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Patterns in Pascal's Triangle - with a Twist - First Twist: What is It?

Kathleen M. Shannon and Michael J. Bardzell

Faculty Notes

Pascal's Triangle (symmetric version) is generated by starting with 1's down the sides and creating the inside entries so that each entry is the sum of the two entries above to the left and to the right. Suppose that, instead of using regular addition to generate the interior entries, you used modular arithmetic (also known as clock arithmetic). You may or may not remember seeing modular arithmetic in school, but you use it regularly when you deal with time. Three hours after 11:00 it is not 14:00 but rather 2:00. You add three to eleven, but when you get to twelve you start over again at 0. Modular arithmetic works the same way, except that we use 0 in place of the last number (which would be 12:00 in clock arithmetic).

[The military and some other organizations use a 24-hour clock and speak of "1100 hours" and "1400 hours". But this is still modular arithmetic with 24 replacing 12. For example, four hours after "2100 hours" is "100 hours" (i.e., 1:00 AM the next day). Furthermore, 0 replaces 24 in the military system -- you never hear a military person say "2400 hours".]

We also can do modular arithmetic with numbers other than 12 for the maximum. For example, we can use 5 as the maximum, in which case we call it mod 5 arithmetic. In mod 5 arithmetic, we use the numbers 0, 1, 2, 3, and 4, and 3 plus 3 is 1, while 2 plus 2 is still 4. If we do arithmetic mod 3, 2 plus 2 is 1, but 1 plus 1 is still 2, etc. You can see the addition tables for a number of different mod numbers by following this link.

One nice thing about modular arithmetic is that there are only a finite number of possible answers. If you assign to each of the possible answers a color, then the triangle can be presented as an array of colored dots or circles. Most of us find it more appealing to look for patterns in this kind of image than in the numbers themselves. The following figures show the first few rows of Pascal's Triangle using mod 3, mod 4, and mod 5 addition, expressed as arrays of colored circles. In these figures red is 0, black is 1, green is 2, blue is 3, and yellow is 4.


Pascal's Triangle Mod 3
27 rows
Pascal's Triangle Mod 4
32 rows
Pascal's Triangle Mod 5
50 rows

Sometimes we can see things differently if we use different colors. Here are the same patterns with different color assignments:

       Faculty Note

Andy's Applets: Would you like to try generating images for Pascal's Triangle mod n on your own?

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - First Twist: What is It?," Convergence (December 2004)