# B3D Chapter 4

## Lisa Eckstein

**Further Reading**
For information on using sunlight and shadows to tell time, check out this page about
sundials.

**Question**

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.*

*moocow@brown.edu*