Question

what is the shading style of the twenty - second figure in the sequence below? sequence of figures: circle (white), triangle (dark), square (striped), star (white), circle (dark), triangle (striped), square (white), star (dark) ... the twenty - second figure in the sequence is dropdown

Explanation:

Step1: Identify the pattern cycle

Looking at the sequence, we can see that the pattern repeats every 4 figures. Let's list the shading styles and shapes:
1. Circle (white)
2. Triangle (solid dark)
3. Square (striped)
4. Star (white)
5. Circle (solid dark)
6. Triangle (striped)
7. Square (white)
8. Star (solid dark)
Wait, maybe a better way: Let's group by the cycle of 4 in terms of shading and shape pattern. Wait, actually, looking at the shading: Let's check the cycle length. Let's list the positions and shading:

Position 1: White (Circle)
Position 2: Solid Dark (Triangle)
Position 3: Striped (Square)
Position 4: White (Star)
Position 5: Solid Dark (Circle)
Position 6: Striped (Triangle)
Position 7: White (Square)
Position 8: Solid Dark (Star)

Wait, maybe the cycle for shading is related to the position modulo 4? Wait, let's check the shading types: White (W), Solid Dark (SD), Striped (S), White (W) for positions 1 - 4? No, position 4 is White (Star), position 5 is Solid Dark (Circle), position 6 is Striped (Triangle), position 7 is White (Square), position 8 is Solid Dark (Star). Wait, maybe the cycle is 4 in terms of the sequence of shading styles: W, SD, S, W, then SD, S, W, SD? No, that doesn't seem. Wait, let's list the shading styles:

1: White (W)
2: Solid Dark (SD)
3: Striped (S)
4: White (W)
5: Solid Dark (SD)
6: Striped (S)
7: White (W)
8: Solid Dark (SD)

Wait, no, position 6 is Striped (Triangle), position 7 is White (Square), position 8 is Solid Dark (Star). Wait, maybe the cycle is 4 for the shape and shading combination. Let's see the shape cycle: Circle, Triangle, Square, Star, then Circle, Triangle, Square, Star... So shape cycle length is 4. Now, shading: Let's check for each shape:

- Circle: Position 1 (W), Position 5 (SD) → so every 4 positions, shading changes? Wait, position 1: Circle (W), position 5: Circle (SD) (5 - 1 = 4)
- Triangle: Position 2 (SD), Position 6 (S) (6 - 2 = 4)
- Square: Position 3 (S), Position 7 (W) (7 - 3 = 4)
- Star: Position 4 (W), Position 8 (SD) (8 - 4 = 4)

Ah, so for each shape (Circle, Triangle, Square, Star), every 4 positions (i.e., cycle length 4), the shading changes in a pattern. Now, let's confirm the cycle for shading within each shape:

- Circle: W (pos1), SD (pos5), then next Circle would be pos9: let's see the pattern. Wait, pos1: W, pos5: SD, so the shading for Circle is W, SD, W, SD... with period 2? Wait, no, pos1: W, pos5: SD (difference 4), pos9:? Let's check the triangle: pos2: SD, pos6: S (difference 4), pos10:? Square: pos3: S, pos7: W (difference 4), pos11:? Star: pos4: W, pos8: SD (difference 4), pos12:?

Wait, maybe the overall cycle is 4, and within each cycle of 4, the shading styles are W, SD, S, W for positions 1 - 4; then SD, S, W, SD for positions 5 - 8? No, that doesn't fit. Wait, let's list the shading styles with their positions:

1: W

2: SD

3: S

4: W

5: SD

6: S

7: W

8: SD

Wait, no, position 6 is S (striped), position 7 is W (white), position 8 is SD (solid dark). Wait, maybe the cycle is 4, and the shading pattern is W, SD, S, W, then SD, S, W, SD, then W, SD, S, W, etc.? No, that seems inconsistent. Wait, let's count the number of figures: we need to find the 22nd figure. Let's find the cycle length. Let's see the sequence:

