(four checkered paths begin / cross within a single S3)

S3 is a programmable logic component (In Field Programmable Gate Array terms) a combinational logic block which contains four tunnels, effectively an array of unconnected switches to be programmed by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous, routes by reconfigurable interconnects. 

                                                MATHEMATICIANS ONLY

Given that a ball must exit at the lowest S3 level: 

What is the minimum number of CUBES necessary to make four QUADRUPLE PATH CUBES (S3s which utilize all four conduits as passageways)? 

How many QUADRUPLE PATH CUBES are possible using twenty-seven S3s?

Remember, the Law of Gravity must be obeyed.

“Description:  The faces of the modular Sommer Cube (S3) feed into twisted conduits that comprise a three-dimensional, tunnelling maze. The website provides example arrangements and construction questions such as "Given that a ball must exit at the lowest S3 level and obey gravity, what is the minimum number of cubes necessary to make four quadruple path cubes, or S3s which utilize all four conduits as passageways?" Available with transparent or opaque walls.”

               “Levels:  High School (9-12), College, Research

        Languages:  English

Resource Types:  Games, Manipulatives

      Math Topics:  Higher-Dimensional Geometry, Topology”



                                                                Princeton’s finest at work on the Quad Problem

Influence among parts and whole:  Open Systems Thinking as a working epistemological premise, within a cubical field of “forbiddingly complex environmental interactions.” (F. Emery, Systems Thinking, 1969)

"'solutions to the narrowly conceived problem have the effect of driving a (deeper) difficulty into some other quarter of the broad problem ” 

                                                   Daniel Dennett  (Hayes):  Cognitive Wheels: The Frame Problem Of AI

       Approaching Complexity (Something You Cannot Predict)



                                                                                Tessellating Space

             Discovering and Representing A System Of Rules

                      S3:  Programming Binary Logic Gates

                                                Is It A Frame Problem?

"In fact, there is less than perfect agreement in usage within the AI research community. McCarthy and Hayes, who coined the term, use it to refer to a particular, narrowly conceived problem about representation that arises only for certain strategies for dealing with a broader problem about real-time planning systems. Others call this broader problem the frame problem-`the whole pudding,' as Hayes has called it (personal correspondence) - and this may not be mere terminological sloppiness. If 'solutions' to the narrowly conceived problem have the effect of driving a (deeper) difficulty into some other quarter of the broad problem, we might better reserve the title for this hard-to-corner difficulty ...

"The frame problem is not the problem of induction in disguise. For suppose the problem of induction were solved. Suppose - perhaps miraculously - that our agent has solved all its induction problems or had them solved by fiat; it believes, then, all the right generalizations from its evidence, and associates with all of them the appropriate probabilities and conditional probabilities. This agent, ex hypothesi, believes just what it ought to believe about all empirical matters in its ken, including the probabilities of future events. It might still have a bad case of the frame problem, for that problem concerns how to represent (so it can be used) all that hard-won empirical information - a problem that arises independently of the truth value, probability, warranted assertability, or subjective certainty of any of it. Even if you have excellent knowledge (and not mere belief) about the changing world, how can this knowledge be represented so that it can be efficaciously brought to bear?"

                                                                    Daniel C. Dennett, Cognitive Wheels: The Frame Problem Of AI

Q.  Is S3 Quad Problem a Frame Problem?                                           

A.  "Interesting questions but they will have to be postponed.”  D.C.D. (May 21, 2014) 

© Michael S. Sommer, Ph.D. 2015