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The Problem

The medieval equivalent of Bob the Builder has a new job offer: He is to build a box with the measures of $0.35$ units by $0.35$ units by $0.35$ units for a king.

There is a little problem with the job offer though: If the king finds out that the box does not have exactly the correct dimensions, Medieval Bob will be executed.

Medieval Bob is only equipped with a stick of the exact length of 1 unit, a ruler without marks on it and has a compass.

Nevertheless, he is aware that there is a so-called error propagation. To be as precise as possible (he still has his own execution in mind) he wants to have as little error propagation as possible.

The Task

Construct the length of $0.35$ unit with the least construction steps, given a ruler of length 1 and a compass. This construction should yield the exact value if executed perfectly.

The Scoring

Each time a length is measured, it will count as one step. Constructing a perpendicular bisector of sides therefore takes 3 steps:

  1. Drawing one circle on one endpoint of the side
  2. Drawing a second circle on the other endpoint of the side
  3. Connecting the two intersections of the circles.

You can decrease your score by $5$ if you post images of the construction so we can follow it step by step. Please indicate this by placing an asterisk behind your score.

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  • $\begingroup$ I have noticed that there is no tag for golf, score or similar. Is it possible to create such a tag for Puzzling.SE? Are there tags that I'm simply unaware of? $\endgroup$
    – Narusan
    Jun 17, 2017 at 10:02
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    $\begingroup$ It sounds as if this is meant to be a ruler-and-compass problem in very light disguise. Can you confirm that that's the intention? (Presumably Bob draws a circle by selecting a string of the given length and putting a pin in one end of it, or something.) Or, e.g., does Bob lack the ability to draw straight lines of arbitrary length, or have the ability to do other non-ruler-and-compass things one could do with string (such as constructing ellipses)? $\endgroup$
    – Gareth McCaughan ♦
    Jun 17, 2017 at 10:50
  • $\begingroup$ @GarethMcCaughan Yes, this is a ruler and compass problem. However, his ruler has a length of 1, so he can't draw lines longer than that in one stroke (matters for scoring)! I have posted a non-completing example answer $\endgroup$
    – Narusan
    Jun 17, 2017 at 10:54
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    $\begingroup$ @Narusan I doubt such a tag is needed. But tags optimizatipn stratergy and open-ended may apply $\endgroup$
    – Beastly Gerbil
    Jun 17, 2017 at 11:18
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    $\begingroup$ you emphasized that we have multiple strings of all length! what does that mean? so i have 0.35 unit string too already? $\endgroup$
    – Oray
    Jun 17, 2017 at 11:42

4 Answers 4

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Score: 6*

My previous answer was an approximate result, but this is an exact result with the highest point possible.

  1. Draw a circle with compass (+1 step)
  2. Draw another circle wherever point you want on the circle you draw previously (+1 step)
  3. Draw the lines shown in the figure below. (+2 steps)

enter image description here

  1. Draw another circle from the mid point of AB using our compass with the radius of $|AG|=2$ (+1 step).
  2. Connect H to G with a straight line shown below (+1 Step).
  3. Adjust our compass from 2 to 1 using the distance from A to B and take 1 unit distance from H to I shown in the figure (+1 Step)
  4. Draw another circle using compass where it intersects the line passing through H as shown, The distance from J to I becomes $.6$ unit (+1 Step):

enter image description here

  1. Draw a straight line connecting the top of the circles we have originally drawn first (tangents from their top), no compass needed for this part. So the distance between intersection point on HG line (K) and G becomes $1.25$ unit. (+1 Step) enter image description here

  2. So take your compass, and adjust it according to $|JI|=0.6$ and mark it on the KG from point K or G (doesnt matter), the point L in the figure (+1 Steps)

  3. Put your stick to the point K through the line HG and cut it from L point and one part of the stick becomes $.35$ unit stick! (+1 Step)

enter image description here

So total point becomes $11$ with shapes $-5$ equal to $6$!

As I stated before, in real life I would use my previous method since every step gives your some error!

