Explorations with Spreadsheets


Moni Olubuyide


Problem: Place four numbers in the first row as follows


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|

Will the process lead to a 0 in all 4 entries for some row? Yes, the process will always lead to zero in all 4 entries!!!

What is the largest number of rows before a zero row is generated? I cannot say for sure but I tried for a very long time and could only get up to 14 rows before I got all zeros.


Here is how I started. I began with something simple such as successive multiples of twos in each column and my result shows only 7 rows.

Then I moved on to try multiples of threes and the results were the same. I began to notice that the last row was always 2*A1 using this method and the most entries I got was 7 rows.



The next strategy was to try squaring each previous cell. Such that B1 = A1^2, C1 = B1^2 etc. Starting with 2. I used negative numbers and high positive numbers and results were the same..only 6 rows!! This strategy was out the window.


After trying to square the cells, I decided to get the sqaure roots instead. Here I used square root of 2, 3, 5, and 11. Finally I was getting somewhere, I got 9 rows. It seemed like using decimals might be the key.


I wondered if using powers with square roots would make a difference. This was my first success, A1= 2, B1= 5, C1 = 2^3 *(sqrt 2), D1 = 2^4 * (sqrt 2). I got 10 rows..finally!!

I took it one step further and tried A1= 2, B1= 2^2 * (sqrt 2), C1= 2^3 *(sqrt 2), D1= 2^4 * (sqrt 2). I finally got over 10 rows. But trying different numbers did not get me any further, I only maxed out at 11 rows!!.

After about another 30 minutes, I gave up using powers and went back to multiplying but this time with decimals. I began with something basic like A1= 2, B1 = 2*2.2, C1= 2*4.2, D1 = 2*8.2. I still only got 11 rows but this was the first attempt.

I tried using starting with 5 instead of two and I got less rows.....



I tried 6 next and I went back up to 11 rows and this was the pattern with different numbers that I used. The second chart shows using 10 for A1 and the numbers went down to 9 rows. Sometimes I got less than 10 rows and sometimes I got over 10 rows.

I tried using other multipliers and then I found another one that worked. Starting with my trusty number 2, B1 = 2*2.25, C1 = 2*4.5, D1 = 2*8.75. Here I got 11 rows!!


So then instead of using 2 as my starting point I started with 2.001 and worked my way up. When I tried 2.006 I got my highest number of rows which was 14 but the entries fluctuated from 11 rows to 14 rows. The ones with the 11 rows always ended with a row of 2*A1.

I went up pretty far but only a few values that gave me 14 rows. Here is another example using 11.731 as my starting point.

Click here to see if you can beat my maximum number if you dare!!