Most Wanted Solutions: 9 Dot puzzle (2024)

Puzzlesare in all cultures throughout time... And the 9Dot puzzle is as old as the hills.Even though it appears in Sam Loyd’s 1914 “Cyclopediaof Puzzles”, the Nine Dot puzzle existedlong before Loyd under many variants. In fact, sucha puzzle belongs to the large labyrinth games family.

9Dot puzzle is also a very well known problem usedby many psychologists, philosophers and authors(PaulWatzlawick, Richard Mayer, Norman Maier, JamesAdams, Victor Papanek...) to explain the mechanismof ‘unblocking’ the mind inproblem solving activities. It is probable that thisbrainteaser gave origin to the expression ‘thinkingoutside the box’.

Solvingit
We hope you don’t mind if we use nice ladybugsinstead of boring dots to make our puzzle demonstrations...Well, below are nine ladybugs arranged in a set of3 rows. The challenge is to draw with a pencil fourcontinuous STRAIGHT lines which go through the middleof all of the 9 ladybugs without taking the penciloff the paper.

Themost frequent difficulty people encounter with thispuzzle is that they tend to join up the dots as ifthey were located on the perimeter (boundary) ofan imaginary square, because:
- they assume a boundary exists since there are no dots to join aline to outside the puzzle.
- it is implicitly presumed that tracing out lines outside the ‘invisible’ boundaryis outside the scope of the problem.
- they are so close to doing it that they keep trying the same waybut harder.Unfortunately, repeating the same wrong process again and again withmore dynamism doesn’t work... No matter how many times they try to drawfour straight lines without lifting the pencil. A dot is always left over!

Trial-and-errorstrategy
Itis easy to connect all the 9 ladybugs with just a CURVEDline (see fig. opposite). Try now to imagine this lineas elastic as a rubber string, and wonder what wouldhappen if one or more curves/bights would be stretchedbeyond the ‘invisible’ boundary, as shownin fig. a and b below.
That intuition turns out, in fact, to be the relevant ‘insight’.Thanks to your imagination, the curved line can bestretched as much as needed to obtain 4 straight lines!(fig. c). Obviously, there are other waysto approach the puzzle...

Seethe final unique solution

Lessonsto be learned from this puzzle
- Analyze the definition to find out what is allowed and what isnot.
- Look for other definitions of problems (if a problem definitionis wrong, no number of solutions will solve the real problem).

Inconclusion, sometimes to solve a problem we needto remove a mental (and unnecessary) constrictionor assumption we initially imposed on ourselves (thelines must be straight, the lines must be drawn insidea ‘subjective’ square, etc.). In fact,mental constrictions always limit our investigationfield.

Here aremore tips and puzzle-solving strategies toconsider.

Alternativesolutions
These solutions seem less mathematical/logicalbut more creative!

3line solution:
From a mathematical point of view, a dot/point hasno dimension, but on the paper, the dots appear likesmall discs... Then, we can use the thickness of thelines to solve the puzzle with just 3 contiguous segments:

Tridimensionalsolution:
The problem is formulated in a way we implicitly assumethat it must be solved in plane geometry... Thoughit might be possible to solve it using a differentsurface, like a sphere or a cylinder, and by drawingonly one single line (see example below).

Theorigami-like solution:
This is our favorite one! Reproduce the puzzle on asquare sheet of paper. By ingeniously folding it, accordingto the example below, it is possible to align the 9dots in order to connect them together with a finalpencil stroke.

Source: MateMagica,Sarcone & Waeber, ISBN: 88-89197-56-0.

SixteenDot Version
Can you solve the Sixteen Dot (4 x 4) puzzle variantshown below? Again, you just have to join the dotstogether without lifting your pencil. What is the MINIMUMnumber of straight lines required to solve it? Do younotice any correlation between number of dots and numberof connecting lines?

Seethe solution

Generalizing
Since 6 straight lines are needed for this variantof the 9 Dot puzzle, then, how many straight lineswould you need to solve the Twenty-Five Dot variant?Exactly 8 lines (you can try to solve this problemby yourself).

Isthere some general formula to find the minimum numberof lines for a given number of dots? By making theNine Dot puzzle as complex as we desire (exponentiallyincreasing the number of dots to 25, 36, 49, 64,etc.), the following pattern appears to emerge throughinspection:

Isthere a proof that one must make the lines go outsideof the dots' boundary to get the minimum line number?We don’t know if there is a proof, but if youstay inside the 'box' and connect all the dots ina simple zig-zag fashion, from the bottom right dotto the top left dot, then the minimum straight linesyou have to trace for a 3x3 pattern is 3 + 2 = 5(generalizing: 2n – 1,see figure below)... That is one stroke more thanwhen using the method of the lines outside of thebox.

Relatedtopological puzzles
1. Try to arrange 7 dots in 4 rows with 3 dots in eachrow.
2. Arrange 10 dots in 5 rows with 4 dots in each row.

Seethe solution