Problem 7: Place four numbers in the first row as follows: ABCD
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 fourth and the first (i.e. cycle)
Will the process lead to a 0 in all 4 entries for some row?
What is the largest number of rows before a zero row is generated?
After many times of trial and error, we have convinced ourselves that this process will always lead to a 0 in all 4 entries for some row. Subtraction causes this to be true. The row above the zeros will obviously have the same number in all four columns.
Although we were supposed to do this on a spreadsheet, we began by practicing on paper. The first entries were 1,2,3,4. The 6th row contained zeros. We remembered from class that the example on the overhead used numbers that were larger than 4, so we tried 10,20,30,40. Nothing changed. We realized that nothing changed because each number increased by the same increment. We decided to drop down to single digits again and used 1,4,7,10. Here again they also increased by the same increment, so needless to say it didn't help much.
At this point we agreed that the spreadsheet would be easier to use. (We had chosen to do it on paper thinking we might be able to understand each row easier). The spreadsheet was no different, just quicker. So we programed the spreadsheet to produce the absolute value of the difference of the entry just above and the entry just to the right in the previous row. We copied the formulas to the first 20 rows on the spreadsheet (positive thinking) and then proceeded to put in numbers.
| 1 | 2 | 5 | 9 |
| 1 | 3 | 4 | 8 |
| 2 | 1 | 4 | 7 |
| 1 | 3 | 3 | 5 |
| 2 | 0 | 2 | 4 |
| 2 | 2 | 2 | 2 |
| 0 | 0 | 0 | 0 |
We knew that the numbers could not increase by the same increment, but could it increase with increments of a pattern? So we tried 1,2,5,9. As you can see that didn't get us very far. We then tried increasing our numbers without a pattern. Ex. 1,2,6,12.
| 1 | 2 | 6 | 12 |
| 1 | 4 | 6 | 11 |
| 3 | 2 | 5 | 10 |
| 1 | 3 | 5 | 7 |
| 2 | 2 | 2 | 6 |
| 0 | 0 | 4 | 4 |
| 0 | 4 | 0 | 4 |
| 4 | 4 | 4 | 4 |
| 0 | 0 | 0 | 0 |
That brought us to nine rows, but the assignment said to try again if we did not get at least 10 rows. This time we experimented with multiplying 1,2,6,12 with some power of 10.
| 10 | 200 | 600 | 1200 |
| 190 | 400 | 600 | 1190 |
| 210 | 200 | 590 | 1000 |
| 10 | 390 | 410 | 790 |
| 380 | 20 | 380 | 780 |
| 360 | 360 | 400 | 400 |
| 0 | 40 | 0 | 40 |
| 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 |
That also gave us 9 rows, but we now had more room for trial and error. We also remembered from class, the hint to pick a row and try changing that number to see the results. So we did. We started with the 10, then the 200, the 600 and finally the 1200. We discovered that 1300 worked better that 1200.
| 10 | 200 | 600 | 1300 |
| 190 | 400 | 700 | 1290 |
| 210 | 300 | 590 | 1100 |
| 90 | 290 | 510 | 890 |
| 200 | 220 | 380 | 800 |
| 20 | 160 | 420 | 600 |
| 140 | 260 | 180 | 580 |
| 120 | 80 | 400 | 440 |
| 40 | 320 | 40 | 320 |
| 280 | 280 | 280 | 280 |
| 0 | 0 | 0 | 0 |
At this point we had passed the 10 row mark, but we were on a roll and did not want to quit. We changed all of the columns for a while and discovered that 1300 still worked just as well if not better than any other. We changed columns A,B and C to 1, 210, 600 and had 13 rows.
Finally, our best result used 1, 210, 590, 1300. See the below spreadsheet.
| 1 | 210 | 590 | 1300 |
| 209 | 380 | 710 | 1299 |
| 171 | 330 | 589 | 1090 |
| 159 | 259 | 501 | 919 |
| 100 | 242 | 418 | 760 |
| 142 | 176 | 342 | 660 |
| 34 | 166 | 318 | 518 |
| 132 | 152 | 200 | 484 |
| 20 | 48 | 284 | 352 |
| 28 | 236 | 68 | 332 |
| 208 | 168 | 264 | 304 |
| 40 | 96 | 40 | 96 |
| 56 | 56 | 56 | 56 |
| 0 | 0 | 0 | 0 |
14 ROWS! We were so excited. This is an excellent way for students to investigate patterns, the lack of patterns, making predictions, and evaluating the results.