Pascal's Patterns

55. Triangle Tribulations

Pascal's Patterns

The numbered triangle (see "Triangle Magic") is commonly known as Pascal's triangle, named for Blaise Pascal, a French philosopher and mathematician from the seventeenth century.

Pascal studied this numbered triangle extensively, but he was not the first to identify it. The Persian poet and mathematician Omar Khayyam (1048-1131) described it in his writings. It also appears in ancient Chinese manuscripts.

Chinese version of Pascal's triangle. MAA Mathematical Treasures

Pascal's triangle is full of interesting number patterns. If you add up the numbers in each row, you get successive powers of 2 (the sum of each row is double the sum of the previous row).

Row 1: 1 = 2⁰

Row 2: 1 + 1 = 2 = 2¹

Row 3: 1 + 2 + 1 = 4 = 2 ✕ 2 = 2²

Row 4: 1 + 3 + 3 + 1 = 8 = 2 ✕ 2 ✕ 2 = 2³

The triangle is also full of geometric patterns. If you shade all the squares numbered with a multiple of 5, for example, you get a pattern of upside-down triangles.

The first nine rows of Pascal's triangle.

TRY IT!

Look for patterns in Pascal's triangle.

You will need:

• several copies of Pascal's triangle (above)
• pencil
• colored pencil (optional)
• calculator (optional)

What to do:

1. Look for number sequences along the triangle's diagonals. The sequence along the first diagonal is 1, 1, 1, 1, 1,…. The sequence along the second diagonal is 1, 2, 3, 4, 5,…. Write down the third diagonal sequence. Do you recognize the sequence?
2. Using a regular or colored pencil, shade all the squares numbered with a multiple of 5. What kind of pattern do you get?
3. Shade all the multiples of 2. How is this pattern different from the pattern in step 2?
4. Try shading multiples of 3, 4, 6, 7, or other numbers and see what patterns turn up. A calculator may be helpful for dividing very large numbers to see which should be shaded.
5. On any of the shaded triangles, use a different color to shade all the ones, and a third color to shade all the squares you have not yet shaded. What sort of pattern do you see?

Answers:

The third diagonal in Pascal's triangle is 1, 3, 6, 10, 15, 21,…, which are the triangular numbers (see "Triangle Tribulations"). The second number is the first number plus 2. The third number is the second number plus 3. The fourth number is the third number plus 4, and so on.

Shading multiples of 5 in Pascal's triangle produces a pattern of triangles in which each shaded triangle is composed of ten numbers.

Pascal's triangle with multiples of 5 shaded in.

Shading multiples of 2 produces triangles of various sizes, which form the special pattern known as the Sierpinski triangle.

Pascal's triangle with multiples of 2 shaded in.

Multiples of 3, 7, and many other numbers also produce triangle patterns. In each case the triangles are "upside-down," meaning they point the opposite way from the original Pascal's triangle.

Shading the squares that are not multiples of a given number often produces a triangle pattern, with the triangles "right-side up," or pointing the same way as the original Pascal's triangle.

NEXT: Pascal's Fractals