Welcome to Passwall! For the curious, or those unable to find a solution to the puzzle on the main page, we have included some of the solutions that seem to work. Re-stating the problem: this is a classic example given to people to demonstrate thinking "outside the box." The question for this puzzle was asked of me like this when I was about half way through elementary school: Using 4 lines or less, pass through each dot at least once. Limits: The start of the first line may be anywhere on the page The end of one line must become the beginning of the next line for each line drawn until 4 or fewer lines have been drawn Lines may be drawn with rulers to make sure they are straight 3x3 dot-matrix image 3x3 dot-matrix ASCII-art ```O O O O O O O O O ``` The expected example from our visitor is included here. If we assume a naming scheme for the point where columns are labeled from left to right as A, B, and C while rows are labeled top to bottom as 1, 2, and 3, we can reference any dot with a coordinate pair. In this way, the top left dot would be .A1, the middle dot would be .B2 and the bottom right dot would be .C3. With the above assumption use just to reference the dots, the desired solution from our visitor would be found by following this procedure: 3x3 dot-matrix image 3x3 dot-matrix ASCII-art ``` O--O--O--, |\ / | \ / O O O | \/ | /\ O O O | / |/ ' ``` Start with pencil on .A1 and draw a line downward to pass through .A2 and .A3 to go beyond the 3x3 matrix as shown here. Make a counter-clockwise turn of 135 degrees and draw a line up and through .B3 and .C2 until you exit the matrix as shown here. Make a counter-clockwise turn of 135 degrees and draw a line left and through .C1, .B1, but stopping on .A1 as shown here. Make a counter-clockwise turn of 135 degrees and draw a line down to the right and through .B2 and .C3 as shown here. I accept this is a solution to this question, but it is not the only solution. Our visitor came to show us how to think "outside of the box" but created a new box to contain our minds and yet again, limit us. (A box within a box!) Nobody likes being told their answer is wrong, especially when their answer fits the rules and is technicaly correct. Here are some things to think about: Nowhere in the rules are "lines" or "points" defined. People assume the geometric definitions would apply here, but that is not explicit. Nowhere in the rules, is the distance between the point specified, nor is the angle of orientation of the 3 by 3 dot-matrix specified. Though the lines may be straight, space may be curved. There is no limit to force this being modeled in only 2 dimentions. There are other possible ways to re-examine the rule to find other ways to exploit unstated or assumed rules in this problem. Some of these may not even be "valid" if you alter you definitions of some key words in the rules. (Many arguements come down to disagreements of definitions.) This solution requires a bit of thought. Imagine a sheet of paper laying flat in 'portrait' fashion. Roll the paper in such a way so as to have the long edges meet precisely along the length of the longest edges of the paper. You will then create a hollow cyllender. Drawing 3 lines along the flattened piece of paper as shown, would form a 3d spiral shape when the paper was joined to form the cyllender. You may choose to imagine the paper as being continuous. If you do, then you may see the joined 3 lines as one continuous line, or as three separate lines. (By the way, the end of the first line is exactly the same point of the beginning of the second line and the end of the second line becomes the start of the thrid line when the flat paper is joined to make the cyllender.) After all three lines are drawn (or one continuous line if you prefer) then you may super impose the 3 by 3 matrix of points in such a way as to have all points crossed as depicted here in ASCI-Art and an animated image. Image: possible 3d solution to assumed 2d problem ASCII art paper (portrait) ``` ---------------- | | | | | | | | | | | | | | | | | | | | | | | | ---------------- ``` ASCII art Cyllen- -der ``` --=*X*=-- | /| | / | | / /| | / / | | / / /| | / / / | |/ / / | | / / | |/ / | | / | |/ | | | ---___--- ``` ASCII art Cyl + Matrix ``` --=*X*=-- | /| | / | | / /| | o / | | o o /| | o o o | |/ o o | | / o | |/ / | | / | |/ | | | ---___--- ``` There are other thoughts on this problem. These should not need visual aids to assist their conveyance. Imagine there is no limit to the size of a point as stated in geometry. It would be trivial to make the points be dots with radius (r) equal to three times the difference in distance between two laterally adjacent points (say measured length/distance between .A1 and .A2 multiplied by 3 as being less than or equal to the radius (r) for dots. If this was true, then a single thin line running through .A2, .B2, and .C2, would also run through .A1, .B1, .C1, .A3, .B3, and .C3 since the size of these "points" also cover the points .A2, .B2, and .C2. Imagine the points are allowed to follow the geometric definition of a point, but no dots are more than 1.00 femptometers apart from each other on a paper. Now take a wide tip felt tip marker and cross through the .B2 point. The marker's point is wider than the cluster of points, and will pass through all points due to the disproprtionate difference in width of the line drawn with the marker and the matrix width. Though it may sound like science fiction, the folding, or compression of space (more than bending of space) can lead to the above two examples as solutions. Go to the main web page
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