Lab 8: Pascal's Triangle (mod 2)
Draw a large square on a piece of graph paper. Make your square 32 boxes on each side. If you're adventurous and you have small enough squares on your graph paper, make your square 64 boxes on each side.
In the first row of your box place a "1" in the upper-left hand corner. In the figures below, we are using a circle to represent the "1", see stage 0. When you complete the construction you will "shade in" all of the boxes with "1"'s. I am showing you the "shaded" figures below.
For each box in the second row, add the contents of the boxes "above" and "above, left". If the sum is "1", place a "1" in your current box. If the sum is divisible by "2", (in this game this means either "0" or "2") place nothing in your current box. See stage 2.
Continue in this way through all 32 (or 64!) rows of your original large square. Shade in every box that contains a "1". We give the first five stages below.
By addition "mod 2", which is an abbreviation for "modulo 2", we mean that the sum of two numbers is equal to the remainder of the ordinary sum of the two numbers upon division by "2". For example:
\[6+7=1 \bmod 2\] \[3+5=0 \bmod 2\]However in our example, we are only using:
\[0+0=0 \bmod 2\] \[0+1=1 \bmod 2\] \[1+0=1 \bmod 2\] \[1+1=0 \bmod 2\]Remember, in the figures below a circle represents a "1", and a blank space represents a "0".
Take a peek at stage 256!
