First, I need to classify the six figures into two categories based on common features. The options are given, so I'll analyze each figure for patterns like the number of sides, symmetry, color distribution, or structural properties.

Let me list the figures again for clarity:

1. Figure ①: A pentagon with a black and white pattern. It has a star-like center with triangles.
2. Figure ②: A trapezoid divided into smaller black and white triangles.
3. Figure ③: A star shape with four points, divided into black and white triangles.
4. Figure ④: A circle divided into sections with a cross and additional lines.
5. Figure ⑤: An octagon with alternating black and white triangles.
6. Figure ⑥: A Star of David (hexagram) with black and white triangles.

Now, I need to find a common feature that groups three of them together and the other three into a different group. The options suggest possible groupings, so I'll test each option.

Starting with option A: ①⑤⑥ in one group, and ②③④ in the other.

• For ①⑤⑥: What do they have in common?

  • ① is a pentagon, ⑤ is an octagon, ⑥ is a hexagram. Different shapes, so maybe it's not about the shape.
  • Color: ① has a mix of black and white, as do ⑤ and ⑥. Not distinguishing.
  • Symmetry: All might have rotational symmetry? Let's check:
    • ①: Pentagonal symmetry? It looks like it might have rotational symmetry of 72 degrees if the pattern matches.
    • ⑤: Octagonal symmetry, likely 45-degree rotational symmetry.
    • ⑥: Star of David has 60-degree rotational symmetry.
      So all three have rotational symmetry.

• Now for ②③④:

  • ②: Trapezoid – does it have rotational symmetry? It seems like it might have reflection symmetry but not rotational. Let's see: the trapezoid is divided into triangles; if it's symmetric, maybe reflection only.
  • ③: Four-pointed star – could have rotational symmetry of 90 degrees.
  • ④: Circle with a cross – might have rotational symmetry of 90 degrees or reflection.

This grouping might not be consistent because ② might not have rotational symmetry. Let's check ② more carefully. If ② is a trapezoid with triangular divisions, it might only have reflection symmetry, not rotational. So perhaps not the best fit.

Next, option B: ①③⑤ in one group, ②④⑥ in the other.

• ①③⑤:

  • ①: Pentagon, ③: Four-pointed star, ⑤: Octagon. Different shapes.
  • Rotational symmetry:
    • ①: Possible 72-degree symmetry.
    • ③: Four points, so 90-degree rotational symmetry.
    • ⑤: Octagon, 45-degree symmetry.
      All have rotational symmetry, but so do others.
  • Maybe they all have an odd number of sides or points? ① has 5, ③ has 4, ⑤ has 8 – mixed.

• ②④⑥:

  • ②: Trapezoid (4 sides), ④: Circle (infinite symmetry, but divided), ⑥: Star of David (6 points).
  • Symmetry: ② might have reflection, ④ has rotational, ⑥ has rotational. Not a clear distinction.

Option C: ①③④ in one group, ②⑤⑥ in the other.

• ①③④:

  • ①: Pentagon, ③: Star, ④: Circle. Different shapes.
  • Color: All have black and white sections.
  • Maybe they have a central symmetry or something else.

• ②⑤⑥:

  • ②: Trapezoid, ⑤: Octagon, ⑥: Star. Again, mixed shapes.

Option D: ①②④ in one group, ③⑤⑥ in the other.

• ①②④:

  • ①: Pentagon, ②: Trapezoid, ④: Circle. Different shapes.
  • Symmetry: ① rotational, ② reflection, ④ rotational – mixed.

• ③⑤⑥:

  • ③: Star, ⑤: Octagon, ⑥: Star. All have pointed shapes or stars? ③ and ⑥ are stars, ⑤ is an octagon – not all stars.

Going back to rotational symmetry, as that was a promising lead. Let's verify rotational symmetry for each figure:

• ①: Yes, pentagonal rotational symmetry.
• ②: Trapezoid – if it's symmetric, it might have 180-degree rotational symmetry. Looking at the figure, if the black and white triangles are arranged symmetrically, it could have 180-degree rotational symmetry.
• ③: Four-pointed star – 90-degree rotational symmetry.
• ④: Circle with cross – 90-degree rotational symmetry if the divisions are symmetric.
• ⑤: Octagon – 45-degree rotational symmetry.
• ⑥: Star of David – 60-degree rotational symmetry.

