □ : ◻× → square with X inside

So adding internal pattern.

Similarly, Row 2: □ : ■ → filling

Row 3: ○ : ◎ → adding dot

Row 5: □ : ◻× → adding X

Row 6: △ : ▽ → flipping

So Row 1 might be similar.

But △ : △□ — what is △□?

Wait — perhaps it’s a triangle with a square attached?

Then □ : □□ — square with a square attached?

So transformation: attach a square

Then □ → □ with a square attached → □□

So answer: □□

But that’s consistent.

So the rule is: add a square to the shape

So triangle gets a square added → △□
Square gets a square added → □□

So answer: □□

Even though it’s not a single shape, it might be acceptable.

Alternatively, maybe it’s a composite.

But in that case, the answer is two squares.

But in the context, perhaps it's meant to be a square with a square inside?

But ◻× is used for cross, not for another square.

So likely: add a square → so answer is two squares.

But let’s accept that.

---

Summary of Answers:

1. △ : △□ :: □ : □□ → Answer: □□ (two squares, or square with square added)
2. □ : ■ :: ○ : ? → Answer: ● (filled circle)
3. ○ : ◎ :: △ : ? → Answer: △ with a dot in center
4. □ : ■ :: ○ : ? → Same as 2 → ●
5. □ : ◻× :: ○ : ? → Answer: ○ with × inside (circle with cross)
6. △ : ▽ :: ■ : ? → Answer: ■ (same, because square is symmetric)

---

But let’s double-check Row 3.

○ → ◎: circle with dot in center

So transformation: add a dot in center

So △ → triangle with dot in center

So answer: a triangle with a small dot in the middle.

Similarly, Row 5: square with X → circle with X

Row 6: triangle → inverted triangle → filled square → inverted filled square → but square looks same → so ■

---

Now, if we had to pick symbols, the answers would be:

1. □□
2. ●
3. △ with dot (perhaps drawn as △ with • inside)
4. ●
5. ○ with × (or ◯×)
6. ■

But since the image is not visible, and we're relying on text, the best we can do is describe.

But perhaps in the original image, the symbols are standardized.

---

Wait — another thought: in Row 1, perhaps △□ means triangle with a square inside?

But that’s not standard.

Alternatively, perhaps it’s a change in orientation?

But △ to △□ doesn’t suggest that.

Another idea: perhaps the second shape is a different type.

Wait — look at Row 6: △ : ▽ → flip

So inversion.

Row 3: ○ : ◎ → add dot

Row 2: fill

Row 5: add cross

So Row 1 might be: add a square or combine with square

But let’s assume it’s consistent.

Given the ambiguity in Row 1, and since Rows 2–6 are clearer, let’s focus on those.

But since the user expects a solution, let’s provide the most plausible answers.

---

Final Answers:

1. □□
2. ●
3. △ with a dot in center
4. ●
5. ○ with × inside
6. ■

But since the question is to "solve the problem", and it's an image, perhaps the missing shapes are to be filled in.

But without seeing the image, we infer based on logic.

However, notice that in Row 1, the second shape might be a filled triangle?

But it’s written as △□ — not ■.

Unless □ is a typo for ■.

But that’s speculation.

Alternatively, perhaps the pattern is:

- Row 1: triangle → triangle with a square inside → so square → square with a square inside? Unlikely.

Another idea: perhaps it’s number of sides?

Triangle (3) → triangle with square (4) — no.

Or perhaps addition of a shape.

But let’s accept the earlier interpretation.

---

Most Plausible Solution:

Assuming the analogies are:

1. Add a square to the shape → □ → □□
2. Fill the shape → ○ → ●
3. Add a dot in center → △ → △ with dot
4. Fill the shape → ○ → ●
5. Add a cross inside → ○ → ○ with ×
6. Flip upside down → ■ → ■ (no change)

So answers:

1. □□
2. ●
3. △ with dot
4. ●
5. ○ with ×
6. ■

Since the user might expect symbols, and assuming standard representations:

- ● = filled circle
- ○ with × = circle with cross
- △ with dot = triangle with center dot
- ■ = filled square

For Row 1, if △□ means triangle with square attached, then □□ is square with square attached.

So final answer set:

Row 1: □□
Row 2: ●
Row 3: △ with dot in center
Row 4: ●
Row 5: ○ with ×
Row 6: ■

But since Row 2 and Row 4 are identical, it’s likely a test of consistency.

---

Conclusion:

The solution is based on identifying the transformation in each analogy:

- Fill the shape → □ → ■, so ○ → ●
- Add a dot in center → ○ → ◎, so △ → △ with dot
- Add a cross inside → □ → ◻×, so ○ → ○ with ×
- Flip upside down → △ → ▽, so ■ → ■ (symmetric)
- Add a square → △ → △□, so □ → □□

Thus, the missing shapes are:

1. □□
2. ●
3. △ with a dot in center
4. ●
5. ○ with × inside
6. ■

These are the solutions to the analogies.