Pascal's Triangle and Its Relatives

Exercises

Other mods

1. (mod 4): Look at Pascal's triangle (mod 4). Shade the squares congruent to 0 (mod 4). Shade the squares congruent to 1 (mod 4). Shade the squares congruent to 2 (mod 4). Shade the squares congruent to 3 (mod 4). Do you see a pattern? Now shade the squares congruent to 1, 2, or 3 (mod 4). Shade the squares congruent to 0 or 2 (mod 4); shade the squares congruent to 1 or 3 (mod 4). How can we explain this pattern? Answer

2. (mod 5): Look at Pascal's triangle (mod 5). Shade the squares congruent to 0 (mod 5). Shade the squares congruent to 1 (mod 5). Shade the squares congruent to 2 (mod 5). Shade the squares congruent to 3 (mod 5). Shade the squares congruent to 4 (mod 5). Shade the squares congruent to 1, 2, 3, or 4 (mod 5). Do you expect to see a clean fractal pattern with any combination of numbers? Why or why not? Answer

3. (mod 6): Look at Pascal's triangle (mod 6). Shade the squares congruent to 0 (mod 6). Shade the squares congruent to 1, 2, 3, 4 and 5 (mod 6). By now you should see the pattern. Which combinations of squares will give clean fractal pictures? What other Pascal's triangle patterns do those produce? Answer

4. (mod 7): Look at Pascal's triangle (mod 7). What should you shade to produce simple patterns? Answer

5. (mod 8): Look at Pascal's triangle (mod 8). What should you shade to produce simple patterns? Answer

6. (mod 9): Look at Pascal's triangle (mod 9). What should you shade to produce simple patterns? Answer

7. (mod 10): Look at Pascal's triangle (mod 10). What should you shade to produce simple patterns? Answer

8. (mod 15): Look at Pascal's triangle (mod 15). What should you shade to produce simple patterns? Answer

General Patterns Do you see any differences between the Pascal's triangle for Z_n when n is prime vs when n is composite? What can you say about the dimension of the shape obtained by shading the squares congruent to 1, ..., n-1 (mod n) when n is prime? What if n is composite? Use these observations to generalize the fact that almost all binomial coefficients are even. Answer

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