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Assignment #12
Nicole Mosteller
EMAT 6680
Investigation #7:  Problem:  Place four  numbers in the first row as follows
A   B   C   D.
For each successive row replace the entries by the absolute value of the difference of the entry just above and the entry
just to the right in the previous row.  In the fourth position use the absolute value of the difference of the fourth and the first (i. e. cycle)
|A - B|   |B - C|   |C - D|   |D - A|.

To begin this investigation, I chose the most obvious entries for A, B, C, and D - 1, 2, 3, and 4 respectively.
Following the instructions, I achieved the matrix in Figure 1.

A
B
C
D
1
2
3
4
1
1
1
3
0
0
2
2
0
2
0
2
2
2
2
2
0
0
0
0
Figure 1.
This general case gives an idea that successive differences will eventually lead to a row of zeroes.
This case also allows us to anticipate the rows preceding the row of zeroes.
Let the row of zeroes be in the ith row, then the (i-1)th row must have the same value,
and in the (i-2)th row, the values in Column A = Column C while the values in Column B = Column D.
In the case were A = 1, B = 2, C = 3, and D = 4, we see that the zeroes appear in then 7th row.

To test this hypothesis, I decided to generate many tables with random values for A, B, C, and D.
Microsoft Excel makes finding random values as well as the recursive differences easy to find.
Below shows several of the tables generated with Excel.

A
B
C
D
11
961
360
301
950
601
59
290
349
542
231
660
193
311
429
311
118
118
118
118
0
0
0
0
Figure 2:  The 6th row gives the row of zeroes.  Notice that the cells in the 5th row have equal values,
and the 4th row has Cell A = Cell C and Cell B = Cell D.

A
B
C
D
40
656
516
369
616
140
147
329
476
7
182
287
469
175
105
189
294
70
84
280
224
14
196
14
210
182
182
210
28
0
28
0
28
28
28
28
0
0
0
0
Figure 3:  The 10th row gives the row of zeroes.  Notice that the cells in the 9th row have equal values,
and the 8th row has Cell A = Cell C and Cell B = Cell D.

A
B
C
D
217
951
744
796
734
207
52
579
527
155
527
155
372
372
372
372
0
0
0
0
Figure 4:  The 5th row gives the row of zeroes.  Notice that the cells in the 4th row have equal values,
and the 3th row has Cell A = Cell C and Cell B = Cell D.