Expert: Scott A Wilson - 9/20/2011

Question
question is in the image

Answer
If A and B are two dimenional positve integer vectors, then 52 = 6^2 + 4^2.
This means A is (4,6) or (6,4).  Call them A1 and A2.  As far as 181, it can be seen that
181 = 9^2 + 10^2.  I'll refer to A+B as C.  This means the C is (9,10) or (10,9).  Call them C1 and C2.

Using this, there are 4 choics for B since B can be C1-A1, C2-A1, C1-A2, or C2-A2,
which is 4 choices.  If the angle of B with the X-axis is called D, then it is known
that the tan(D) = Cy/Cx where C = (Cx,Cy).  The choices for B are
(5,4), (3,6), (6,3), or (4,5).  The angles are arctan(5/4), arctan(3/6), arctan(6/3),
and arctan(4/5).  That is the same as arctan(1.25), arctan(0.5), arctan(2), or arctan(0.8).
As written, these are the four most likely choices for B.

However, to make matters worse, A and C could have positive or negative values,
like A could be (4,-6), (-4,6), or (-4,-6).

In case this wasn't enough choices, let root(x) be short for the squareroot(x).
In this case, we could have A as (1,root(50)), (root(2),root(49)), (root(3),root(48)), etc.
Throw in the fact that either Ax or Ay could be positive or negative, and suddenly we have even more choices.  Not only that, we could use fraction, as in A could be (root(5.7),root(45.3)).

Now if that wasn't bad enough, A and B could be 3-dimensional.
For example, C might be (1,6,12), for 1^2 + 6^2 + 12^2 = 181.
If C was this way, A could be (3,3,root(34)) or (4,4,root(20)).