Texas Tech University |
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Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu |
Texas Tech University |
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Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu |
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The sequence 1, 2, 4, 8, 16, . . . is generated as follows: | |||||
| For each integer n, consider n distinct distinguished points on the boundary of a circle. Next, draw all possible secant lines between all such distinguished points. Count the number of regions into which the interior of the circle has been divided by the drawn secant lines. The nth value of the sequence is the maximum number of such regions over all possible distributions of n distinct distinguished points on the boundary of a circle. | |||||
| For n = 1, one point is placed on the boundary of the circle. No possible secant lines can be drawn. The maximum (and minimum) number of regions into which the interior of the circle can be divided is 1. |  | ||||
| For n = 2, two points are placed on the boundary of the circle. Exactly one secant line can be drawn. The maximum (and minimum) number of regions into which the interior of the circle can be divided is 2. |  | ||||
| For n = 3, three points are placed on the boundary of the circle. Exactly three secant lines can be drawn. The maximum (and minimum) number of regions into which the interior of the circle can be divided is 4. |  | ||||
| For n = 4, four points are placed on the boundary of the circle. Exactly six secant lines can be drawn. The maximum (and minimum) number of regions into which the interior of the circle can be divided is 8. |  | ||||
| For n = 5, five points are placed on the boundary of the circle. Exactly ten secant lines can be drawn. The maximum (and minimum) number of regions into which the interior of the circle can be divided is 16. |  | ||||
| For n = 6, six points are placed on the boundary of the circle. Exactly 15 secant lines can be drawn. The maximum number of regions into which the interior of the circle can be divided is 31. (If the points are symmetrically distributed then the center triangular region may degenerate into an intesection point of the secant lines and only 30 regions will exist.) |  |  | |||
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