A solitaire game is played with the following rules:
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On a line segment (of arbitrary length, set it as long as you wish, but for convenience/reference sake, let's say it extends from 0 to 1 on the number line), you place a point anywhere you like on it.

Now place a second point, such that either of the two points is within a different half of the line segment. (The halves are taken to be "open intervals", which means that the end points are not considered "inside" the interval.)

Place a third point so that each of the three is in a different third of the line.

At this point, you may notice that the first two points can't be just anywhere. They cannot, for example, be close together in the middle of the line or close together at one end. They must be carefully placed so that when the third point is added, each will be in a different third of the line.

You proceed in this way, placing every nth point so that the first n points always occupy different 1/nth parts of the line.
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If you choose locations carefully, how many points can you put on the line?

(In reply to Solution (I think !) by Penny)

I suspect that theoretically n can be infinite, but your thoughts do suggest that it is just possible that there is a limit and it is relatively low. Either way, your solution as posted is flawed.

Your first error is the assumption that "The first two have to be placed at least 1/2 apart." If the first is placed just before .5 and the second just after, they satisfy the conditions of the game and are far less than 1/2 apart. (Since, in this case, both would be the middle third of the line, you could not progress to n=3, but then you would need only choose a point immediately after .666... -- which is only slightly more than 1/6, and still far less than 1/2 -- and there will still be somewhere to place the third point.)

Your second error concerns the algorithm for placing the points. It fails at n = 6 regardless of your formula. Since, starting with point 4, each point is placed between the last point placed and 1, there is never a point placed between 0 and .333... Once the length of a segment drops to .1666... or lower there are segments in that range without points.

Note: For reasons SilverKnight explains in the previous response, the points in your algorithm should be irrational numbers arbirarily close to the numbers listed. (As long as n remains finite, a chosen point can be rational, p/q [reduced], provided it is not true that p ≤ q ≤ n, but for infinite n that eliminates all rationals

Edit: added following paragraph

While I was composing this, Penny and SilverKnight both posted new responses. My reference to SK's "last" reponse refers to his last previous response. (His second)
Edited on January 17, 2004, 3:04 am

Posted by TomM on 2004-01-17 02:58:28