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Pascal’s Triangle Fundamentals

Structure: Each term is the sum of the two terms directly above it. Rows start and end with 1.

Term Notation: tn,r denotes the term in row n (starting from 0) at position r.

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Patterns in Pascal's Triangle

1. Pascal's method

t(n+1,r+1)=t(n,r)+t(n,r+1)

2. Row sums

Row sum=2^n (n: the sum of nth row)

3. Divisibility

Find the divisibility of two terms within the Pascal's Triangle

Concepts: Odd-numbered rows (odd numbers: 1,3,5,7,...)

Even-numbered rows (even numbers: 2,4,6,8,...)

All rows that have x or more terms: Exclude 0~(x-2) rows

4. Triangular numbers

Finding the number of dots in a triangle (equilateral) with nth rows: t(n+1,2)

5. Perfect Squares

n^2=t(n,2)+t(n+1,2)

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Research question: Pathway problems

Conditions:

1. Starting and ending points are defined

2. Only can move into 2 direction in one step

Specific Applications

A: Path to spell a world: easy application of Pascal Triangle, the sum of value in the last row is the number of paths

B: Checkerboard

Features: Limited area

There may be only one path to a grid. If so, only the data of the previous layer needs to be retained without adding them.

*Special circumstances: Blocked grid: Just skip them, there is only one path leading to the diagonally adjacent grids on the next layer.

C: Paths in two locations

“Rhombus” Pascal's Triangle: Narrow-Wide-Narrow——The push-down principle remains unchanged

*Special circumstances: Impassable Road

Treat them as edges (see example)