You are put in charge of testing the durability of a new cellphone. You have two cellphones and a 100 story building. You want to know at which floor the cellphone will break. If the cellphone breaks when dropped from a a certain floor it will also break at any floor higher. If a cellphone survives a drop it will survive if dropped from a lower floor. How can you minimize the amount of times you have to drop the cellphone?
Can you come up with a scheme for dropping the cellphones so you don't have to stop at every floor? How would you skip floors? We'll be revealing the answer in a later post, so stay tuned. Hint: It involves magical triangular numbers!
Speaking of triangles and their mystical properties, in our geometry class we looked at sets of three natural numbers that add up to 18. Then we dusted off our compasses and got to work constructing triangles with edges of the lengths of these triples. A strange property emerged before our mathematicians own eyes!
Triangle architecture aside, we ventured to a town with the problem of not knowing which bridges to construct. I'm talking of course about the famous Seven Bridges of Königsberg Problem. Our mathematicians travelled the same path as Euler in the early 18th century down the bridges of Konisberg to see if they could construct a route through the city crossing each bridge once and only once. The thinkers dipped their toes into the world of graph theory.
Just like the bridges in a city in Prussia, what else can we use graph theory to represent? How about the paths that mathematicians could take to get to the Reem-Kayden Center each morning for C.A.M.P. Let's say you were to park in the Reem-Kayden Center Parking lot and eventually make your way to the Reem-Kayden Center Building, but you want to cross through every path on central campus once and only once. Can you do this?
Does the parity of the nodes and edges matter? Which areas of campus can you traverse using this method?
In computer science we dabbled in a different sort of pairwise relationship -- the relationship between the command you would give a computer program and the output. Mathematicians coded for shapes to see that making an ellipse isn't so different from making a circle once they figured out the geometric difference between them.
Can we use those same geometric properties to make artwork? In our last module of the day, the thinkers took our their trusty compasses again to make tessellation's and fractals out of a variety of polygons and circles. In doing so they used properties like equidistance and symmetry.
Walking around the class, I happened to speak to a mathematician whose art looked like this:
Which poses an interesting question! If each circle in this drawing contains a set of triples that adds up to 18, can we fill it in without writing numbers twice? How would the triples from Geometry look in this artwork? Is there any aesthetically appealing symmetry?
That's all for now, keep posted for more updates throughout a week of mathematical explorations.