So all six have rotational symmetry? That can't be right because the question implies a distinction. Maybe some have reflection symmetry instead or in addition.

Reflection symmetry:

• ①: Pentagon – might have reflection symmetry if the pattern is symmetric.
• ②: Trapezoid – likely has vertical or horizontal reflection symmetry.
• ③: Star – has reflection symmetry across axes.
• ④: Circle with cross – has reflection symmetry.
• ⑤: Octagon – has reflection symmetry.
• ⑥: Star of David – has reflection symmetry.

Again, all might have reflection symmetry. So that's not the distinguishing feature.

Another idea: the number of triangular sections or how the black and white areas are divided.

Looking at the figures:

• ①: Divided into triangles.
• ②: Divided into triangles.
• ③: Divided into triangles.
• ④: Divided into quadrants and smaller shapes, not just triangles.
• ⑤: Divided into triangles.
• ⑥: Divided into triangles.

So ④ is different as it's not solely triangles; it has a circle and cross, which includes other shapes. But the other five seem to be triangle-based. But the options group three and three, so that doesn't fit.

Wait, maybe it's about whether the figure has a central point or not.

• ①: Central pentagonal area.
• ②: No clear central point; it's a trapezoid.
• ③: Central intersection point.
• ④: Central point where lines cross.
• ⑤: Central octagonal area.
• ⑥: Central intersection point.

Not clear.

Let's look at the answer choices again. Option B is ①③⑤ and ②④⑥. Let's see if there's a pattern in the number of sides or points.

• ①: 5 sides (pentagon).
• ③: 4 points (star).
• ⑤: 8 sides (octagon).
• ②: 4 sides (trapezoid).
• ④: Circular (infinite sides).
• ⑥: 6 points (Star of David).

No obvious numerical pattern.

Another approach: count the number of black and white regions.

• ①: Multiple black and white triangles.
• ②: Fewer sections, maybe 4 black and 4 white?
• ③: 4 black and 4 white?
• ④: Multiple sections, hard to count.
• ⑤: 8 black and 8 white?
• ⑥: 6 black and 6 white?

Not helpful.

Perhaps it's about the presence of a square or rectangular element. But ④ has a cross, which might involve squares, but others don't.

Wait, looking at the figures more carefully, maybe it's about whether the figure can be divided into smaller congruent shapes.

• ①: Can be divided into triangles.
• ②: Divided into triangles, but not all congruent?
• ③: Divided into congruent triangles?
• ④: Divided into quadrants, which are congruent.
• ⑤: Divided into congruent triangles.
• ⑥: Divided into congruent triangles.

Still not clear.

Let's consider the symmetry type. Maybe some have only rotational symmetry, while others have both rotational and reflection symmetry.

• ①: Pentagonal – has both.
• ②: Trapezoid – likely only reflection symmetry (if isosceles trapezoid).
• ③: Star – both.
• ④: Circle with cross – both.
• ⑤: Octagon – both.
• ⑥: Star of David – both.

No, all seem to have both.

Perhaps the key is the number of axes of symmetry.

• ①: 5 axes (if regular pentagon).
• ②: 1 or 2 axes (trapezoid).
• ③: 4 axes (square symmetry).
• ④: 4 axes (symmetrical cross).
• ⑤: 8 axes (regular octagon).
• ⑥: 6 axes (hexagram).

Still mixed.

Going back to the options, option B is ①③⑤ and ②④⑥. Let's see if ①,③,⑤ have an odd number of sides/points, while ②,④,⑥ have even.

• ①: 5 (odd)
• ③: 4 (even) – wait, no, ③ is a four-pointed star, so 4 points (even).
• ⑤: 8 (even)
• ②: 4 (even)
• ④: Infinite (even in a sense)
• ⑥: 6 (even)

No, ① is odd, others even – not consistent.

Another idea: the central shape. ① has a pentagon, ③ has a diamond-like center, ⑤ has an octagon – all regular polygons. ② is a trapezoid (irregular), ④ is a circle (not polygon), ⑥ is a star (not regular polygon). But ③ and ⑥ are stars, so not consistent.

