Pascal’s Triangle Prime Rows, Hexagon Sums, Fractal of Prime Multiples

One of the amazing properties of Pascal’s Triangle is that the prime rows (2,3,5,7,11,13,17,19,23,29…) are the ONLY rows of Pascal’s in which all numbers (except for the “1s”) are multiples of that prime number. For example, row 7 has 21, 35, and 42; row 11 has 55,165,330,and 462. The attached pdf not only highlights those rows but also shows the number of the multiple. For example, in row 7: x1 for 7, x3 for 21, x5 for 35, and x6 for 42. If you can find a pattern in the multiple numbers, maybe a Fields Medal is in our future.

The second page of the pdf shows that all seven cells of any hexagon within Pascal’s adds up to double the cell below the hexagon. The example on the pdf is 4,6,5,10,10,15,20 which adds to 70 or double the cell below the 15 and 20 which is 35. This works for all hexagons in Pascal’s. Why does it work? There are extra Mathlete dollars for anyone who finds the algorithm to support this.

Finally, my favorite discovery in Pascal’s are the fractals first found by Sierpinski in 1915 (almost 300 years after Pascal). Last week we colored in the even numbers to find Sierpinski’s Triangle and I showed the children the 128 row Pascal’s Triangle I colored with this pattern. Then I thought about what happens if you color in other multiples; would it form a fractal (a self-similar object) or just a pretty symmetrical picture? It turns out that if you color in prime number multiples such as 3, 5, 7, 11, 13 and 17, you always get a beautiful fractal. 

The objective of this lesson was not only to reinforce the concept of fractals but to teach them the divisibility rules for these critical primes. The divisibility rules are as follows:

A number is divisible by 2 if its last digit is also (i.e. 0,2,4,6 or 8).

A number is divisible by 3 if the sum of its digits is also. Example: 534: 5+3+4=12 and 1+2=3 so 534 is divisible by 3.

A number is divisible by 5 if the last digit is 5 or 0.

Test for divisibility by 7. Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also.

There are similar rules for the remaining primes under 50, i.e. 11,13, 17,19,23,29,31,37,41,43 and 47.

Test for divisibility by 11. Starting with the first digit of the number, take every other digit of the number (the first digit, the third digit, the fifth digit, etc) and add them up. Now take the digits you didn’t use the first time (the second digit, the fourth digit, etc.) and add THEM up. Now subtract these two sums from each other. If the difference between the two sums is divisible by 11, then the number is divisible by 11.

A lot of the time, two sums will be equal, so their difference equals 0, which is divisible by everything, so certainly is divisible by 11. (note: this rule only works for divisibility by 11!)

Test for divisibility by 13. Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary. 

Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13.

Test for divisibility by 17. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary. 

Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17.

The pdf allows students to try all 7 fractals as well as one on their own. They may want to challenge my conjecture that other multiples such as 4, 6, 8, 9, 10, will not produce fractals on Pascal’s Triangle. I consider the multiples of 2, 3, and 5 easy and 7, 11, 13, and 17 advanced. Third graders may want to try up to 7 and 5th through 7th graders may want to try all the way to 17. Look for algorithms that support inverted triangles of these prime multiples and show your work. A third grader found a mistake in my multiples of 3 fractal because I made an incorrect assumption about how the fractal would look and she proved it by adding the digits and showing that the sum was not a multiple of 3 (my first mistake was 12,870 which I colored in). I love it when students find mistakes that I have made. It makes it easier to prove to other students that you should always show your work. Children are naturally pushing boundaries (this is very normal), but trying to do all the work mentally will always (I MEAN ALWAYS) end in easily avoidable mistakes.