By Cat. #3610001, the Alternating
Row Sum Identity, we know that any pattern that goes +-+-+-... all the way across a row adds up to zero; the
0 row has only one entry, so it can't alternate and its sum is 1. Let's
consider some row where the signs alternate and put a break in it, splitting
the alternating sum into two alternating sums this and that.
{+-+-....+}+{-+-+...}
= 0 this
+ that = 0
-or- this = -that
Since that starts with a negative sign and
alternates, -that would start with a positive sign and alternate;
this proves that if the Hockey Stick Theorem works for a stick whose handle
begins on the left edge of Pascal's Triangle, it will also work for sticks
that start on the right edge.
Like with the Christams Stocking Theorem, the Hockey
Stick theorem is a proof by induction based on the length of the handle,
where we split the stick into three parts: the handle, the heel and the
toe as follows
{+} <- toe {...handle-+-+-}{+}
<- heel Note that the toe and heel always have a positive
value, while the number in the handle nearest the heel has a negative value. Step 1: Handle length = 0. This means both the heel
and the toe are on the edge of the Triangle, so that says the equation
would be 1=1, which is obviously true; if this doesn't convince you we have
the right to continue, check a few steps more by hand. (We did the
same thing with the proof for the Christmas Stocking, since the first case
seemed kind of, well... duh!)
Step 2: Assume the stick works for handle length = n;
see if this information is enough to prove it also works for handle length
= n+1. Here the old stick will be in red, while the new heel and toe will be in blue.
{+}{+} {...handle-+-+-}{+}{+}
To make the new blue handle, we need to negate the
red handle and heel; since we get to assume that the Hockey Stick Theorem works
for this length of stick, -(handle+
heel) = -toe; but since we now have two consecutive
values in a row and value from the row below that stands between
them, we can turn toe + toe = heel into the
equation toe = heel
- toe, which will suffice to prove the statement
for the next length of hockey stick.