Wait, maybe it's about the outer shape being a polygon with an odd number of sides. ① is pentagon (5, odd), ③ is a star (4 points, even), ⑤ is octagon (8, even). No.

Perhaps the classification is based on whether the figure contains a square or not. ④ has a square in the cross, but others don't. But ② might have a square-like shape? Not really.

Let's look for a different pattern. Maybe the number of triangles in the figure.

• ①: Multiple triangles, maybe 5 or more.
• ②: 4 triangles.
• ③: 4 triangles.
• ④: Not just triangles; has squares.
• ⑤: 8 triangles.
• ⑥: 6 triangles.

Not helpful.

Considering the answer choices, option B is often correct in such problems, but I need a better reason.

Another thought: the direction of the black and white areas. For example, in ①, the black triangles are in a certain orientation; in ③, they are different; in ⑤, another. But hard to define.

Wait, perhaps it's about the presence of a "corner" or "pointed" shape. ① has a pentagon (pointed corners), ③ is a star (pointed), ⑤ is an octagon (not as pointed). ② is a trapezoid (not pointed), ④ is a circle (no points), ⑥ is a star (pointed). So ③ and ⑥ are pointed, but ① and ⑤ are mixed.

Not consistent.

Let's try to visualize folding or unfolding, but that might not apply.

Perhaps the key is the number of colors or the contrast. All have black and white, so no.

Another idea: the area ratio of black to white. But without measurements, hard to tell.

Looking at the figures again, maybe it's about the inner shape. ① has a central white pentagon, ③ has a central black or white area, ⑤ has a central white octagon. ② has no central area, ④ has a central cross, ⑥ has a central hexagon.

Still not clear.

Wait, considering the options, option B is ①③⑤ and ②④⑥. Let's see if ①,③,⑤ have a triangular base, while ②,④,⑥ have a different base. ① is pentagon, not triangle; ③ is star, not triangle; ⑤ is octagon, not triangle. No.

Perhaps it's about the number of layers or levels. ① has two layers (black and white), same for others.

I recall that in some classification problems, the distinction is based on whether the figure has a prime number of sides or not. ① has 5 (prime), ③ has 4 (not prime), ⑤ has 8 (not prime). No.

Alternatively, ① has 5 (prime), ② has 4 (not), ③ has 4 (not), ④ infinite, ⑤ 8 (not), ⑥ 6 (not). Only ① is prime, so not.

Let's consider the answer choices and see which one makes sense. Option B is ①③⑤ and ②④⑥. Maybe ①,③,⑤ have an odd number of triangles, while ②,④,⑥ have even. But counting triangles:

• ①: 5 black triangles? Maybe.
• ③: 4 black triangles.
• ⑤: 8 black triangles.
• ②: 4 black triangles.
• ④: Not triangles.
• ⑥: 6 black triangles.

No, ① is odd, others even – not consistent.

Perhaps it's about the outer edge being a circle or polygon. ④ is a circle, others are polygons. But ④ is in ②④⑥, so not.

Another approach: check the connectivity of the black regions.

• ①: Black regions are connected.
• ②: Black regions are connected.
• ③: Black regions are connected.
• ④: Black regions are connected.
• ⑤: Black regions are connected.
• ⑥: Black regions are connected.

All connected, so no.

I think I need to look for a different feature. Let's consider the symmetry again, but this time, whether the figure has a center of symmetry (point symmetry).

• ①: Pentagonal – no point symmetry (odd number of sides).
• ②: Trapezoid – if isosceles, has point symmetry only if it's a rectangle, which it's not. So no.
• ③: Star – has point symmetry.
• ④: Circle with cross – has point symmetry.
• ⑤: Octagon – has point symmetry.
• ⑥: Star of David – has point symmetry.

So ① and ② do not have point symmetry, while ③,④,⑤,⑥ do. But the options group three and three, so not matching.

Option D is ①②④ and ③⑤⑥. ①②④: ① no point symmetry, ② no, ④ yes – mixed.

Not working.

Perhaps it's about the number of sides of the outer shape being even or odd.

• ①: 5 (odd)
• ②: 4 (even)
• ③: 4 (even) – if we consider the star as having 4 sides.
• ④: Infinite (even)
• ⑤: 8 (even)
• ⑥: 6 (even)

Only ① is odd, so not.

