Some R & R (Research and Recreation)

Rick Mabry,
Crosscut Convex Quadrilaterals,
Math. Mag. 84, no. 1 (Feb. 2011), 16–25.
 Erratum: Terrible typo in Figure 5—at point A_{1} replace (1t)u with (1t)v.
 Update: Peter Ash wrote to inform me of a similar article,
Constructing a Quadrilateral Inside Another One, by J. Marshall Ash, Michael Ash, and Peter Ash, in the Mathematical Gazette 93, no. 528 (2009), 522–532. The article by the Ash's contains some of the same main results in mine, though we then go in different directions. The authors point out that there are even more independent discoveries of these results, all fairly recent.
 Richard D. Mabry, Stretched shadings and a Banach measure that is not scaleinvariant,
Fund. Math. 209 (2010), no. 2, 95–113 [Copyright IM PAN (Institute of Mathematics  Polish Academy of Sciences)]
 Steven D. Smith (student) and Rick Mabry, "The determinant of a Kronecker Product" (Solution to Problem 1833), Math. Mag. 83, no. 5 (Dec. 2010), 394–395.
 A little "Quickie" of mine (Q1003) is printed in the October 2010 issue of Mathematics Magazine (VOL. 83, NO. 4). Tom LaFaro of Gustavus Adolphus College wrote me with another solution, one that I prefer very much to my own, using the Leibniz rule for differentiation products. Thanks, Tom!
 Rick Mabry, Solution to "Congruent cakes" (Puzzle Corner 17), Gaz. Aust. Math. Soc., 37, no. 2 (May 2010), p. 72.

Rick Mabry,
The Hardest Straightin Pool Shot,
College Math. J., 41, no. 1 (Jan. 2010), 49–57.
 Rick Mabry, "An infinite sum of a function with its Taylor polynomial" (Solution to Problem 893, with extension), College Math. J., 41, no. 1 (Jan. 2010), 67–69.

Rick Mabry and Laura McCormick (student),
Square products of punctured sequences of factorials,
Gaz. Aust. Math. Soc., 36, no. 5 (Nov. 2009), 346–352.

Rick Mabry,
A Result of Strauss,
Amer. Math. Monthly, 115, no. 8 (Oct 2008), p. 756.

Rick Mabry and Paul Deiermann,
Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results,
Amer. Math. Monthly, 116, no. 5 (May 2009), 423–438.

Rick Mabry,
Problem 1772,
Math. Mag., 80, no.3 (June 2007), p.230.

The solution appeared here: Taylor minima,
Math. Mag., 81, no.3 (June 2008), 222.

The original proposal and solution are here, along with pretty pictures.

Rick Mabry and Debbie Shepherd,
Problem 11327,
Am. Math. Monthly, 114, no.10 (Dec 2007), p.925.

[Note: "Game of chance"? "Connect the dots"? I barely recognize the problem. Our original proposal is HERE.]
 The solution appeared here: A Chancy Function,
Amer. Math. Monthly 116, no.7 (Aug.–Sep. 2009), 655–656.
 Our solution to our proposed problem is HERE.

Richard D. Mabry,
"No nontrivial Hamel basis is closed under multiplication",
Aequationes Mathematicae, 71 (2006), no. 3, 294–299.

Edwin Clark (Prof. Emeritus, USF in Tampa) emailed me (on May 8, 2009) a much shorter and sweeter proof of the result in my paper, which extends it quite a bit. With his permission, I have posted it here as a text (.txt) file.

Also in 2009 a paper ("Algebras having bases consisting entirely of units") appeared (PDF available here) that similarly "augmented" the result, using similar idea(l)s. The result comes from Jeremy Moore's very nice PhD dissertation, here.

Rick Mabry,
Problem 11001,
Am. Math. Monthly, 110, no.6 (June–July 2003), p.543.

[Note: The editors originally changed this problem completely due to "typesetting" errors, and the botched version appeared in the March issue (no. 3), p. 240, as seen HERE. That rendered the problem trivial (even silly) but the corrected version in the June–July issue is okay. The solution appears as "Powered Terms in a Series: 11001," Am. Math. Monthly, 111, no.10 (Dec 2004), p.920. JSTOR.]

