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arithmetic progression

See two attachments.

Each point on the plane will be counted sooner or later, starting from
the origin and following the "path" suggested in the figure. Note
that the counting starts from 0 not 1 (which is not a big deal).

For example, the origin, (0,0), will be counted first. When you plug
(0,0) into the formula, you will get 0, which is correct b/c (0,0) is
the first pointed that is counted (note again, the counting starts
from 0 not 1). Look at the point (1,2) in the figure, when you plug
(1,2) into the formula, you will get 7; when you actually count toward
(1,2), starting from (0,0), in the figure, you can see that (1,2) will
be reached at 7th, which shows that the formula gives the correct result.
The problem is to show that for a general point (i,j), the
order when that point is counted is m (the formula.)

(m is represented in a fraction form, but this fraction will always
evaluate to an integer....)

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Solution Preview

The explanations are in the attached file.

Prove/disprove that the method of counting N x N indicated in the figure (see attachment, figure 1) visits the pair (i,j) mth, where


Note the particular details that the counting/numbering starts from 0 and that the assignment of the axis is different from ...

Solution Summary

This solution thoroughly demonstrates arithmetic progression.