__Ulrich Berger’s argument rephrased__

We introduce a special type of line integral, the “contour integral”, in which the differential `dc` is an infinitesimal vector along the tangent. [With `s` the length along the contour, we have

`dc` = (cos.`φ` `ds`, sin.`φ` `ds)`.]

We only consider clockwise integrals along *closed* contours; hence the law ∮`dc` = (0,0).

We now consider for a vector field `f` and some contour the contour integral ∮`f`.(`x`,`y`)•`dc`, where • denotes the scalar product. We don’t need to consider “bagels” with a hole in them

for the above two figures have equal contour integrals because the two contributions along the cut cancel.

Similarly, when we consider —like a jigsaw puzzle— a figure partitioned into a finite number of pieces, then the contour integral of the whole equals the sum of the contour integrals of the pieces. (We refer to this as “the Law”.)

__Note__ For `f`.(`x`,`y`) = (`y`,0) or `f`.(`x`,`y`) = (0,`–x`), the contour integral equals the area enclosed by the contour, and indeed: the area of the whole jigsaw puzzle equals the sum of the areas of the pieces. (End of Note.)

^{*} _{*} ^{*}

We only consider rectangles with horizontal and vertical sides. A rectangle is called *nice* when in at least one direction it has sides of integer length. The theorem to be proved is that in the case of a large rectangle `R` partitioned in a finite number of little rectangles `r`, rectangle `R` is nice if all rectangles `r` are nice.

We demonstrate that, with all rectangles `r` nice, `R` has an integer width `w` when its height `h` is noninteger. We do this by introducing an `f` such that

(i) | the contour integral of R equals w |

(ii) | the contour integral of each r is integer. |

The Law then allows us to conclude that `w`, a sum of integers, is integer.

When we place `R` with its bottom at `y=0` (and, hence, its top at `y=h`), some reflection shows that this is, for instance, realized by

`f`.(`x`,`y`) = (**if** `y` is integer → 0 ▯ `y` is not integer → 1 **fi**, 0)

ad (i). Only the top side, of length `w`, contributes to the contour integral of `R`, 0 being integer and `h` not.

ad (ii). For a rectangle whose vertical sides are of integer length, the contributions along its horizontal sides (zero or not) cancel, and for a rectangle `r` whose horizontal sides are of integer length, the contributions along its horizontal sides (zero or not) are integer; in either case, nice rectangle `r` has a contour integral of integer value.

Nuenen, 8 September 1999

prof.dr. Edsger W.Dijkstra

Department of Computer Sciences

The University of Texas at Austin

Austin, TX 78712–1188

USA