## Math Research Project — Grid Walking

Each Math Research Project outline consists of a Seed QuestionQuestions to Build OnExtensions, and Basic Background.  Click here to see a list of available Math Research Project outlines.

The Seed Question is a question that appears in a typical math curriculum.  Questions to Build On are simple extensions, re-interpretations, and generalizations of the Seed Question that a student can build a simple math research paper around, with help from a mentor (be it student or teacher).

The Extension Questions are bigger, more challenging extensions of the Seed Question.  If a student can make some headway into understanding these, great!  If not, they are good questions to include as part of the “Where do we go from here?” section of the paper.

The Basic Background consists of some of the mathematical leg-work that underlies the investigation.  It can be worked on simultaneously with the investigation, which gives it context.

### Grid Walking

How many distinct paths are there from (0,0) to (5,3), if movement is only allowed “North” and “East” along the lines y = 0, y = 1, y = 2, and y = 3, and x = 0, x = 1, x = 2, …, x =5?

Questions to Build On

How many distinct paths are there from (0,0) to (5,4)?

How many distinct paths are there from (0,0) to (6,3)?

How many distinct paths are there from (0,0) to (m,n)?

How many distinct paths are there from (0,0) to (5,4) that pass through (2,2)?

Extension Questions

How many distinct paths are there from (0,0,0) to (5,4,3)?

How many distinct paths are there from (0,0) to (5,4) if in addition to moving “North” or “East”, you are allowed at most one move “West”?

How many distinct paths are there from (0,0) to (5,4) if in addition to moving “North” or “East”, you are allowed to move along diagonal line segments?

How many distinct paths are there from (0,0) to (5,5) if you are not allowed to cross the line y = x?

Basic Background

Derive the formulas for combinations.

Explore the basic features of recursion and some recursive sequences.

Demonstrate some relationships between combinations and Pascal’s triangle.

Click here to see a list of available Math Research Project outlines.