For information on using sunlight and shadows to tell time, check out this page about sundials.
A few weeks ago, a math teacher friend of mine showed me another pattern he found in a k -cubes in n -cubes chart like the one at the bottom of page 76: Each number in the table is equal to the sum of all the numbers to its left in its row and in the previous row. (For the top row to work, it is necessary to imagine a 1 in a row above the top row.) Can you figure out why this is true?
As I mentioned in the back of B3D, my sundial consultant was Prof. Bill Hausdoerffer, who gave the first college math lecture I ever heard some forty years ago. I don't know whether or not he is on line, but I'll try to pass the location on to him and see what he has to say.
Thanks too for bringing up that alternate way of looking at the k-cubes in an n-cube chart. There are all sorts of interesting patterns that show up there, and although Pascal's triangle has been explored to death, I think that there is still more to be seen in the cubes chart that isn't just another Pascal-type observation. I think that the particular statement from your math teacher friend can be established by mathematical induction, but that would not in itself show "why" it is true, and that is the kind of question that is very appropriate to ask in this course. Perhaps we can get some discussion going on this.