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A COMPLEX MAZE In the maze above, you must go from the green dot to the red dot.  You start with a running total (a+bi) of 4+i.  When you move one space (diagonally or orthogonally) to a new number (c+di), you have four choices:
 1.  Add the new number to your running total. (a+bi) + (c+di) = (a+c) + i(b+d) 2.  Subtract the new number from your running total. (a+bi) - (c+di) = (a-c) - i(b-d) 3.  Multiply your running total by the new number. (a+bi)(c+di) = (ac-bd) + i(bc+ad) 4.  Divide your running total by the new number. (a+bi)/(c+di) = ( (ac+bd) + i(bc-ad) )/(c2+d2)
One condition is placed on your running total.  At every step, it must be in one of the following forms, where a is a non-zero integer (positive or negative).  If you cannot make one of the following forms, you've hit a dead end.
 1.  1+ai 2.  -1+ai 3.  a+i 4.  a-i
Thus, every step must be of the form of a double circled number on the diagram below.  As an example, you start with (4+i).  You can move diagonally down and multiply by (1+3i), to obtain a running total of (1+13i).  Then you can move diagonally back, dividing.  (1+13i)/(4+i) = (1+3i).  You are now back at the starting square, with a different running total.  You could follow by moving right and subtracting.  But there are other starting paths.  When I started creating this maze, I intended to make it slightly larger. Then I figured out a state diagram, in the tradition of Robert Abbott.  It took hours, and became unbelievably complicated.  I found at least eight different solutions.  In one of the nicer solutions, all the orthogonal moves involve addition or multiplication, and all the diagonal moves involve subtraction or division.  If you can find any solution, write me.  If anyone writes a computer program for this, can you find an eleven cell grid that has many false paths but a unique solution? The maze is complex in more ways than one.  The mathematical term complex was developed by Carl Friedrich Gauss (1777-1855).  Complex numbers of the form a+bi with a and b both integers are called Gaussian Integers.  Gauss discovered that these integers can be uniquely factored into Gaussian Primes.  Every nonzero Gaussian integer is uniquely expressible as a product of a power of i and powers of "positive" Gaussian primes.  Using the Mathematica definition, the first few positive Gaussian Primes are in green. "Normal" primes of the form 4a+1 are not Gaussian primes, it turns out.  2 = (1+i)(1-i), 5=(2+i)(2-i), 13=(3+2i)(3-2i), ... but 3, 7, and 11 are Gaussian Primes.  1999-2000i = i(1+4i)(2+3i)(9+4i)(7+18i) is a sample factorization.  Using complex numbers, can you find the equations for the line and circle in blue?  I hope to develop complex equations for all the major mathematical curves.

The book Visual Complex Analysis by Tristan Needham starts with this quote: "The present book openly challenges the current dominance of purely symbolic logical reasoning by using new, visually accessible arguments to explain the truths of elementary complex analysis."  I heartily agree with this philosophy, and highly recommend the book.

Joseph Devincentis:
I found these calculations tedious so I wrote a program to solve your maze.  It exhausted the search space (pruning paths that lead to the same cell with the same running total -- there were *lots* of these) and found these solutions:

4+i - 3-2i / 1+i + 2+5i / 1+i - 10 + 7-3i + 2+3i = 1-i
4+i - 3-2i / 1+i - 7-3i + 10 - 1+i + 7-3i / 2+3i = 1-2i
4+i * 1+3i / 4+i - 2+5i - 1+3i + 1+i * 7-3i + 2+3i = 1+20i
4+i * 1+3i / 4+i - 2+5i - 1+3i + 10 - 7-3i + 2+3i = 3-i
4+i * 1+3i / 4+i - 2+5i - 1+3i + 10 - 7-3i - 2+3i = -1+7i
4+i * 1+3i / 4+i - 2+5i - 1+i + 10 - 7-3i + 2+3i = 3+i
4+i * 1+3i / 4+i - 2+5i - 1+i + 10 - 7-3i * 2+3i = 8-i
4+i * 1+3i / 4+i - 2+5i - 1+i + 10 - 7-3i - 2+3i = -1+5i
4+i * 1+3i - 2+5i / 3-2i + 1+i - 3-5i - 7-3i / 2+3i = -1+2i
4+i * 1+3i - 2+5i / 3-2i + 1+i - 10 + 7-3i + 2+3i = 1+i
4+i * 1+3i / 4+i + 3-2i - 3-5i + 2+5i / 1+i * 7-3i - 2+3i = -1+20i
4+i - 3-2i / 1+i + 2+5i / 1+i + 10 / 2+5i * 1+3i + 10 / 7-3i * 2+3i = 1+8i

