The Problem:

I am to place four numbers in the first row on a spreadsheet 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

|A-B| |B-C| |C-D| |D-A|

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Ok, LetÕs start by substituting arbitrary values in for A, B, C, and D and see what happens. LetÕs try: 1, 2, 3, 4

 0 1 2 3 4 1 1 1 1 3 2 0 0 2 2 3 0 2 0 2 4 2 2 2 2 5 0 0 0 0

The process ends in zeros after 4 rows. Not bad for the first try. Now letÕs continue, hoping to find some sort of pattern.

LetÕs try alternating between zeros:  11, 0, 29, 0.

 0 11 0 29 0 1 11 29 29 11 2 18 0 18 0 3 18 18 18 18 4 0 0 0 0

Wow, that obviously is not the pattern. It now ends in zeros after only 3 rows. LetÕs try some larger numbers with zeros at the end : 10000, 14, 0, and 0 .

 0 10000 14 0 0 1 9986 14 0 10000 2 9972 14 10000 14 3 9958 9986 9986 9958 4 28 0 28 0 5 28 28 28 28 6 0 0 0 0

It now ends in zeros after 5 rows. It does seem to always end in zero.

LetÕs try decimals:  4.2, 5.3, 6.4, 7.5.

 0 4.2 5.3 6.4 7.5 1 1.1 1.1 1.1 3.3 2 8.88e-16 8.88e-16 2.2 2.2 3 0 2.2 0 2.2 4 2.2 2.2 2.2 2.2 5 0 0 0 0

Again, the process ends in zeros after 4 rows.

LetÕs try numbers that have some sort of relationship. LetÕs try: 3, 5, 7, 9.

 0 3 5 7 9 1 2 2 2 6 2 0 0 4 4 3 0 4 0 4 4 4 4 4 4 5 0 0 0 0

Well, that did not improve things. I am now back to ending in zeros after 4 rows.

LetÕs try:  2, 4, 8, 16.

 0 2 4 8 16 1 2 4 8 14 2 2 4 6 12 3 2 2 6 10 4 0 4 4 8 5 4 0 4 8 6 4 4 4 4 7 0 0 0 0

Well, it looks like IÕm making progress and that some sort of pattern is the key. The process ends in zeros after 6 rows. It looks like the pattern could be multiplying by two, or successive powers lets try multiplying by two first.

LetÕs try 6, 12, 24, 48.

 0 6 12 24 48 1 6 12 24 42 2 6 12 18 36 3 6 6 18 30 4 0 12 12 24 5 12 0 12 24 6 12 12 12 12 7 0 0 0 0

Again, the process ends in zeros after 6 rows.

LetÕs try decimals: 0.625, 1.25, 2.5, 5

 0 0.625 1.25 2.5 5 1 0.625 1.25 2.5 4.375 2 0.625 1.25 1.875 3.75 3 0.625 0.625 1.875 3.125 4 0 1.25 1.25 2.5 5 1.25 0 1.25 2.5 6 1.25 1.25 1.25 1.25

Again, the process ends in zeros after 6 rows.

LetÕs try the successive exponents, which means that B=A^2, C=A^3, D=A^4.

I have created a formula in excel to perform the exponential calculations, I only need to enter a value for A. LetÕs try A=7.

 0 7 49 343 2401 1 42 294 2058 2394 2 252 1764 336 2352 3 1512 1428 2016 2100 4 84 588 84 588 5 504 504 504 504

Well, my previous conjecture seems to be a little flawed.

LetÕs try a decimal, A = 3.1

 0 3.1 9.61 29.8 92.4 1 6.51 20.181 62.5611 89.2521 2 13.671 42.3801 26.691 82.7421 3 28.7091 15.6891 56.0511 69.0711 4 13.02 40.362 13.02 40.362 5 27.342 27.342 27.342 27.342 6 3.55e-15 0 3.55e-15 0 7 3.55e-15 3.55e-15 3.55e-15 3.55e-15 8 0 0 0 0

Well, at least I am making progress. Now I have 7 rows before I reach zero. It appears that decimals are the key. But where do I start? Is there a particular pattern? LetÕs try another decimal, A= 1.62.

