This geometric construction challenge is a set of components to be placed in order, $\begingroup \def \s #1{{ \small\sf #1 }} \def \AB { \overline {\s{AB}} } \def \line #1{{ \small \overleftrightarrow {\s{#1}} }} $ dividing line segment $\AB$ of length 6 into 6 subsegments, each of length 1.

hexasect – transitive verb – to divide into 6 equal parts

How can $\AB$ be hexasected by placing 5 circles and 7 lines that produce just 1 lines-only node?

Construction guidelines
• Circles and lines are placed sequentially, in any order that accords with nodes that exist at times of placement.
• A circle may be placed where a node exists for the circle’s center.
• A line may be placed where it crosses at least two existing nodes.
Nodes are endpoints A and B as well as intersections among circles, lines and/or $\AB$.
• A lines-only node is an intersection of lines in the completed solution. No circles pass through a lines-only node. (Nodes along $\AB$ are ineligible because $\AB$ is technically a segment, not a line.)

quadrisect – transitive verb – to divide into 4 equal parts
 A different $\AB$, of length 4, can be quadrisected by placing 4 circles and 6 lines that produce just 1 lines-only node.

• Step 1 places circles centered at nodes A and B. These circles’ intersections produce nodes C and D.
• Step 2 places line $\line{BC}$ and a circle centered at C, whose intersection produces node E, and places $\line{CD}$, whose intersection with $\AB$ produces node F.
• Step 3 places a circle centered at E. This circle’s intersection with $\line{BC}$ produces node G.
• Step 4 places lines $\line{AD}$ and $\line{FG}$, whose intersection produces node H.
• Step 5 completes this quadrisection by placing lines $\line{CH}$ and $\line{DG}$. The desired lines-only node is H.


  • $\begingroup$ This quadrisection example is indeed suboptimal (and more complex than the puzzle's solution, which is tied for optimal hexasection). Optimal quadrisection uses fewer components but doesn't demonstrate a lines-only node. $\endgroup$
    – humn
    Commented Mar 7, 2023 at 16:06

1 Answer 1


One way I believe:

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This is the easiest to understand I could find.


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