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.

(In reply to another trick by Larry)

I cannot visualise what your 1/4 of the circum. and so pass thru 9 points.

That you went to the sphere is an excellent thought; I recall that it is possible to force an increase in the inimum number of colours required to shade a map by applying the map to the surface of a torus. 

And .. similarly I believe it is possible to cross every line segment (below) only once, if one of the cells contains the inner ring of a torus:
______________
|      |          |     |
|           |           |

Problem in hand:
A straight line is the shortest distance between two points (Larry is following this thought).

I'm going to use the Earth as my point of reference:
If I start at the lowest left dot and proceed North along a line of longitude to 0 deg N, I return thru the middle column of dots to 0 deg S and continue by a third line of longitude so that I pass thru the dots of the third column.

This uses 3 straight lines (as defined above) and because I cannot prove an axiom, and I have fulfilled the task, there is nothing to prove.

Else, if I am limited to a 2D coord system (flat), exploring Pascal, Desargue, etc, I see no plausible solution.

Posted by brianjn on 2005-07-15 08:19:02