Do you know how to play Rock-Paper-Scissors?
Two players simultaneously use hand gestures to
play one of three possible weapons.
Rock smashes scissors, scissors cuts paper, and paper covers rock.
In Japan, a variation has a tiger beaten by a chief
while the chief is beaten by the mother of the chief.
In Indonesia, a version has an elephant beating a man,
the man beating an ant, and the ant beating the elephant
(because the ant crawls into the elephant's ear and drives it insane)
Rock-paper-scissors is not completely a game of chance.
With inexperienced players, women usually start with scissors,
while men usually start with rock.
Moreover, most players have predictable strategies, so a computer can
learn their style.
Of course Rock-paper-scissors can be represented more abstractly with a
graph where each vertex has an incoming and outgoing edge.
This arrangement is also represented by the famous Borromean rings where each ring is
above a different ring but is below the other one.
These can't be circular rings though; they must bend somehow to avoid intersecting.
These rings, originally used by the Borromeo family of Northern Italy, have
also been used in religious symbolism, medallions, beer
logos, Escher-like mathematical art and Seifert surfaces
which connect the three rings.
Even molecular Borromean rings have been constructed from DNA.
Rock-paper-scissors can end in a tie one third of the time.
To lower this annoyance, play the five weapon game Rock-paper-scissors-lizard-Spock.
Notice how each weapon, for example Spock, beats two and loses to two.
Let's explain the ten comparisons:
The Borromean Rings for this setting has each ring above two others and below two
others.
In this new game, there is only a one in five chance of having a tie.
We could represent this five weapon game with a graph, but what if
we make a copy and switch the arrows a bit while still having two
incoming and two outgoing arrows for each vertex?
Do we get a fundamentally different graph?
No, we can simply rearrange some of the colors and see that the two graphs are
still the same.
For example, red points to yellow and blue and is pointed
at by green and orange.
Note also that we can sometimes remove two vertices to get
the graph we saw for rock-paper-scissors.
There are five of these 3-cycles within the 5-graph.
Since the 5-weapon game is better than the 3-weapon game,
is there a 7-weapon game that is better still?
We can make a graph where each vertex points to three
others and is pointed at by the remaining three.
In fact, there are two other seemingly different graphs with the same
structure; one based on a hexagon where the 6 outside points represent the
same seventh point (it's drawn like this to look symmetric),
and another graph based on what's called the Fano plane.
Like in the 5 weapon case, we ask if these graphs are equivalent
if we switch the labels.
Each of these graphs has exactly 14 3-cycles but the number of 5-cycles
varies.
Since these cycles are structural and would not change with relabeling,
these graphs must be fundamentally different from each other.
Matrix analysis shows that these are in fact the only three
possibilities with seven weapons.
These graphs correspond to three sets of rings where
each ring is above three and below three rings.
So there are three fundamentally different 7-weapon games.
As the number of weapons increases, the number of
fundamentally different games rises dramatically.
So, perhaps its best to let Rock-Paper-Scissors-Lizard-Spock be our tool for challenges.