Real MATHEMATICS – LIfe vs. Maths #7

Game of H&G

I selected this name for the game as it reminds me of the story of Hansel and Gretel. (If you haven’t already, pretty please with sugar on top, read it!)

In H&G there is only one goal: Finding the shortest path. Though this game is not a board game; you have got to participate physically. You can find the answer if and only if you analyze your experience.

I have to mention it now: We learned this game from bugs, ants in particular. I will get back to that inside the article.

What is H&G? How can you play it?

  • Players walk between two places.
  • Start and finish are the same locations for all players.
  • There is more than one path for the walk.
  • Goal is to find the shortest one among those paths.
  • During the game it is forbidden to use any technological device. Yes, including watches and phones.
  • Only tool allowed is a pen.
  • Each player draws a line every time he/she hits the starting and finishing ends of the path.
  • In order to maintain same (or at least similar) speed for all players, it is forbidden to run.

Game #1

Assume that there are two paths for H&G as follows:


In the beginning players on paths A and B walk the same distance. But as walks progress players using the path A arrives to the finishing point way before than the players on path B:


When the players on path B arrive to the finishing point, they see a mark that is left from the players who use path A. This means that the path A is shorter than the path B. Most of the players from B would prefer path A for return. Some of them (stubborn ones) would follow B and see that they were behind of everyone else.

After some time everyone chooses the path A.

Game #2

Let’s say that we add obstacle on the path A:


After a decent amount of time some of the players will try the path B in order to see if it is the shortest path now:

There is a small but significant difference between B and A.

Careful players will realize that the numbers of lines on path B are increasing faster than the path A which would only mean that path B had become the shortest path.

In time all of the players will realize the fact that path B became the shortest path after the obstacle.

Game #3

Let’s add a third path; we’ll call it path C, to the existing game:


At least one player will be curious and try the path C. Just like the previous games, within time players will realize that the lines from path C increase faster than the other two. Hence players slowly understand that the path C is the shortest one among those three.

Best H&G Players: Ants

In the beginning I mentioned that ants taught us how to find the shortest paths. Back in 1992 a scientist named Marco Dorigo was researching the behaviors of ants. Dorigo soon discovered that ants choose particular paths from their nests to the food supplies.

Assume the map of a nest and a food supply looks like the following:


And assume that ants start using the following path:


Pheromone: A chemical secreted by an animal that shapes its social behavior. For instance ants leave this chemical with their footprints which can be traced by other ants.

Ants are in the search of pheromone when they decide their paths between their nest and food supply. If there are more ants on a path, there will be more pheromone. This will cause more ants to use that particular path. Here, pheromones are just like our pen marks:


Let’s add an obstacle to the path:


At first some of the ants will use the path I as others use the path II:


Since path I is shorter than the path II, after some time there will be more pheromone on path I. This is why ants will abondone path II will choose to use path I in the long run:


Ant Colony Optimization Algorithm (ACOA)

Modern people are impatient: They need to run their things in the fastest way, in the shortest time. Ant Colony Optimization Algorithms help people a lot for this cause. And this algorithm’s logic comes from the method ants use in order to find the shortest path.

A warehouse of Alibaba where robots share a workspace.

For instance ACOA is crucial for robotic moves. Robots imitate ants when they learn how to move from one place to another. If there are a decent number of robots in a workspace, ACOA helps robots to avoid collisions.

M. Serkan Kalaycıoğlu


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