Assignment 12: The Spreadsheet in Mathematics Explorations

Problem: Place four numbers in the first row as follows

A B C D

For each succesive row replace the entries by the absolute value of the difference of the entry just above and the entry just to teh right in the previous row. In the fourth position use the absolute value of teh difference of teh 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?

What is the largest number of rows before a zero row is generated?

This is a good problem for middle school students when they are studying the absolute values. Technology is used in the form of a spreadsheet. Students can begin by randomly selecting the first row of numbers. Then they can set up the spreadsheet using the formula for the absolute value. Since it is unclear how many cells will be non zero, students should carry their formulas down 15 to 20 rows. Now, Students can count the non zero rows. They will see that this process will lead to a row of 0s across the four columns. It is important to let students change the original numbers and develop theories about what is happening. The goal is to get the most non zero rows.

Here is an example. Notice that the row before the zero row has the same number. This must be the case in order to have a row of zeros. Students can work backwards to see if they can find a pattern or a relationship between the numbers. If they discover something, then they can change the original numbers and see if the relationship holds.

 177 366 279 270 189 87 9 93 102 78 84 96 24 6 12 6 18 6 6 18 12 0 12 0 12 12 12 12 0 0 0 0 0 0 0 0

What would happen if the middle columns had the same number?

For example,

 A B C D 1 0 0 5 1 0 5 4 1 5 1 3 4 4 2 2 0 2 0 2 2 2 2 2 0 0 0 0

What if the middle columns had the same number and the end columns had the same number?

 A B C D 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0

Students can use different numbers to see if any relationship or pattern holds. Students might want to look at large numbers.

 A B C D 2515 3840 28405 55054 1325 24565 26649 52539 23240 2084 25890 51214 21156 23806 25324 27974 2650 1518 2650 6818 1132 1132 4168 4168 0 3036 0 3036 3036 3036 3036 3036 0 0 0 0 0 0 0 0