Question

Pascal's triangle has 1 in the first row, then 1, 1 in the second rod, then 1, 2, 1 in the third row, etc. One can place these numbers in a triangle, with "rows" as diagonals, as

drawn below.
111111
1.2345
1 3 6 10
1410
15
1
Etc.
Turn these numbers into black and white dots, depending on whether they are even (black) or odd (white). This produces an image like the following:
hoo
XX
X
Now perform this in PIL, to create an image that fills the top-left triangle in an image of at least 200 x 200 pixels. You should not be computing giant numbers like 200
factorial! Each row of Pascal's triangle can be computed from the previous by addition... and you can perform this "mod 2" all the way through.
Extra fun: instead of black/white dots for even/odd, try colors based on remainders after dividing by three, by four, etc.

Answer 1

The question involves utilizing PIL to create an image representing Pascal's triangle with colors or shades representing even or odd numbers, or other modulus divisions. The process leverages combinatorics and modulo operations to avoid large calculations.

Step-by-step explanation:

The student's question pertains to creating a visual representation of Pascal's triangle using the Python Imaging Library (PIL) to differentiate between odd and even numbers. The task requires generating an image where even numbers correspond to black dots and odd numbers to white dots, or potentially using colors to represent different remainders when divided by 3, 4, etc. This exercise helps visualize patterns in Pascal's triangle and serves as a programming challenge in manipulating images based on mathematical properties.

One must not compute large factorials as the rows of Pascal's triangle can be derived from the previous row by addition and perform these operations with modulo arithmetic (mod 2 for even/odd, mod 3 for coloring, etc.), to reduce the computational complexity. It's not only an application of combinatorics but also demonstrates how certain mathematical properties can be translated into a visual medium.