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5.Q. NUMBER OF REGIONS DETERMINED BY N LINES OR PLANES
Mittenzwey. 1880. Prob. 200, pp. 37 & 89; 1895?: 225, pp. 41 & 91; 1917: 225, pp. 38 & 88. Family of 4 adults and 4 children. With three cuts, divide a cake so the adults and the children get equal pieces. He makes two perpendicular diametrical cuts and then a circular cut around the middle. He seems to mean the adults get equal pieces and the children get equal pieces, not necessarily the same. But if the circular cut is at 2/2 of the radius, then the areas are all equal. Not clear where this should go -- also entered in 5.T.

Jakob Steiner. Einige Gesetze über die Theilung der Ebene und des Raumes. (J. reine u. angew. Math. 1 (1826) 349 364) = Gesam. Werke, 1881, vol. 1, pp. 77 94. Says the plane problem has been raised before, even in a Pestalozzi school book, but believes he is first to consider 3 space. Considers division by lines and circles (planes and spheres) and allows parallel families, but no three coincident.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. "3. A man marks 6 straight lines on a field in such a way as to enclose 10 spaces. How does he manage this?" Solution begins: "III. To inclose ten spaces by six ropes fastened to nine pegs." Take (0,0), (1,0), ..., (n,0), (0,n), ..., (0,1), as 2n+1 points, using n+2 ropes from (0,0) to (n,0) and to (0,n) and from (i,0) to (0,n+1-i) to enclose n(n+1)/2 areas.

Richard A. Proctor. Our puzzles. Knowledge 10 (Nov 1886) 9 & (Dec 1886) 39-40. Describes several ways of solving previous problem and asks for a symmetric version.

G. Chrystal. Algebra -- An Elementary Text-Book. Vol. 2, A. & C. Black, Edinburgh, 1889. [Note -- the 1889 version of vol. 1 is a 2nd ed.] Chap. 23, Exercises IV, p. 34. Several similar problems and the following.

No. 7 -- find number of interior and of exterior intersections of the diagonals of a convex n-gon.

No. 8 -- n points in general position in space, draw planes through every three and find number of lines and of points of intersection.

L. Schläfli. Theorie der vielfachen Kontinuität. Neue Denkschriften der allgemeinen schweizerischen Gesellschaft für die Naturwissenschaften 38:IV, Zürich, 1901, 239 pp. = Ges. Math. Abh., Birkhäuser, Basel, 1950 1956, vol. 1, pp. 167 392. (Pp. 388 392 are a Nachwort by J. J. Burckhardt.) Material of interest is Art. 16: Über die Zahl der Teile, ..., pp. 209 212. Obtains formula for k hyperplanes in n space.

Loyd, Dudeney, Pearson & Loyd Jr. give various puzzles based on this topic.

Howard D. Grossman. Plane- and space-dissection. SM 11 (1945) 189-190. Notes Schläfli's result and observes that the number of regions determined by k+1 hyperspheres in n space is twice the number of regions determined by k hyperplanes and gives a two to one correspondence for the case n = 2.

Leo Moser, solver. MM 26 (Mar 1953) 226. ??NYS. Given in: Charles W. Trigg; Mathematical Quickies; (McGraw Hill, NY, 1967); corrected ed., Dover, 1985. Quickie 32: Triangles in a circle, pp. 11 & 90 91. N points on a circle with all diagonals drawn. Assume no three diagonals are concurrent. How many triangles are formed whose vertices are internal intersections?

Timothy Murphy. The dissection of a circle by chords. MG 56 (No. 396) (May 1972) 113 115 + Correction (No. 397) (Oct 1972) 235 236. N points on a circle, in a plane or on a sphere; or N lines in a plane or on a sphere, all simply done, using Euler's formula.

Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Slicing cakes, pp. 33 & 61. Cut a circular cake into 12 equal pieces with 4 cuts. [From this, we see that N full cuts can yield either 2N or 4(N-1) equal pieces. Further, if we make k circular cuts producing k+1 regions of equal area and then make N-k diametric cuts equally spaced, we get 2(k+1)(N-k) pieces of the same size.]

Looking at this problem, I see that one can obtain any number of pieces from N+1 up through the maximum.