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Drawing a long line

Question

Suppose you have a single sheet of an 81/2" x 11" paper. Your task is to draw the longest continuous line possible using an EXPO marker that has infinite ink. The line can curve but it cannot be picked up off the paper or overlap and of your previously drawn line. Since you are drawing a line with a marker, your line has a defined width. For the sake of argument, presume your line width is 1 inch. What is the longest possible line that can be drawn?

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Drawing a continuous line beginning at the edge of the paper running down the length and then making a 90 degree turn at each corner without crossing the line will yield the following lengths in inches before you run out of space. The total length is 91.5 inches.

11
7.5
10
6.5
9
5.5
8
4.5
7
3.5
6
2.5
5
1.5
4
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There is ambiguity to define "line length" when the line has non-zero width and is not all straight. If it curves there is both an inside and outside circumference. Similarly at corners different parts of the line have different lengths.

I'll take a simple-minded approach and say first that it's clear the entire page can be covered without lifting the pen or retracing. Then I'll say area = length x width.

Length = 11 x 8.5 / 1 = 93.5 inches.

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Actually, I think newbie has the right idea, but numbers are off a little...so the answer would be:

77 + 15*(Pi/4).

You would have to start 1/2 down from the corner so the pen is completely on the paper, travel along the longe edge 10". Then turn 90 deg. The arc formed would never cross itself (although the bottom corner would not move). The diameter is the 1" width of the pen, and the center of the pen, arc length is Pi/4.

Then travel down the paper 6.5". (you started 1" down from the edge), then make another arc, then travel 9", arc, 5.5", arc, 8" arc, 4.5" arc....

then it gets interesting at the end.. you travel 4" then make an arc, then travel 0.5" then make another arc and travel 3", then to maximize it, you get another arc, then another 0.5" before you are maxed out.

This strategy leaves you slightly better off than the option of just going back and forth from the top edge across the long way, back and forth, because at the end of that method, you loose 1/2" strip[ across the entire length of the bottom! this way, you only loose a 2" strip!

Good teaser though, got me thinking! Thanks
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The paper is a 3d object (or else my books wouldn't take up so much space). I'll take shammaa's approach in reverse--starting at the center. Arriving at the corner of one large face after 77+15pi/4 inches I'll continue over one edge to cross an 8.5-by-epsilon face, pivot over another edge, and apply shammaa's original approach to the second large face.

(Of course, if BMAD wants to split hairs---or sheets---we can probably splay the original sheet into an arbitrarily-large drawing surface, let the kindergarteners go to town on it, and return when they've all passed out from the infinite Expo fumes spewing forth.)
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There is ambiguity to define "line length" when the line has non-zero width and is not all straight. If it curves there is both an inside and outside circumference. Similarly at corners different parts of the line have different lengths.

I'll take a simple-minded approach and say first that it's clear the entire page can be covered without lifting the pen or retracing. Then I'll say area = length x width.

Length = 11 x 8.5 / 1 = 93.5 inches.

Agree! Another Aha puzzle.

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There is ambiguity to define "line length" when the line has non-zero width and is not all straight. If it curves there is both an inside and outside circumference. Similarly at corners different parts of the line have different lengths.

I'll take a simple-minded approach and say first that it's clear the entire page can be covered without lifting the pen or retracing. Then I'll say area = length x width.

Length = 11 x 8.5 / 1 = 93.5 inches.

Agree! Another Aha puzzle.

But still ... if I had to pick the most satisfying approach, I would define (zero-width) line segments that run midway from the edges of the 1-inch line, and then sum their lengths. I think this was the approach taken in previous posts. Or, take the average of the left and right edges of the 1-in-thick line.

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Would the solution change if we consider that the initial mark is from a curved end and not square?

like the picture below:

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After filling in the first side and ending up in the centre of the paper sheet bring the corner of the sheet close to the pen and without removing the pen from the paper move the pen onto the reverse side. Then fill in the second side giving a total length line of about 176 inches.

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