|
Division Claim: On the top and bottom of the rectangle the distance to the corner ahead is of the form {$mb-na-x$}, where {$n$} and {$m$} are natural numbers. On the sides of the rectangle, the distance to the corner ahead is of the form {$na+x-mb$}. Furthermore, with each step the coefficients do not decrease and with each diagonal step one coefficient increases and the other stays the same. Proof: This is certainly true at the start.  We only have to look at what happens in all cases. But we'll just look at two. First, we've been moving along the wall and we're on the bottom (of the rectangle).  Then we do a diagonal.  Then we go along the side.  At each stage, the Claim holds true. Here's another case, which could happen if {$b$} is much larger than {$a$}:  Now the diagonal hits the opposite side.  And the Claim continues to hold. |