Pascal's Christmas Tree

Blaise Pascal was a French mathematician in the 16th century, whose name has become attached to an "arithmetic triangle" that we now call Pascal's Triangle.

The start of the triangle looks like this:

The first and last number on each line are 1.  The other numbers are the sum of the two numbers in the line above, just to the left and just to the right.  This famous triangle has many interesting connections in mathematics.

But today, I want to share a Christmas activity for young mathematicians and artists:  Pascal's Christmas Tree.

Here's what to do:

1. Print a copy of  Pascal's Christmas Tree Template 
2. On a blank piece of paper, create the first 8 or 10 lines of Pascal's Triangle.  The first 5 lines are above, to get you started.



The first 5 lines

3. Assign a colour to the odd numbers (numbers ending in 1, 3, 5, 7, 9), and a different colour to the even numbers (numbers ending in 0, 2, 4, 6, 8).
4. On the printed tree, colour the squares that correspond to an odd number in the odd-number-colour, and the squares that correspond to an even number in the even-number-colour. See sample at right.
5. After a while you should see the pattern, so that instead of using the numbers from the triangle you created in Step 2, you can use the colours above a square to figure out the what colour is next.  For example, in the sample, when both squares above are green, the new square is red (ie. green + green = red.  Colour addition!).
6. Once you know the colour addition rules, continue to fill in the entire tree.  
7. How many different patterns can you find on your Christmas tree?  
8. Decorate your tree if you want.   Maybe a star at the top, gifts under the tree? 
9. Enjoy your mathematical Christmas creation!

Want to try a more complex version, with a somewhat different pattern?  This time choose 3 colours to use.  One colour will be for numbers that are multiples of 3 (eg. 3, 6, 9, 12 . . .), one will be for numbers that are one more than a multiple of 3 (eg. 1, 4, 7, 10 . . .)  and the third colour will be for numbers that are two more (or one less) that a multiple of 3 (eg. 2, 5, 8, 11 . . .).  You should be able to sort out the new "colour addition rules" and will see a more complicated pattern develop on the tree.  You can do the same with using 4 colours (using multiples of 4, one more, two more and 3 more than multiples of 4) or 5 colours.

Sneak preview of 3 colour tree

Here are two more templates:
A 27 line tree for 3 colours
A 64 line tree for 4 (or 2) colours)
Can you figure out why it is nice (but not necessary) to use the 27 line tree for 3 colours, the 32 or 64 line tree for 2 colours or the 64 line tree for 4 colours?

Have fun, and Merry Christmas and Happy Holidays!

12 Days of Christmas Gifts

A tradition, in our home, following in my mother's footsteps, is to have Pork Tourtiere for dinner on Christmas Eve.  For one of my sons this is the best dinner possible, and he has never understood why I don't make pork tourtiere throughout the rest of the year. (It wouldn't be as special then, would it?)

At 28, he has finally figured out a solution - he wants to make a minor modification to the 12 days of Christmas song:

On the first day of Christmas my mother gave to me, a very big delicious pork tourtiere.
On the second day of Christmas my mother gave to me, 2 pork tourtieres and a very big delicious pork tourtiere.
On the third day of Christmas my mother gave to me, 3 pork tourtieres, 2 pork tourtieres and a very big delicious pork tourtiere.

And so on.

So we decided to have some fun and do a bit of math to calculate how many gifts you would get (in his case, how many pork tourtieres) if you got all those gifts.  In my son's case, let's just say he would need a very large freezer and would have food for a year. Almost.

The sum is:
 1 +  (2 + 1) +  (3 + 2+ 1) + (4+ 3 + 2 + 1) + . . . + (12 + 11 + 10 + 9 +  . . . + 1)

I have shown the calculations on a separate document.  Take a minute to have a look!

Much to our surprise, the number of gifts you would receive, or the number of pork tourtieres my son would have, after the 12 days of Christmas, would be 364 - one less than the number of days in a year - one for every day EXCEPT Christmas day.  We liked that little coincidence.

And enjoyed a little recreational math along the way.

And needless to say, he's not getting all those dinners!