 0 1.62 2.62 4.25 6.89 1 1.00468 1.62771 2.63709 5.26948 2 0.623027 1.00938 2.63239 4.2648 3 0.386355 1.62301 1.63241 3.64177 4 1.23665 0.009404 2.00936 3.25541 5 1.22725 1.99996 1.24605 2.01876 6 0.772709 0.753902 0.772709 0.791517 7 0.018808 0.018808 0.018808 0.018808 8 2.22e-16 2.22e-16 2.22e-16 2.22e-16 9 0 0 0 0

Wow, IÕm not up to 8 rows before zeros, letÕs try A=1.85.

 0 1.85 3.43 6.36 11.8 1 1.57889 2.92469 5.41759 9.92116 2 1.34579 2.4929 4.50358 8.34227 3 1.14711 2.01068 3.83869 6.99648 4 0.863569 1.82802 3.15778 5.84937 5 0.964447 1.32977 2.69159 4.9858 6 0.365321 1.36182 2.29421 4.02135 7 0.996498 0.932396 1.72714 3.65603 8 0.064101 0.794742 1.92889 2.65953 9 0.730641 1.13415 0.730641 2.59543 10 0.403511 0.403511 1.86479 1.86479 11 0 1.46128 0 1.46128 12 1.46128 1.46128 1.46128 1.46128 13 0 0 0 0 14 0 0 0 0

Hooray, I now have 12 rows before I reach zeros. LetÕs keep it going. LetÕs try 1.84.

 0 1.84 3.39 6.23 11.5 1 1.5456 2.8439 5.23278 9.62229 2 1.2983 2.38888 4.3895 8.07669 3 1.09058 2.00062 3.68718 6.77838 4 0.910049 1.68656 3.0912 5.68781 5 0.776509 1.40464 2.59661 4.77776 6 0.628132 1.19197 2.18115 4.00125 7 0.563835 0.989184 1.8201 3.37312 8 0.425349 0.830915 1.55302 2.80928 9 0.405565 0.722104 1.25626 2.38393 10 0.316539 0.534159 1.12767 1.97837 11 0.21762 0.59351 0.850698 1.66183 12 0.37589 0.257188 0.811131 1.44421 13 0.118702 0.553943 0.633078 1.06832 14 0.435241 0.079135 0.435241 0.949617 15 0.356106 0.356106 0.514376 0.514376 16 0 0.158269 0 0.158269 17 0.158269 0.158269 0.158269 0.158269 18 0 0 0 0

Can you believe it?  I now have a whopping 17 rows before it reaches zeros.

LetÕs try 1.86

 0 1.86 3.46 6.43 12 1 1.5996 2.97526 5.53398 10.1088 2 1.37566 2.55872 4.57486 8.50923 3 1.18306 2.01614 3.93438 7.13358 4 0.833072 1.91824 3.1992 5.95051 5 1.08517 1.28096 2.75131 5.11744 6 0.195791 1.47035 2.36613 4.03227 7 1.27456 0.895776 1.66614 3.83648 8 0.378785 0.770367 2.17034 2.56192 9 0.391582 1.39997 0.391582 2.18313 10 1.00839 1.00839 1.79155 1.79155 11 2.22e-16 0.783164 2.22e-16 0.783164 12 0.783164 0.783164 0.783164 0.783164 13 0 0 0 0

Interesting, it decreases to 12 rows before zeros.

LetÕs try going back down to A=1.83

 0 1.83 3.35 6.13 11.2 1 1.5189 2.77959 5.08664 9.38513 2 1.26069 2.30706 4.29849 7.86623 3 1.04637 1.99143 3.56774 6.60554 4 0.94506 1.57631 3.0378 5.55917 5 0.631255 1.46149 2.52137 4.61411 6 0.830231 1.05989 2.09274 3.98286 7 0.229658 1.03285 1.89012 3.15263 8 0.803194 0.857267 1.26251 2.92297 9 0.054073 0.405243 1.66046 2.11978 10 0.35117 1.25522 0.459315 2.0657 11 0.904049 0.795904 1.60639 1.71453 12 0.108146 0.810485 0.108146 0.810485 13 0.702339 0.702339 0.702339 0.702339 14 0 0 0 0

Now, I have 13 rows before it reaches zero. It appears that it reaches the maximum number of rows when A is near 1.84.  The maximum number of rows that I discovered before a zero row is generated was 17. But, I think that this number is much larger. In fact, I think that this number may approach infinity.