Επιστροφή στην Κεντρική Σελίδα
3-2i 3-5i
* 4+i 2+5i 1+i 7-3i 2+3i *
1+3i 10
In the maze above, you must go from the green star to the red star. 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.