Fabio Buffoni:
I saw your complex maze and I wrote a little program to find the solutions. I didn't spend much time checking it,but it should be right. The number before the solution is the path length.  The attachment is the analysis with the max depth of 18.  [And he sent 3129 solutions]

* False paths average length: 16.575655
* False paths: 1771440
* Max depth (18) reached 1300875 times
* Total solutions found : 3129

John Bollinger:

I have had a lot of fun with your complex maze.  I found my first solution (which turns out to be the shortest one) by hand while working on the state diagram for the maze.  I decided that the problem of elucidating the state diagram was more suitable for a computer than for me, so I wrote a program to do it.  It found a total of 235 accessable states (each characterized by a position in the maze and a particular running total), and 411 allowed transitions between pairs of states.

When I extended my program to actually trace through the state diagram to find solutions, I found (as expected) that there are cycles and multiple solutions.  In fact, there are many, many, many cycles, and many, many cycle-less solutions.  (And infinately many solutions that do contain cycles.)  There are 16 distinct final states possible for paths through the maze; the shortest paths leading to each are below.  I suspect that generating a nontrivial maze of this kind that has only one solution is a very difficult problem.

The 16 possible final totals are 1+i,  3+i,  -1-5i,  8-i, 1-2i,  1-i, 1+20i, 1+8i, -1+2i, -1-20i, -1+i, -1-i,  -1-8i, 1+5i, -3-i, -8+i

J B Gill:

I was intrigued with your Complex Maze puzzle, so I thought I'd give it a whirl.  After finding a few solutions, I discovered that there are probably dozens of ways to solve this, most of which involve a handful of common "states" (I define a "state" as being on a specific square  with a specific running total).  I also read your comments on a "state diagram" for this puzzle and I was curious as to how many possible states there are for this maze, so I wrote a simple program that helped me figure it out.  I attached the output and source code in case you're interested.  The way I determined how many unique states there are is count the number of lines in the output, then count the number of lines that say "been in this state", multiply it by 2, and subtract from the total lines.

From this I determined that there are 237 unique states for this puzzle (unless I miscounted).  This output also shows that there are solutions of at least 63 steps which don't re-use a state - there may be longer ones.  (I didn't keep track of unique states per solution, just unique states
overall.)  There are also 16 unique solution states.

The output can also be used to list all possible solutions to the puzzle (if you know how to read it correctly).  [He also sent a Java Program and output.

Gareth Rees:

The shortest solution takes five steps

(4+i)   - (3-2i)
=> (1+3i)  / (1+i)
=> (2+i)   - 10
=> (-8+i)  + (7-3i)
=> (-1-2i) + (2+3i)
=> (1+i)

There are some loops in the maze, so there are infinitely many paths in all.  There's a loop of length 9 that is accessible from the start and which can be extended after any number of cycles to reach the finish:

Here are the values that you can have when you reach the finish (I think this is exhaustive, but I don't have a proof -- the state diagram is huge):

-8+i -3-i -1-20i -1-8i -1-5i -1-i  -1+i -1+2i
1-i  1-2i 1+i    1+5i  1+8i  1+20i 3+i  8-i

[Gareth also sent me some Emacs LISP code.]