Typical "stars" are drawn in connected, but not repeated, line segments. For example, a 5-point star is drawn as such - line segments AC, CE, EB, BD, DA, with the order of the points going clockwise being A, B, C, D, and E.
The segments must always skip a constant number of points which is at least one.
Given the information that there is only 1 way to draw a 5-point star, and that there is NO way to draw a 6-point star (in continuous lines, that is), and there are 2 ways to draw a 7-point star.
How many different ways are there to draw a 1000-point star?
Each step must go more than one position around the circumscribing circle and less than 500 positions. The number must be relatively prime to 1000, and thus must be odd and not a multiple of 5.
Each decade has 4 such numbers, ending in 1, 3, 7 or 9. There are 50 decades and 50 * 4 = 200. However 1 itself was counted but is not allowable, leaving 199 ways, not counting using a reverse path as different.
Verifying with:
a=3:499;
b=setdiff(a, 2:2:498);
c=setdiff(b, 5:5:495);
length(c)
>> thousandPointStar
ans =
199
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Posted by Charlie
on 2023-10-25 08:55:29 |
solution