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  • $\begingroup$ That's a very neat solution! Maybe my wording was ambiguous. Each time something is drawn (for which you need some sort of measurement: radius, angle etc.) it counts as +1 step. The Third Step therefore is another +2. $\endgroup$
    – Narusan
    Jun 18, 2017 at 10:06
  • $\begingroup$ @Narusan you can edit the points if you wish, I could not understand stepping part. $\endgroup$
    – Oray
    Jun 18, 2017 at 10:07
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    $\begingroup$ @Narusan am I going to get one more point if I put the stick starting from K point through L, instead of adjusting the compass from K to L, G to L twice? $\endgroup$
    – Oray
    Jun 18, 2017 at 10:35
  • $\begingroup$ Sorry, that was a mistype. The current version also only counts as +1 step. Whenever you draw something, it is 1 step. You have to draw the 0.65 to get 0.35 somewhere $\endgroup$
    – Narusan
    Jun 18, 2017 at 10:36
  • $\begingroup$ @Narusan I may have misunderstood your stepping again, fix it accordingly if you disagree what I have just said. $\endgroup$
    – Oray
    Jun 18, 2017 at 10:39
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Score: 12*

  1. Draw a line and mark 10 unit lengths on it - (+10 steps)

  2. Draw a line parallel to that and mark one unit length on it (Drawing three circles of unit length radius on the long line and connecting the intersection points). This takes 4 steps, but the steps of drawing the circle were covered in step 1, so just (+1 step)

  3. Connecting the ends of the line segment formed in Step 1 and the line segment formed in step two to form a triangle (+2 steps)

  4. Draw a line from the triangle vertex to the longer line segment 7 units from either side (you have the markings already) (+1 step)

This will divide the segment of unit length (formed in step 2) into 0.7 and 0.3

  1. Bisect the 0.7 segment - (+3 Steps)

Image below: Image Sample

Score: 17-5=12

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    $\begingroup$ Great Work! I am wondering: Could you save steps by drawing 5 circles of 2 units radius? This will also give you the mid-point and you could essentially save 4 steps. I like your approach! $\endgroup$
    – Narusan
    Jun 18, 2017 at 8:29
  • $\begingroup$ You're right, I thought about that later. I could have done five circles, and then marked 3.5 by joining opposite intersecting points of the circles. $\endgroup$
    – sanketalekar
    Jun 18, 2017 at 22:47
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Score: 21 (Non-Competing)

  1. Bisect the unit length. +3 steps
  2. Bisect the half again. +3 steps

We now have a segment with the length of 0.25 units.

  1. Draw a circle with the size of 0.25 from one endpoint of the segment.+1 step
  2. Choose any point on the circle and connect it to one endpoint by drawing a line, not a segment.+1 step
  3. Prolong this line 4 more times with a segment of equal length +4 steps

We now have a diagonal to the line with 5 equal parts

  1. Connect the endpoint of the diagonal line with the endpoint of the line +1 step
  2. Draw a parallel line to this line through the second-to-last point +6 steps

See this method of constructing a parallel using 6 steps.

This leaves us with a line of the length of 0.25 and a segment on the line with 1/5 of the length, 0.05 units.

  1. Adding the segment two times to the line yields 0.35 as final length +2 steps.

To be updated with images as soon as I get my hands on a computer.

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Draw a unit length, and construct a unit length perpendicular to an endpoint. Perpendicular to the other endpoint, construct a segment length 6. The line through the two endpoints extends the base unit length by 0.2. Halve this and halve again, add the bits together to give a sum of 0.35.

point35

My picture is a bit rough and I'm not sure how many steps this is.

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  • $\begingroup$ how do u make a line perpendicular to the original line without using compass? $\endgroup$
    – Oray
    Jun 19, 2017 at 11:32
  • $\begingroup$ i used a compass - it's a pretty well known technique. You can get a unit length by drawing a unit circle from the centre dot. the paint image uses a line at exactly $45^\circ$ to preserve the length, but you should use a compass. $\endgroup$
    – JMP
    Jun 19, 2017 at 11:44

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