Base+Group+Activitie

=Base Group Activities= Author: Tom Hayden (thayden@gmail.com) Date: 5/5/2011

These are small (10-15 minute) brain teasers or activities that the students worked on in groups of 4-5. These were given to both my students in the accelerated and the lower-level classes, so the lessons should be relatively independent of student ability. Some of these were done in the classroom and a few were just done with small student groups over the summer.

Rectangle Free Coloring (aka "G17")
Math/Computational Idea: Computational Complexity, NP-hardness Below is a link to a game where the idea is to find a coloring of the grid such that there are no single color rectangles. The below link only goes up to 12x12 but the real unsolved problem is 17x17. If you can find a coloring for the 17x17 grid then you'll win $289 (17x17). [|Rectangle Free Coloring HTML5 Game]

Here is a link to the original post on the computational complexity blog about the problem: [|17x17 Challenge (Bill Gasarch & Lance Fortnow's Blog)]

Pi=4 Proof
Math/Computational Idea: Iterative methods, proofs and fallacies Below is a link to a "troll proof" that made the rounds on the internet in the winter of 2010. The claim is that there is an iterative method that can be used to show that the value of pi is actually 4 rather than the 3.141... that we commonly associate with the constant. My challenge to the students: prove to me what is wrong with the proof.

[|Troll Pi=4 Proof]

The correct answers vary: there are actually quite a few things wrong with the proof. The most obvious, to me, was that when you create the sharp squared edges, the points on the corners have no derivative. Then, in the process of iteratively making the circle out of those edges, there will be no points on the circle which have a derivative. This violates the definition of a circle. There are also other counterexamples too; some people claim the process actually converges on a triangle, not a circle (which I cannot prove or disprove at the moment)

Rope Burning Puzzler
Math/Computational Ideas: Non-linearity, proofs Here is the puzzler, the blog linked basically covers everything that I would say: [|Rope Burning Logic Puzzler]