Towers of Chicago
Robert Swartz
robert@mathapplets.net
   
I am a mathematician and computer programmer from Chicago.  I have a BA in Mathematics from
Northeastern Illinois University, and I'm also Sun Certified.  Since 2000, I've been working on the
Towers of Chicago puzzle; I programmed the multipeg version in Java.  The multipeg version involves
4 or more pegs, as opposed to the usual 3.  These pegs are enumerated as follows:  the initial peg,
the final peg, and 2 or more auxiliary pegs.  This applet can display up to 50 discs by 10 pegs with
the default window size (1300  x  1000).
  
The Towers of Chicago uses a dynamic algorithm to compute the optimal partition numbers, and
then a recursive algorithm to compute the moves.  The program features sequence variation through
successive randomization.  The Towers of Chicago is an example of Automated Reasoning.
  

The puzzle software can be found at the following link:  Towers of Chicago   The software was

programmed using Java 20; from here, the JDK Installer (Windows 10 version) can be downloaded.
Run this batch file after installing Java.  Also, adjust the High DPI settings for the Javaw executable
located in the system folder.  Here is the JDK Documentation.  The original version of my
Towers of Chicago software won 5th prize in the Quest for Java Contest.

     



     
                 
    


                            


                   

     
Here are some screenshots, and here is an applet that calculates the number of moves in
the multipeg problem.  Here is another applet that calculates the number of top-level
partition values in the multipeg problem.  Furthermore, here is a scholarly paper about
the mathematics behind Multipeg Towers of Chicago.
   
  
Also, I wrote software that solves problems in Boolean Logic:  Meta Theorem

Curiously enough, any statement in mathematics can be reduced to a statement of
Boolean Logic.  My treatment of Boolean Logic includes 64 variables, truth constants,
6 operations, and perfect syntax checking.  The program features a postfix form
polynomial time speed boost in this exponential time problem.
Here is a list of logical sentences that can be used to test Meta Theorem.

  
   
       
  
  
Meta Theorem was developed at Argonne National Laboratory under the
Automated Reasoning Project (Research Project #289).  It was used to
resolve some important hypotheses in mathematics:
Continuum Hypothesis
Twin Primes Conjecture
Rationality of Pi to the Pi
   
 
My math applets are generating 500 Gigawatts of D-D-T Fusion Power:
   
D + D   →    T + H + 4.03 MeV
        
D + T   →   He4 + n + 17.6 MeV
   
This power source will last for 50 million years!

The 4 countries that run nuclear fusion are America, Canada, Japan, and Haiti.
     
Here is the
  
and here is the imperial city skyline, Chicago, USA.
  
Here is a list of Chicago's top 20 skyscrapers.
    
It turns out that two supertowers were built in Chicago
 using my Automated Reasoning software:
   
(1)    —   Completed in 2009 @ 1388 ft
   
(2)    —   Completed in 2020 @ 1191 ft

This is the new symbol of Christendom.   

     
The mass of neutrinos has been verified to be zero by Multipeg Towers of Chicago.
The 3D volume of neutrinos has been observed to be 10 times smaller than that of photons;
according to the ancient neutrino-photon graph, at this factor, the neutrino mass is insignificant.

85% of the mass of the universe is thought to consist of dark matter.
Since neutrinos are massless, therefore dark matter cannot be composed of neutrinos.
There are no other known matter particles that are candidates for dark matter.
Perhaps the inertial field of space has mass, and thus holds the universe together.

The inertial field would be a continuous manifold that produces effects such as
the centrifugal force.  It would contain a set of equivalent inertial reference frames as
prescribed by Galileo.  This would distinguish it from the electromagnetic ether which
transmits light waves and has a preferred reference frame for the speed of light,
as prescribed by Maxwell.

The masses of the universe seem to be distributed as follows:  Pie Chart
The cardinality of the inertial field seems to be 2ℵ0 (the power of the continuum), whereas
the cardinality of the electromagnetic ether seems to be in between ℵ0 and 2ℵ0 (fractal).
    
As Emperor of North America, I propose the construction of a national high speed rail system:
  
North America has an inexhaustible power source that can be used to run high speed trains.    

Here is a map of the proposed ;  this system shall be powered by  

nuclear fusion.  This will be a grid network of dedicated TGV railways that will run at 200 mph,
and will cover all regions of the continent.  Maglev can be used for downtown to airport
transit in key cities.
                               



In order to build a national high speed rail system, we need the U.S. military to
withdraw from a variety of foreign lands and restore development on American soil.
The Federal infrastructure package is insufficient in terms of intercity high speed rail;
electrification needs to proceed westward of the northeast corridor for us to compete
with China.  At least $5 trillion needs to be appropriated in order to fund the entire
Acela National Railway System.

Imperial Edict #1:
Rebuild the following Amtrak Railroads as Dedicated TGV Railways:
1) Borealis  →  Chicago — Milwaukee — St. Paul
2) Lincoln Service  →  Chicago — St. Louis — Kansas City
3) Lake Shore Express  →  Chicago — Toledo — Cleveland / Detroit
   
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Last updated July 2025

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