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.
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’.
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.
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!
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...
the final unique solution
to be learned from this puzzle
- Analyze the definition to find out what is allowed and what is
- Look for other definitions of problems (if a problem definition
is wrong, no number of solutions will solve the real problem).
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
more tips and puzzle-solving strategies to
These solutions seem less mathematical/logical
but more creative!
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:
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).
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
Sarcone & Waeber, ISBN: 88-89197-56-0.
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?
of possible solutions
lines are required, at a minimum, to solve the Sixteen
Dot puzzle. Unlike the 9 Dot puzzle, this variant
has 15 possible solutions (excluding compositions
obtained by rotation and reflection).