Waage

Get acquainted with Shannon's formula by playing a game


A project of the physics education department of the university of Karlsruhe, Germany
Programme: Immanuel Halupczok


Deutsche Version

 

The game

Objective of the game

The twelve animals have the same weight – except one. By means of the beamscales, find out which is the “wrong one”.

When making a weighing operation you are asking a question, to which the balance can give one of three answers. In other terms: the balance answers with three different symbols:
 
1. right side down
2. left side down
3. equilibrium

With each symbol you get a certain amount of data H. According to Shannon's formula

Shannonsche Formel

the amount of data is maximum, when the probabilities of the three symbols are equal, i. e. when

p(1) = p(2) = p(3) = 1/3.

In this case, the amount of data is:

Gleichung

Thus, the best strategy is: Place the animals on the scales in such way as to get the three answers with nearly equal probabilities.

The computer as a partner and as a teacher

The computer plays three different roles.  

The computer as the opponent:
On the screen it diplays a balance as well as the choosen animals. The computer knows, which of them is the “wrong” animal. You don't.

The computer as an assistent:
It makes the bookkeeping for you. The result of each weighing is displayed on the screen in such a way that you know about every weighing, in which direction the balance had moved and which balls had been placed on which scale.

The computer as a teacher:
It evaluates each of your moves. Before each of your weighings, it tells you how many bits you get when playing the best strategy. After each weighing it tells you much you got effectively: the maximum value, if your stategy was the best one, or less if your strategy was bad. The computer also tells you how many bits you have “given away”. Only in case you take a risk, there is space for good or bad luck. If for instance, as a first move, you place one animal on each scale, and it happens that one of these animals it the wrong one, you are lucky although your strategy is bad. However, the better your strategy, the less space there is for good or bad luck.
Chance and intelligence sometimes exclude one another.