Assignment #12Nicole MostellerEMAT 6680

: Problem: Place four numbers in the first row as followsInvestigation #7A 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 inFigure 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 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 theFigure 1.ithrow, then the(i-1)throw must have the same value, and in the(i-2)throw, the values inColumn A = Column Cwhile the values inColumn B = Column D. In the case wereA = 1,B = 2,C = 3, andD = 4, we see that the zeroes appear in then7throw.

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

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 TheFigure 2:6throw gives the row of zeroes. Notice that the cells in the5throw have equal values, and the4throw hasCell A = Cell CandCell 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 TheFigure 3:10throw gives the row of zeroes. Notice that the cells in the9throw have equal values, and the8throw hasCell A = Cell CandCell 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 TheFigure 4:5throw gives the row of zeroes. Notice that the cells in the4throw have equal values, and the3throw hasCell A = Cell CandCell B = Cell D.

To make your own investigationClick Here!for the Random Numbers forA, B, C, and DonExcel.

From this investigation, I became interested to know if it was possible to arrive at the zero row after more than 10 recursions. And after a paper and pencil investigation, I found that it is possible. The case that I found was just one case out of 4,294,967,296 cases. Click Here! to see a the work into this investigation as well as an explanation.

After discussing this problem with another UGA Student - Thanks to Signe Kastberg! - since we are not restricted in choosingA, B, C,andDfrom the positive intergers, we see that it is possible to find cases where the zero row occurs after10recursions. SeeFigure 5below.

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