It is well-known the solution to the problem of connecting nine dots, arranged in three rows of 3 dots, with four straight lines, without lifting up the pencil from the paper where they are drawn, and without any tricks at all, like folding the paper, etc...

           o        o        o


           o        o        o


           o        o        o

The question is: given the nine dots above, is it possible to connect them with only 3 straight lines ? The restrictions are the same, that is, without lifting up the pencil from the paper where they are drawn, no tricks allowed, and if you retrace a line, you must count one more line.

Prove your answer!

Note: this is a revisit to the problem Nine Dots already posted in this site and you can use that drawing for reference.

Contrary to what the preceding comments supposed, suppose the 9 dots are geometrical points of zero radius.

No (straight) line can contain more than 3 of the 9 points.

In order for 3 lines together to contain all the points it must be the case that

a) None of the 3 lines may contain less than 3 points.

b) No two of the 3 lines may contain the same point.

c) The 3 lines must either all be vertical or all be horizontal, and hence must all be parallel.

The conclusion then is that the 9 points cannot be connected by 3 lines drawn according to the rules set forth by the problem statement.

Posted by Richard on 2005-07-14 20:45:56