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From the same Author
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Libro MateMagica
MateMagica: Giochi d'ingegno con la matematica

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Most Wanted Puzzle Solutions

The Nine Dot Puzzle by Gianni A. Sarcone

Solutions for your logic and mechanical puzzles

nine dot puzzleshoked face"Do you know this puzzle where you have to connect all 9 dots arranged in a square with only 4 straight lines? It’s a puzzle my boy-friend got from a friend. We have played with it and we think there is no solution. Please get back to me if you find the solution!" -- Mary

Category: Topological game / Unicursal path game.
Name: Nine Dot puzzle, Dot-Joining puzzle, or Christopher Columbus's Egg Puzzle.
In other languages: Neun-Punkte-Problem (Ger), problème des neuf points (Fr), problema dei nove puntini (It), rompecabezas de los nueve puntos (Sp), trencaclosques dels nou punts (Cat).

Material: Pencil, piece of paper.
Configuration: Nine dots arranged in a square matrix, each of them being equidistant from its neighbors.
Aim of the game: Join all the dots up with not more than 4 continuous straight lines.
Origin of the puzzle: Unknown. Probably an old European puzzle.


Puzzles are in all cultures throughout time... And the 9 Dot puzzle is as old as the hills. Even though it appears in Sam Loyd’s 1914 “Cyclopedia of Puzzles”, the Nine Dot puzzle existed long before Loyd under many variants. In fact, such a puzzle belongs to the large labyrinth games family.

9 Dot puzzle is also a very well known problem used by many psychologists, philosophers and authors (Paul Watzlawick, Richard Mayer, Norman Maier, James Adams, Victor Papanek...) to explain the mechanism of ‘unblocking’ the mind in problem solving activities. It is probable that this brainteaser gave origin to the expression ‘thinking outside the box’.

Solving it
We hope you don’t mind if we use nice ladybugs instead of boring dots to make our puzzle demonstrations... Well, below are nine ladybugs arranged in a set of 3 rows. The challenge is to draw with a pencil four continuous STRAIGHT lines which go through the middle of all of the 9 ladybugs without taking the pencil off the paper.

9 dot problem with ladybugs

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

Trial-and-error strategy
9 dot joined by a curved lineIt is easy to connect all the 9 ladybugs with just a CURVED line (see fig. opposite). Try now to imagine this line as elastic as a rubber string, and wonder what would happen if one or more curves/bights would be stretched beyond the ‘invisible’ boundary, as shown in fig. a and b below.
That intuition turns out, in fact, to be the relevant ‘insight’. Thanks to your imagination, the curved line can be stretched as much as needed to obtain 4 straight lines! (fig. c). Obviously, there are other ways to approach the puzzle...

resolution of the 9 dot puzzle

arrow See the final unique solution

Lessons to be learned from this puzzle
  - Analyze the definition to find out what is allowed and what is not.
  - Look for other definitions of problems (if a problem definition is wrong, no number of solutions will solve the real problem).

In conclusion, sometimes to solve a problem we need to remove a mental (and unnecessary) constriction or assumption we initially imposed on ourselves (the lines must be straight, the lines must be drawn inside a ‘subjective’ square, etc.). In fact, mental constrictions always limit our investigation field.

arrow Here are more tips and puzzle-solving strategies to consider.

Alternative solutions
These solutions seem less mathematical/logical but more creative!

3 line solution:
From a mathematical point of view, a dot/point has no dimension, but on the paper, the dots appear like small discs... Then, we can use the thickness of the lines to solve the puzzle with just 3 contiguous segments:

alternative solution 1

Tridimensional solution:
The problem is formulated in a way we implicitly assume that it must be solved in plane geometry... Though it might be possible to solve it using a different surface, like a sphere or a cylinder, and by drawing only one single line (see example below).

alternative solution 2

The origami-like solution:
This is our favorite one! Reproduce the puzzle on a square sheet of paper. By ingeniously folding it, according to the example below, it is possible to align the 9 dots in order to connect them together with a final pencil stroke.

alternative solution 3
Source: MateMagica, Sarcone & Waeber, ISBN: 88-89197-56-0.

Sixteen Dot Version
Can you solve the Sixteen Dot (4 x 4) puzzle variant shown below? Again, you just have to join the dots together without lifting your pencil. What is the MINIMUM number of straight lines required to solve it? Do you notice any correlation between number of dots and number of connecting lines?

16 dot puzzle

arrow See the solution



brain and gearsGeneralizing
Since 6 straight lines are needed for this variant of the 9 Dot puzzle, then, how many straight lines would you need to solve the Twenty-Five Dot variant? Exactly 8 lines (you can try to solve this problem by yourself).

Is there some general formula to find the minimum number of lines for a given number of dots? By making the Nine Dot puzzle as complex as we desire (exponentially increasing the number of dots to 25, 36, 49, 64, etc.), the following pattern appears to emerge through inspection:

No. of Dots Required Straight Lines
3 x 3
4 x 4
5 x 5
6 x 6

n x n
(3 + 1) = 4
(4 + 2) = 6
(5 + 3) = 8
(6 + 4) = 10

[n + (n - 2)] = 2(n - 1)

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

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

arrow See the solution

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