C. Find the missing number in the circle.

INSIGHT

Пошаговое решение:

1. Observe the numbers in the circle: 4, 10, 13, ?, 17, 11, 8, 9.
2. Let's look for a pattern between adjacent numbers.
3. From 4 to 10, the difference is +6.
4. From 10 to 13, the difference is +3.
5. From 13 to ?, let's assume a pattern in the differences. The differences are 6, 3. This doesn't seem immediately obvious.
6. Let's try another approach. Consider the sum of numbers.
7. Let's consider the pattern by skipping numbers.
8. Starting from 4, going clockwise: 4, 10, 13, ?, 17, 11, 8, 9.
9. Let's examine the relationship between consecutive numbers in a different way.
10. Consider the number of lines or segments. The number of segments in each slice are not explicit.
11. Let's look for a pattern in the differences:
12. 4 to 10: +6
13. 10 to 13: +3
14. 13 to ?: Let's assume the differences themselves follow a pattern. Perhaps decreasing by 3? Then 13 + (3-3) = 13. But that doesn't make sense.
15. Let's try adding consecutive numbers in the sequence.
16. Let's consider the starting number of each segment and the increment to the next.
17. Look at the numbers as they increase and decrease.
18. Let's assume a pattern where we add a number and then subtract a number, or add numbers that increase/decrease in a certain way.
19. Let's consider pairs of numbers.
20. If we look at the numbers: 4, 10, 13, ?, 17, 11, 8, 9.
21. Let's try to find a relationship between numbers that are opposite each other or separated by a fixed number of segments.
22. 4 and 17: difference is 13.
23. 10 and 11: difference is 1.
24. 13 and 8: difference is 5.
25. ? and 9: difference is ? - 9.
26. This opposite number relationship does not seem consistent.
27. Let's revisit the consecutive differences: +6, +3.
28. What if the pattern is related to prime numbers or other number sequences?
29. Let's try to find a pattern in the jumps.
30. 4 --> 10 (+6)
31. 10 --> 13 (+3)
32. 13 --> ?
33. ? --> 17
34. 17 --> 11 (-6)
35. 11 --> 8 (-3)
36. 8 --> 9 (+1)
37. The differences are +6, +3, ?, ?, -6, -3, +1. This sequence of differences is not immediately obvious.
38. Let's look at the sequence from 8 to 4: 8 -> 9 (+1), 9 -> 4 (-5). This is not helpful.
39. Let's assume the pattern is consistent around the circle.
40. Consider the differences: 6, 3, x, y, -6, -3, 1.
41. If we look at the sequence: 4, 10, 13, ?, 17, 11, 8, 9.
42. Let's try to find a pattern that uses the numbers themselves.
43. What if we add the numbers in a specific order?
44. Let's reconsider the differences: +6, +3. This suggests a possible halving or decreasing pattern. If the next difference is +0, then ?=13. Then +13, then -6, -3, +1. This doesn't fit.
45. Let's assume the pattern of differences is: +6, +3, +0, -3, -6, -3, +1. If the difference after 13 is +0, then ?=13. Then the next difference is -3, so 13-3=10. But the next number is 17.
46. Let's look at the sequence again: 4, 10, 13, ?, 17, 11, 8, 9.
47. Consider the possibility of adding the digits of the previous number, or some operation with the numbers.
48. Let's try to find a pattern related to the shape of the slices. There are 8 slices.
49. What if the pattern is: number + increment = next number, and the increment changes?
50. Let's assume the increment sequence is symmetric or follows a pattern.
51. Look at the numbers from 17: 17 -> 11 (-6), 11 -> 8 (-3). This is the reverse of the first two differences.
52. This suggests a pattern where the increments are symmetrical.
