We had our math circle meeting Oct.30 at the regular time and place (despite Hurricane Sandy and closed city schools).
Our topic was some extra exploration of some of our recent projects. Please feel free to add ideas to the comments below.
From MC 1 we looked at a triangle (3 dots and 3 lines) with 4 dots inside. When we had filled in all the possible lines, we had 4 dots, and 9 triangles. After a few false starts we found the Tee formula: (3 T +3)/2=L relating the number T of triangles and the number L of lines. Then we noticed that 4 dots and 9 triangles came with 15 lines. Charles suggested the BeeGee formula: L-D+1=S relating the number D of dots and the number L of lines, and the number S of shapes (they didn’t have to be triangles!). What happens if we try to combine these formulas (in the cases when all the shapes are triangles)? Why are these formulas true?
From MC 2 we tried to find some rules for constructing 4 by 4 sumptuous squares (and other even by even sumptuous squares). We tried to recreate our 1514 example from before, but it is not so easy to do by trial and error, even using the fact that we remembered it could be done so that “opposite numbers add to 17″… But we got it!!!
[latexpage]
$$left[begin{array}{c c c c}
* & 3 & 2 & * \
* & * & * & * \
* & * & * & * \
* & 15 & 14 & *
end{array} right]$$