Figures 1 - 4: Circle (W), Triangle (SD), Square (S), Star (W)

Figures 5 - 8: Circle (SD), Triangle (S), Square (W), Star (SD)

Figures 9 - 12: Circle (W), Triangle (SD), Square (S), Star (W)

Ah! Now I see. The cycle length is 8? No, wait, figures 1 - 4 and 5 - 8: let's check the pattern. Figure 1: Circle (W), Figure 5: Circle (SD) → same shape, different shading. Figure 2: Triangle (SD), Figure 6: Triangle (S) → same shape, different shading. Figure 3: Square (S), Figure 7: Square (W) → same shape, different shading. Figure 4: Star (W), Figure 8: Star (SD) → same shape, different shading. Then Figure 9: Circle (W) (same as Figure 1), Figure 10: Triangle (SD) (same as Figure 2), Figure 11: Square (S) (same as Figure 3), Figure 12: Star (W) (same as Figure 4). So the cycle length is 8? Wait, no, because Figure 9 is same as Figure 1, Figure 10 same as Figure 2, etc. So the cycle is 8? Wait, no, let's check the number of figures in a full cycle. Let's see:

Cycle 1: Figures 1 - 8

Cycle 2: Figures 9 - 16 (same as 1 - 8)

Cycle 3: Figures 17 - 24 (same as 1 - 8)

But we need Figure 22. Let's find 22 modulo 8. 8*2=16, 22-16=6. So Figure 22 is the same as Figure 6.

Figure 6: Triangle (Striped)

Wait, but let's confirm the cycle length. Alternatively, maybe the cycle is 4 in terms of shape, and 2 in terms of shading. Wait, shape cycle: Circle (1,5,9,13,17,21), Triangle (2,6,10,14,18,22), Square (3,7,11,15,19,23), Star (4,8,12,16,20,24). So for Triangle, the positions are 2,6,10,14,18,22. Now, let's check the shading of Triangle at these positions:

Position 2: Solid Dark (SD)

Position 6: Striped (S)

Position 10:? Let's see the pattern. From position 2 to 6: difference 4, shading changes from SD to S. Then position 10: difference 4 from 6, so shading would be W? Wait, no, let's check the sequence:

Position 2: Triangle (SD)

Position 6: Triangle (S)

Position 10: Triangle (W)?

Wait, but let's check the other shapes:

Circle:

Position 1: W

Position 5: SD

Position 9: W (since 9-5=4, same as 5-1=4, so W→SD→W)

Square:

Position 3: S

Position 7: W

Position 11: S (7-3=4, 11-7=4, so S→W→S)

Star:

Position 4: W

Position 8: SD

Position 12: W (8-4=4, 12-8=4, so W→SD→W)

Ah! Now I see. For each shape, the shading cycles every 2 positions (i.e., period 2) within their shape's position cycle. Wait, no, the shape's position cycle is every 4 positions (since shape repeats every 4: Circle at 1,5,9,13,17,21; Triangle at 2,6,10,14,18,22; etc.). So for each shape, the shading changes every 4 positions? Wait, no, Circle: 1 (W), 5 (SD), 9 (W), 13 (SD), etc. So period 2 for shading within the shape's position cycle (since 5-1=4, 9-5=4, and shading alternates W, SD, W, SD...). So the number of times the shape has appeared before position 22: for Triangle, position 22 is the (22 - 2)/4 + 1 = (20)/4 +1 = 5 +1 = 6th time? Wait, no, position 2: 1st Triangle, position 6: 2nd Triangle, position 10: 3rd Triangle, position 14: 4th Triangle, position 18: 5th Triangle, position 22: 6th Triangle. Now, the shading for Triangle: 1st (pos2): SD, 2nd (pos6): S, 3rd (pos10): W, 4th (pos14): SD, 5th (pos18): S, 6th (pos22): W? Wait, that doesn't match the earlier thought. Wait, let's look at the sequence again:

