1000 Point Star
Logic puzzles require you to think. You will have to be logical in your reasoning.
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. The segments must always alternate a constant number of points (in the above case, skipping 1 point in between).
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?
HintIt's not guess work. Think of factors and mirror images. Try drawing out examples.
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Answer
199.
First, let's examine the 5-point star and why there is only that one, described, way to draw it. Starting at A, you cannot go straight to point B, otherwise you will have a pentagon, not a star. Same with point E. AC and AD (and their progressions) yield identical pictures (notice that the last segment in the AC one is DA, literally reverse all the letters to get AD ending in CA).
Now, for a 6-point star, there are points A, B, C, D, E, and F. Starting at A, we cannot go to B or F, because then we have a hexagon. If we go AC, then we must go CE, then CA... making a triangle, not a 6-point star. Going backwards, AE, EC, EA; mirror image. If we skip 2 points, we go AD, then DA, a straight line, hardly a 6-point star.
For a 7-point star, A-G, AB and AG are not possible, as that's a heptagon. AC, CE, EG, GB, BD, DF, FA is one way to draw the star, skipping one point in between. Also, AD, DG, GC, CF, FB, BE, EA, skipping two points in between each, is the other way. The last two line possibilities, AF and AE are mirror images of the two working scenarios.
Examining the above information a little more deeply, we see that we must divide all possible drawings by two, to get rid of the mirror images. Also, we cannot use the point immediately to the left, that is AB as a segment. This explains why there is only 1 way for a 5-point, and 2 for a 7-point, but does not explain the 6-point. Of those remaining points (dealing only with the first half, again, mirror images), those that are factors of the number of total points do not work, because you will end up back at the starting point before hitting every number. AC is a factor in six because it passes point B, so it's 2. AD is a factor of 3 (B, C and D). Not only factors, but multiples of factors, as well (for example, skipping 4 points in the 6-point still does not work).
Using the knowledge of mirrors, first point impossibility (AB) and factors, we see that the number 1000 is made up of 10^3, or 2^3*5^3. Without going through every factor, it's quickly and easily discernible that of the numbers 1-1000, 2 or multiples of it encompass 500 numbers, all evens. Any multiple of 5 includes 200 numbers, half of which end in 0 and are already included with the 2's, so there are 600 unique factors and multiples between 1 and 1000. That leaves us with 400 that are *not*, which should explain how many unique stars we can make. Remember to cut it in half because of mirroring, leaving us with 200, and to discount the AB segment (factor of 1), leaving us with 199. (If you discount AB, and THEN halve it, remember you needed to remove A[Z], the mirror image of that point).
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Comments
HiImDavid   
Sep 04, 2006
| Wow, very nice. It looks liek you invested a lot of time into that one!!! Very nice |
Punk_Rocker
Sep 05, 2006
| Someone has wayyy too much free time. xD
Interesting, though...That was cool. |
soccerfreak 
Sep 05, 2006
| how ling did it take u to do this?
 |
jazzmusician46 
Sep 06, 2006
| My goodness!  |
Infrared
Sep 18, 2006
| Waaaaay too complicated or me.
Great job. Must've been very time-consuming. |
googoogjoob
Sep 20, 2006
| Got me on this one! I've been interested in multi-pointed stars ever since a 9-pointed star showed up in a math contest I once entered in middle school. So this one was a lot of fun for me, even though I came up with a wrong answer. |
ulan
Oct 19, 2006
| Good puzzle, not so diffucult of one know about phi  |
Matio_Mario
Oct 29, 2006
| Ugh! Math is my best subject, but at the end I'm like "WHAT?!?!?!? Is this English???"  |
NomadShadow
Nov 08, 2006
| In the course of my job I usually have to do alot of modulus arithmetics, so this took me less than 10 seconds to figure out.
Cool one though, this is the first time I think of stars as ranges.
Keep up the good work |
qwertyopiusa
Nov 29, 2006
| Nice problem! |
stil
Dec 02, 2006
| Confusing use of "star" and "repeated line segment."
Hope to effect correction. |
McBobby1212
Mar 02, 2007
| ...what?... |
sftball_rocks13
Mar 25, 2007
| wow...that was hard
the answer was long. very long. it scared me
good teaser 
scary teaser  |
Odessius
Apr 26, 2007
| i dunno, a million? yet i think that defeats the purpose of brain logic. still just grab a pencil and that ones pretty easy. for some people |
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