Are you ready to challenge your mind? If you enjoy analyzing complex patterns and you want to be among others who share your enthusiasm—spend a week at Math Camp!

Click here for online registration form.

Analyzing games to find winning strategies, discovering and understanding patterns associated with Pascal’s triangle and competing in team problem-solving contests will help build confidence in your problem- solving skills!

By spending time with Houghton faculty and students at Math Camp, you can experience math as a living, growing area of discovery.

The camp is open to all high school students who have completed a year of high school algebra.

Math Camp is directed by Kristin Camenga, a Houghton mathematics faculty member. http://www.houghton.edu/academics/programs/math-computer-science/kristin_camenga.htm  Two Houghton seniors, Laura Stutzman and Rob Zima will also be working with the camp in class sessions and problem settings.  If you have any questions about Math Camp, please contact camp administrator Ginny Jacobsen at 585-567-9649 or excellencecamps@houghton.edu or Kristin Camenga (kristin.camenga@houghton.edu).

When we are working on math (and not out having fun with people from different camps!), we will focus on three different areas: analyzing games, patterns associated with Pascal’s triangle, and team problem-solving competitions.  Below are some examples of each of these three areas.

Games:
We will spend some of our time investigating and analyzing a few mathematical games to learn about them and try to discover winning strategies.  One of the games we will consider is called Sprouts, which is explained below. 

Start with a few dots (or vertices). (For your first few games, start with 2-4 vertices. ) Players take turns connecting two vertices with a curve (or edge) and placing a new vertex along this edge. This is done following two rules:

1. Each vertex can have at most three edges coming from it
2. Edges must be drawn so that they do not cross or touch any other vertices than the two they are connecting

If a player is not able to draw an edge according to the rules, the other player wins.
NOTE: you can draw an edge connecting a vertex to itself, such as:

A game might start like:

Since we are at math camp, we won’t stop at just playing the game; we want to find out more about it!  We will consider questions like the following:

·       Does the game always end? (At each stage we add a vertex, so couldn’t we just keep playing forever?)

·       What can the ending look like?

·       Does one player have a winning strategy – a way of playing so that they can guarantee that they will win?

Patterns in Pascal’s Triangle:

(image from http://www.roma.unisa.edu.au/07305/pascal.htm)

The Chinese may have first used what we now call Pascal’s triangle as early as 1261.  The picture above is an illustration from a Chinese text.  We will consider patterns in this triangle and a variety of applications.  Some of these applications are related to the following problems: 

·         How many different paths from the M in the top left corner to the N in the bottom right corner can you take to spell out “MATH CAMP IS FUN”?  (Each path must go from one letter to a letter that is either below or to the right at each step.)

·         If you flip a coin 20 times, what fraction of the time do you get exactly 13 heads?  13 or more heads?

·         What polynomial do you get if you expand (x + 3)⁸ ?

 Problem-Solving:

Every day we will have a team problem-solving competition.  Most of these problems will not be based on the other two areas or on any specific math course, but are intended to make you think!  Some of the problems might be like the following:

• You are given a 2 x 10 board and dominoes, each of which would cover 2 adjacent squares on the board.  How many different ways can you lay dominoes on the board so you can cover every square?
• What is the largest number of zeros that can occur at the end of 1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ for any positive integer n?