EWD 1288

Ulrich Berger’s argument rephrased

EWD1288 figure 1

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

EWD1288 figure 2

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”.)

EWD1288 figure 3

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.

EWD1288 figure 4

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