Points on a circle

36 points are marked, equally spaced, on the circumference of a circle. Some of the points are marked with crosses in such a way that the distances between every two consecutive crosses are all di ffrent. What is the maximum number of crosses that can be made?

Ive tried individual cases but, i havnt been able to do much else. I tried finding a pattern with 1 point, 2 points, 3 points etc but couldnt find anything.

• If there are 3 crosses, then the total distance around the circle is at least $1+2+3=6$, which is OK, since the distance is actually 36. If there are 4 crosses, $1+2+3=4=10$, still OK. When do we run out of luck with this approach? Commented Jul 6, 2013 at 7:48
• why do you add successive tries, i.e. the 6+4 = 10 Commented Jul 6, 2013 at 7:57
• Because you start at one marked point and go to the next and then from there to the next and then from there to the next, so the distance you go is the sum of the distances between consecutive marked points --- but there's a limit to har far you can go before you run out of circle. Commented Jul 7, 2013 at 3:17

Hint:

• You want to put as many crosses as you can, so the distances has to be as small as possible.
• The smallest distance is $1$ (that is the length of one side of your 36-gram).
• When you have used distance $1$, the next smallest distance you can use is $2$ and then $3$ and so on...
• What can you say about the total sum of distances between crosses?

I hope this helps ;-)

The answer is 7.

Since we need to have as small distance as possible between succeeding points try incrementing each distance by one starting with 1.

X-1-X-2-X-3-X-4-X-5-X-6-X-8

1+2+3+4+5+6+8=29

29+7(x's)=36 points.

since 1+2+...+7+7(x's) is less than 36 (initial try) try adding one unit distance to the last distance making it 8.