Mathematical Questions

Allen Klinger, © 12/20/2016

0. What started math? Many things: 1) Stretching ropes; 2) Making altars; 3) Creating calendars; 4) Measuring property.

Math changes: new questions are asked and answered.

Facts that are discovered might be expressed by an if, then pair: if a 1st thing then a 2nd thing. This puzzle takes one into reasoning about relating a 1st and 2nd thing.

Arthur, Betty, Carmen, and David are thinking about an event. If Arthur goes, Betty will go. If Betty goes, Carmen will go. If Carmen goes, David will go. Two of these four went. What are their names?

4 People May Attend

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}."** However Gardner should have added "restricting the cuts to the circle or crescent in its original or intact position."

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.
*Drawing by Gavin Wu based on 3 solutions (A. Klinger).

**Gardner, Martin, Mathematical Puzzles of Sam Loyd, NY: Dover, 1959, p. 158.

Nahin, Paul An Imaginary Tale: The Story of √ -1

12/20/2016 Version
©2009 Allen Klinger