[itachi-san]

Usually hand-written 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 following info:

1)there is only 1 way to draw a 5-point star

2)there is NO way to draw a 6-point star (in continuous lines, that is)

3)there are 2 ways to draw a 7-point star

how many different ways are there to draw a 1000-point star?

[Guest]

495 ish I think

[Guest]

199 (number of integers under 500 coprime to 1000)

[itachi-san]

199 (number of integers under 500 coprime to 1000)

Yep. Good job

[Guest]

Pardon me, but I'm missing something here; maybe it's an incorrect assumption on my part. If you are drawing a star with continuous line segments, isn't there only one way to draw any given star, unless you count stroke direction, in which case there are two ways for stars with odd-numbered points and still no ways (line segments cannot be continuous) for stars with even points? What am I missing?

[Guest]

Pardon me, but I'm missing something here; maybe it's an incorrect assumption on my part. If you are drawing a star with continuous line segments, isn't there only one way to draw any given star, unless you count stroke direction, in which case there are two ways for stars with odd-numbered points and still no ways (line segments cannot be continuous) for stars with even points? What am I missing?

You can test this out yourself. Draw seven points evenly spaced on a circle. Starting at any point, draw a line to the next next point (so you're skipping one). Keep doing this until you've drawn a star. Now draw the same seven points, but this time skip two points when connecting them (so connect it to the next next next point). You'll see that you've drawn a different star. There are many different stars to be drawn for other numbers.

[Guest]

Thanks, flowstone, I get it now. That's pretty cool.

[Guest]

My answer goes to 199 ways.

I found that for any star with number of points = x, and skipping y points in between in order to form a star, there is 2 conditions:

1) the x and (y+1) cannot have common factors;

2) y+1 should not equal or more than x/2, otherwise it just the mirror image of the stars that already drawn.

For example, a 5 points star have only 1 combination of (x, y+1) which is (5, 2) - skip 1 points and (5,3) is just the mirror image of (5,2)

6 points star has no way as (6,2) has 2 as common factor, (6,3) has 3 as common factor

7 points star has 2 ways which is (7,2), (7,3), which is skip 1 point and skip 2 points in between.

8 points star has only 1 way which is (8,3) - skip 2 points. (8,2) and (8,4) has 2 as common factor

So for 1000 points star, maximum possible number of ways should be 499 (less than x/2)

and since 1000 = 2x2x2x5x5x5, any multiplier of 2 and 5 should not form 1000 point stars

hence, set { 2, 4, 6, 8,......498} has 249 numbers

set {5, 15, 25, 35,....495} has 50 numbers.

499 - 249 - 50 = 200

And also the y+1 cannot be 1 as well, otherwise you will only form a poligon (ha! This is the part I missed!)

So 200 - 1 = 199

Am I right?

Edited May 30, 2008 by woon