Wait, maybe the classification is based on the inner shape's sides. ① has a pentagon inside, ③ has a square, ⑤ has an octagon. ② has no clear inner shape, ④ has a cross, ⑥ has a hexagon.

Not helpful.

Let's look at the answer choices and see which one is correct by elimination. Option B is ①③⑤ and ②④⑥. Maybe ①,③,⑤ have a triangular pattern, while ②,④,⑥ have a different pattern. But ① is pentagonal, not triangular.

Another idea: the orientation. ① is rotated, ③ is rotated, ⑤ is rotated, while ②,④,⑥ are "upright." But ④ is a circle, so no orientation.

Not clear.

Perhaps it's about the presence of a horizontal or vertical axis. All have such axes except maybe ① if it's irregular, but it's likely symmetric.

I think I need to consider that the correct classification is based on the number offold or something, but that's not visual.

Wait, looking at the figures, maybe ①,③,⑤ have a central black or white area, while ②,④,⑥ do not. ① has a white center, ③ has a black center, ⑤ has a white center, ② has no central area, ④ has a cross (central), ⑥ has a central hexagon. So ④ and ⑥ have central areas, which contradicts.

Not.

Let's count the number of black triangles:

• ①: 5
• ②: 4
• ③: 4
• ④: Not applicable (not triangles)
• ⑤: 8
• ⑥: 6

No pattern.

Perhaps it's about the sum of black and white areas. All are equal, so no.

I recall that in some such problems, the distinction is based on whether the figure is a regular polygon or not. ① is regular pentagon, ③ is a regular star (if it's a square-based star), ⑤ is regular octagon. ② is a trapezoid (not regular), ④ is a circle (not polygon), ⑥ is a regular star (hexagram). So ①,③,⑤ are regular polygons or stars, while ②,④,⑥ are not. But ③ and ⑥ are stars, which are not polygons, and ④ is a circle. So not a clear distinction.

However, ①,③,⑤ are all "regular" in the sense of symmetric division, while ② is irregular, ④ is a circle (not divided into regular polygons), ⑥ is regular. So not consistent.

After careful consideration, the most plausible classification is option B: ①③⑤ and ②④⑥, where ①,③,⑤ have an odd number of sides or points in their outer shape, but ③ has 4 (even), so that's not it.

Wait, ① has 5 sides, ③ has 4 points, ⑤ has 8 sides – no.

Another possibility: the number of triangular sections. ① has 5, ③ has 4, ⑤ has 8 – sum 17, not helpful.

Perhaps it's about the year or a code, but that's not logical.

I think I need to look for a different feature. Let's consider the area of the black regions relative to the whole. But without measurements, hard.

Finally, after re-examining the figures, I notice that ①,③,⑤ have a "pointed" or "star-like" outer shape, while ②,④,⑥ do not. ① is a pentagon (pointed corners), ③ is a star, ⑤ is an octagon (not as pointed, but still has corners), ② is a trapezoid (no points), ④ is a circle (no points), ⑥ is a star (pointed). So ③ and ⑥ are stars, but ① and ⑤ are not. So not consistent.

However, ①,③,⑤ all have a "multi-pointed" or "angular" outer boundary, while ②,④,⑥ have a "smooth" or "less angular" boundary. ② is a trapezoid (angular but not pointed), ④ is a circle (smooth), ⑥ is a star (pointed). So ⑥ is pointed, which contradicts.

This is tricky. Maybe the correct answer is B, as it's a common choice in such problems.

Alternatively, considering the number of triangles: ① has 5, ③ has 4, ⑤ has 8 – no pattern. ② has 4, ④ has not, ⑥ has 6.

Wait, ①,③,⑤ have a number of triangles that are not multiples of 2? 5 is odd, 4 is even, 8 is even – no.

I think I need to conclude that the correct classification is based on rotational symmetry with a specific degree, but I'm not sure.

After careful analysis, the most consistent feature is that ①,③,⑤ have a central symmetrical axis that is vertical, while ②,④,⑥ have a different axis. But not clear.

Given the time I've spent, I'll go with option B, as it's a standard choice in such problems.