Figure 1: Circle (W)

Figure 2: Triangle (SD)

Figure 3: Square (S)

Figure 4: Star (W)

Figure 5: Circle (SD)

Figure 6: Triangle (S)

Figure 7: Square (W)

Figure 8: Star (SD)

Figure 9: Circle (W)

Figure 10: Triangle (SD)

Wait, no! Wait, Figure 9 should be Circle (W) like Figure 1, Figure 10: Triangle (SD) like Figure 2? But that contradicts Figure 6 being Triangle (S). Wait, maybe I made a mistake in the sequence. Let's re-express the sequence with positions:

1: Circle, White (W)

2: Triangle, Solid Dark (SD)

3: Square, Striped (S)

4: Star, White (W)

5: Circle, Solid Dark (SD)

6: Triangle, Striped (S)

7: Square, White (W)

8: Star, Solid Dark (SD)

9: Circle, White (W) [same as 1]

10: Triangle, Solid Dark (SD) [same as 2]

11: Square, Striped (S) [same as 3]

12: Star, White (W) [same as 4]

13: Circle, Solid Dark (SD) [same as 5]

14: Triangle, Striped (S) [same as 6]

15: Square, White (W) [same as 7]

16: Star, Solid Dark (SD) [same as 8]

17: Circle, White (W) [same as 1,9]

18: Triangle, Solid Dark (SD) [same as 2,10]

19: Square, Striped (S) [same as 3,11]

20: Star, White (W) [same as 4,12]

21: Circle, Solid Dark (SD) [same as 5,13]

22: Triangle, Striped (S) [same as 6,14]

Ah! Now I see. The cycle length is 8? No, wait, from 1-8, then 9-16 (same as 1-8), 17-24 (same as 1-8). Wait, no, 9 is same as 1, 10 same as 2, 11 same as 3, 12 same as 4, 13 same as 5, 14 same as 6, 15 same as 7, 16 same as 8. So the cycle length is 8? Wait, no, 1-8: 8 figures, then 9-16: another 8, etc. So to find the 22nd figure, we calculate 22 divided by 8. 8*2=16, 22-16=6. So the 22nd figure is the same as the 6th figure.

Figure 6: Triangle, Striped (S)

Wait, but let's check with the shape cycle. The shape cycle is 4: Circle (1,5,9,13,17,21), Triangle (2,6,10,14,18,22), Square (3,7,11,15,19,23), Star (4,8,12,16,20,24). So position 22 is a Triangle (since 22 mod 4 = 2, because 4*5=20, 22-20=2; so remainder 2, which corresponds to Triangle, as position 2 is Triangle). Now, for Triangle, the positions are 2,6,10,14,18,22. Let's list the shading of Triangle at these positions:

Position 2: SD

Position 6: S

Position 10: SD

Position 14: S

Position 18: SD

Position 22: S

Ah! So the shading for Triangle alternates between SD and S every 4 positions (i.e., every time the Triangle appears, the shading changes: SD, S, SD, S, etc.). So the number of times Triangle has appeared before position 22: (22 - 2)/4 + 1 = (20)/4 +1 = 5 +1 = 6th time. Wait, no, position 2: 1st Triangle (SD), position 6: 2nd Triangle (S), position 10: 3rd Triangle (SD), position 14: 4th Triangle (S), position 18: 5th Triangle (SD), position 22: 6th Triangle (S). So the shading is S (striped) for the 22nd figure.

Step2: Confirm the cycle

Alternatively, let's use modular arithmetic. The sequence repeats every 8 figures? Wait, no, looking at the sequence, figures 1-8 and 9-16 are identical. So the period is 8. So 22 divided by 8: 8*2=16, remainder 6. So figure 22 is the same as figure 6. Figure 6 is Triangle with striped shading.

Answer:

Striped (or the specific term for the shading style, which is striped as per figure 6)