1. Partition.Cutting up a circular entity. Exponent
notation and physical meaning. See Pizza*. Also note:
"It has been observed that for a circle, the maximum number of
pieces that can be produced by n cuts is {[(n2+n)/2]+1}, but
for a crescent, the number increases to {[(n2+3n)/2]+1}."**
For n = 1, 2, 3 the values of {[n(n+1)/2]+1} are 2, 4, 7. It is easy to
see the first two; the value 7 comes when the 3rd cut intersects three
sectors (e.g., 3 of the 4 areas that reults from quartering a circle).
For n = 1, 2, 3 the values of {[n(n+3)/2]+1} are 3, 6, 10. It is easy to
see the first - a crescent is divided 3 by a single cut. Only 5 areas result
from 2 cuts unless the parts are moved. Either within the plane, or
out of it - as in moving them on top of one another - yields 6.
However, if that stacking process is valid, it would be possible to get
12 parts from 3 cuts, not 10.
The table summarizes the known situation for the first few n when no
motion of parts is allowed. However, it seems clear that only relocating
the 3 pieces that result from a single line intersecting a crescent
enables the n = 2 result of 6.
2. Games Is Laskers determined (player
with first move can always win)? Mnatsakanian, Mamikon, "Corner the King" 3x4
Chessboard,
CV,.......... Caltech Site,......... MamiChess
3. Approximation. The base of natural logarithms is approximated by many finite numbers
as in about
e
4. Divisibility of a ratio: result a whole number? When is the ratio of the nth power of 2, reduced by
3, to n, a whole number? Four Values Now and Mod n tell about the movement forward from the
single value 4 billion 7 hundred million and change known when I was
introduced to this problem.
5. Integer valued. Davis, Philip J., "Are There Coincidences in
Mathematics?" American Mathematical Monthly 88 (May 1981):
311-20. Nahin, 1998, pp. 250 and 235-237 discuss (-1)^√(-163) =
262,537,412,640,768,743.99999999999925007 and [√(-1)]^[√(-1)]=
i^i = 0.20787 95763 50761 90854 69556 19834 97877 00338 77841 63176
96080 75135 88305 54198 77285 48213 97886 00277 86542 60454 40521 77330
72350 21808 19061 97303 74663. The former involves - 1 = e^( ± iπ ) and
√-163 =±i√163.
6. Geometry. Smith, David E. and Latham, Marcia L., The
Geometry of Rene Descartes, Dover 1954 is "the third illustrative
appendix to Descartes" Discourse on the Method of Reasoning and
Seeking Truth in Science" ... called "the greatest single step ever
made in the progress of the exact sciences" by John Stuart Mill ... it
is a masterpiece. (Nahin, 1998, p. 241).
7. Discovery. Leibniz & Huygens were astounded to learn that:
√(1+ √(-3)) +
√(1- √(-3)) = √6. See McClenon, R.B. "A Contribution
of Leibniz to the History of Complex Numbers," American Mathematical
Monthly 30 (November 1930):369-74, as noted by Nahin, 1998 pp. 241 &
25.
References
*Drawing by Gavin Wu based on 3 solutions (A. Klinger).
**Gardner, Martin, Mathematical Puzzles of Sam Loyd, NY: Dover,
1959, p. 158.