(Quad:  four checkered paths begin / cross within a single S3 Utility = 1.0)

"'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

The Sommer Cube (S3) is the story of exploration in depth of the contrapuntal possibilities inherent in a cubical maze module; it is at once fundamental and obscure — leveraging Binary and Analogical reasoning, as both hybrid toy (Tate Museums) and Cognitive Tool (a switching of “true" and “false" logical arguments presented visually and tactilely — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity. 


                                                                             Rolling Ball “Tilt-Switch

                                                                      (Resets according to orientation.)

It’s about patterns — always patterns.


                                  CHALLENGE CONSTRUCTION BLOCKS (Analog-Binary Processor


Think like J.S. Bach:  see the “music”.  (Scubedom and fuguedom) 

A continuous interweaving, user created, network of tunnel paths (inner voices -- relating to bass line -- cantus firmus), imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony.

A story of  "Distributed Adaptive Message Block Network”, in Paul Baran’s Internet terms, in a low-tech format.  A fractal command and control, strategic and tactical planning challenge.


(Think of the ball as “ping”:  a network test for reachability; think graph theory.)

S3 is a combinational (technically “permutational”, like a combination lock) logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches / routers to be programmed (with or without a ball) by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous routes (network theory), depicted physically or logically.  An architecture of nested dichotomies.

(From child to super-mathematician, the Smanipulator has only one path:  reason / intuit up the original "evolution of mathematics” ladderfrom things, to abstractions of things, to binaryness, to geometryness, to algebraness, to setness -- create more sophisticated problem models, know what the laws are.)

A topology of paradox"contradiction that follows correct deduction from consistent premises”.  Reasoning as play. 

                                                MATHEMATICIANS ONLY

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

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

How many QUADRUPLE PATH S3s 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”


On the other hand 

"The S3 readily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play …

                                                                        Laura E. Berk, Child Development, 8th Edition (May 13, 2008, cc mss)

Thus, the full engagement of the physical and mental universe, Hands and Mind; a dynamic structure of nested ideas, their context, and their logical expression a counterpoint of logic and intuition with myriad and changing dead-ends and thruways.  

A cubical maze module leveraging Binary and Analogical reasoning:  true and false logical arguments presented visually and tactilely — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity:  mathematician's delight. 


Princeton’s finest at work all day in the Common Room on the Quad Problem — April 23, 2002.  He began by carefully setting up several S3, then put the ball into an opening; then he frowned, sat back, and said, “That’s odd — I just proved that that could not happen.”  


Attacking problem by making cardboard model of Quad Problem Space:  Group Theory (symmetry).


In other words, Spatterns of 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)

            Discovering and Representing A System Of Rules

                                                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 Quad Problem a Frame Problem?                                           

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


                                                      (virtual reality paths confirmed)

       Approaching Complexity (Something You Cannot Predict)



                                                                                Tessellating Space

Note other interesting Sproblem spaces:  Boolean Construction, Atomic Warfare (surgical elimination of command / control node-links), Euler’s Revenge at Challenges / Games, for anyone.

                        Ball Path (trajectory) Packing Problem

                 Sommer Polyhedron:  Porous Space-Filler


                                      Hungry Balls  +  Cube (jello) = Tunnels

Configure four tunnels, of unlike symmetry, to allow a ball to enter and exit in any of twenty-four perpendicular orientations, where at least one low-quadrant exit is always possible.  

                                      In Calculus Terms

Four tunnels (trajectories) are as closely packed as possible within a cube (they attempt to “kiss” each other and the faces of the cube) without overlapping and / or kinking and are configured to allow a ball to enter and exit in any of twenty-four perpendicular orientations, with the constraints of gravity and / or specific exits.  In other words:

Problem.   “Given [four sets of] two points [tunnel openings in a hollow cube] A and B in a vertical [inclined] plane, what is the curve [four curves, i.e., four tunnels, closely packed and intertwined in harmony] traced out by a point [ball] acted on only by gravity, which starts at A and reaches B in the shortest [longest] time?”    

                                            Bernoulli's Brachistochrone problem [with a twist], calculus of variations.

                                                                                     Note Paul J.Nahin, When Least is Best (2004)

Solution.    S3

© Michael S. Sommer, Ph.D, 2018