A gives B a natural number n . Then B is given a line segment of length 1, a piece of paper, straightedge, compass and pencil. He says, "I can't divide the line into n equal segments." Then
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solution please
It should be a Lvl 1 thing in my opinion... Basic affine constructions like this should be standard geometry bread-and-butter!
Here's a beautiful thing: dividing line SEGMENTS (lines are infinitely long) into n equal segments is not even something you need a compass to do. Here are the steps to do such a thing in affine geometry (here, we assume the axiom that there is a unique line through a given point that is parallel to a given line, i.e. the two lines do not meet):
Construct the line segment A B , given points A and B in affine n -space.
Take one of the points on the line segment, say A , and draw a line through A not parallel to the line A B .
Pick a point C 1 on the line constructed in Step 2.
Draw a line through A not parallel to the line A B or A C 1 , and pick a point D 1 on this line.
Construct a parallel line to the line A C 1 through the point D 1 .
Construct a parallel line to the line A D 1 through the point C 1 , and define the meet (intersection) of this line with the line in Step 4 to be the point D 2 .
Draw the line D 1 C 1 and construct a parallel line to this line through the point D 2 .
Define C 2 to be the meet of the line constructed in Step 6 with the line A C 1 .
Repeat Steps 5-7, now with C 1 , C 2 and D 2 instead of A , C 1 and D 1 ; we do this n − 1 times, so that we get n copies of the line segment A C 1 .
Suppose then that the last point we have is C n ; we then draw the line B C n .
Through each of the points C i , for i ranging from 1 to n − 1 , draw the lines through each of the points, parallel to the line drawn in Step 9.
The meets of each of the lines in Step 10 and the line segment A B will then be denoted by E i , for i ranging from 1 to n − 1 .
As a result of this construction, we then have that the points E i , for i ranging from 1 to n − 1 , divide the line segment (\overline{AB}) into n equal line segments.
I did a construction (see picture above) where I just divided a simple line segment into 3 equal segments.
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This isn't a solution but if you want me to give one I can, just write a comment asking me to.
I'm just asking, why is this lvl 2? It should be like lvl 3 at least...
Solution:
Suppose the initial line segment (of length 1) is P Q , wher P and Q are points. Then from P construct a segment, P N , with length n which is divided into n segments of length 1 . Construct lines parallel to N Q through the division of P N .
All of these constructions are pretty standard (such as constructing parallel lines through a point).