53. So, if the differences are +6, +3, then the next difference should lead to 17.
54. Let the differences be d1, d2, d3, d4, d5, d6, d7, d8.
55. d1 = 10-4 = 6
56. d2 = 13-10 = 3
57. d7 = 11-17 = -6
58. d6 = 8-11 = -3
59. This suggests a symmetry in the differences: d1 = -d5, d2 = -d6.
60. So, we have +6, +3, d3, d4, d5, -3, -6, d8.
61. This does not seem to fit because d5 should be -d1 = -6, but we have 11-17 = -6. This implies d5 = -6. So, 17 + (-6) = 11. Correct.
62. And d6 = -d2 = -3. So, 11 + (-3) = 8. Correct.
63. So the pattern of differences is +6, +3, d3, d4, -6, -3, d7, d8.
64. Wait, the differences are between consecutive numbers.
65. 4 (+6) 10 (+3) 13 (?+d3) ? (?+d4) 17 (?+d5) 11 (?+d6) 8 (?+d7) 9 (?+d8) 4.
66. Let's re-evaluate the sequence of differences:
67. 10 - 4 = 6
68. 13 - 10 = 3
69. ? - 13 = d3
70. 17 - ? = d4
71. 11 - 17 = -6
72. 8 - 11 = -3
73. 9 - 8 = 1
74. 4 - 9 = -5 (This is not a consecutive difference, it's closing the circle)
75. The differences are: 6, 3, d3, d4, -6, -3, 1.
76. It looks like the absolute values of differences are decreasing and then increasing, and there's a sign change.
77. Let's consider the pattern of differences: +6, +3, ...
78. If we assume symmetry around the circle, then the difference from 17 to 11 is -6, and from 11 to 8 is -3. This matches the start of the sequence but in reverse and with opposite signs.
79. This implies that the pattern of differences is: +6, +3, d3, d4, -6, -3, -d3, -d4.
80. However, the last difference is 8 to 9 (+1). This breaks the symmetry.
81. Let's rethink. Consider pairs of numbers: (4, 10), (10, 13), (13, ?), (?, 17), (17, 11), (11, 8), (8, 9), (9, 4).
82. Let's look at the numbers in relation to each other in segments.
83. Segment 1: 4
84. Segment 2: 10
85. Segment 3: 13
86. Segment 4: ?
87. Segment 5: 17
88. Segment 6: 11
89. Segment 7: 8
90. Segment 8: 9
91. Let's try to find a pattern by adding or subtracting specific values.
92. Consider the possibility of adding the segment number, or its square, or some other relation.
93. Let's go back to the differences: 6, 3. The next difference is likely to be smaller than 3, or related.
94. What if the pattern is related to adding something and then subtracting something?
95. Let's look at the numbers 4, 10, 13. The sum is 27.
96. What about 17, 11, 8, 9? The sum is 45.
97. Let's assume a pattern where you add a value, then add another value, and so on.
98. Consider the pattern: Add 6, then add 3. What if the next step is to add 0? Then 13+0 = 13. But the next number is 17.
99. What if the pattern is related to the previous numbers in the sequence, not just consecutive ones?
100. Let's look at the numbers 4, 10, 13, ?, 17, 11, 8, 9.
101. Try to find a pattern that fits.
102. Consider the relationship between 4 and 10, and 8 and 9.
103. Let's test a hypothesis: The sum of opposite numbers is constant.
104. 4 + 17 = 21
105. 10 + 11 = 21
106. 13 + 8 = 21
107. ? + 9 = 21
108. If this is the pattern, then ? = 21 - 9 = 12.
109. Let's check if this is consistent.
110. The numbers are arranged in a circle. Let's number the positions 1 to 8 clockwise.
111. Position 1: 4
112. Position 2: 10
113. Position 3: 13
114. Position 4: ?
115. Position 5: 17
116. Position 6: 11
117. Position 7: 8
118. Position 8: 9
119. Opposite pairs are (1, 5), (2, 6), (3, 7), (4, 8).
120. 1+5: 4 + 17 = 21
121. 2+6: 10 + 11 = 21
122. 3+7: 13 + 8 = 21
123. 4+8: ? + 9 = 21
124. This implies that ? = 21 - 9 = 12.
125. The missing number is 12.