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 One way by Bob Smith)

How large does the paper have to be?

The solution depends on the fact that a dot is not a point, but rather, has a finite size.  Let's say the dots' diameters are 1/10 the distance separating their centers.  An almost-horizontal line going through three horizontally aligned points can be tilted somewhat. To make calculations easy, let's assume we want to make the line go through the top of the leftmost dot and the bottom of the rightmost dot in a row. (We could have made the line tilt even more by making the line tangent to the two end dots.)

The slope of the line is (1/10)/2 = 1/20. To connect with the line sloping the other way from the next row, the line need go down only 1/2 the distance between the rows, which can be accomplished by going to the right of the center, 10 times the inter-dot distance.  Then the next line zigzags the other direction and goes out 10 times in the opposite direction from the center, and the third line then goes in the original direction.

So if the dots are 1 mm in diameter and separated by 1 cm, the width of the paper needs to be 20 cm.  And if tangent lines were allowed, the paper could be narrower still.

Posted by Charlie on 2005-07-14 19:08:24