Rick Mabry,
Problem 10990,
Am. Math. Monthly, 110, no.1 (2003), p.59 JSTOR.

Note: The problem deals with envelopes of Bernstein polynomials. Funny story: The editors changed this problem from its original form (a change to which I acquiesced after a short, feeble struggle). The original problem is here and is much more subtle, in my humble opinion.
 The solution appeared here: "The Envelope, Please: 10990,"
Am. Math. Monthly, 111, no.9 (2004), 825–826 JSTOR. The editors seem to have paid lipservice to my intention, illustrated by their own — unfortunately — incorrect graphic.
 My own solution is here.

Paul Deiermann and Richard Mabry,
"Asymptotic Symmetry of Polynomials,"
Mathematics Magazine, 75, no. 2 (April 2002), 131–135.

Rick Mabry,
"Mathematics Without Words,"
College Journal of Mathematics,
32, no. 1 (2001), p.18. (This is reprinted on p.65 of "Mathematical Delights" by Ross Hernsberger, MAA, 2005. See HERE for more of the same.)

Michael Avidon, Rick Mabry and Paul Sisson,
Enumerating Row Arrangements of Three Species,
Mathematics Magazine, 74, no. 2 (April 2001), 131–135.

Paul Deiermann and Rick Mabry,
Problem 676,
College Math. J., 31, no.3 (2000), p.219.
JSTOR

Paul Deiermann and Rick Mabry,
"Intersecting curves" (Problem 10712),
Am. Math. Monthly, 106, no.2 (Feb. 1999), p.166
(posed)

Rick Mabry,
"Going for the stars" (FFF #151),
College Math. J.,
30, no. 5 (1999), p.383.

Rick Mabry,
Shades of the Cauchy functional equation and Segre functions,
a 25minute talk at the
37th International Symposium on Functional Equations
, at
Marshall University
on May 18, 1999, during
Functional Equations Week in Huntington, West Virginia.

Rick Mabry,
The measure theoretic approach to the density of quasiarithmetic sequences,
a 20minute talk at the
1999 Seaway Number Theory Conference, March 27–28, 1999, at
Pennsylvania State University, State College, PA.

Richard D. Mabry,
Some remarks concerning the uniformly gray sets
of G. Jacopini,
Rend. Acc. Naz. Sci. XL, Mem. Mat., vol. XXII (1998), fasc. 1, pagg. 43–49
[ISSN: 03924106].

Rick Mabry,
"Thirds of a Triangle" (my suggested title, not used),
Mathematics Magazine, 72, no. 1 (Feb. 1999), p. 63. This is reprinted in various places (listed below) and will probably have to end up on (or as) my tombstone.
 Page 46 of Archimedes: What Did He Do Besides Cry Eureka? by Sherman K. Stein, MAA, 1999.
 The cover of and page 111 of Proofs Without Words II: More Exercises in Visual Thinking by Roger B. Nelsen, MAA 2000.
 CRC Standard Mathematical Tables and Formulae, 30th ed., Daniel Zwillinger (Editorinchief), CRC Press, 2002.
 Page 32, The role of visual representations in the learning of mathematics, Abraham Arcavi (Weizmann Institute of Science, Israel), Proceedings of the XXI Conference on the Psychology of Mathematics Education, North American Chapter, Mexico, 1999, 26–41.
 Page 199 of "Real Infinite Series" by Daniel D. Bonar, Micheal J. Khoury, Jr., and Michael J. Khoury, MAA, 2006.
 Pages 73–74 of Math Made Visual by Alaudi Alsina and Roger B. Nelsen, MAA, 2006.
 Visual Computation of Three Geometric Sums from The Wolfram Demonstrations Project, contributed by: Soledad Mª Sáez Martínez and Félix Martínez de la Rosa.
 A Master's thesis by Raquel Barrera at UNIVERSITÉ DENIS DIDEROT – PARIS 7 (Dec. 2012), "MULTIPLICATION DE FRACTIONS AU COLLÈGE : « Le rôle d'un processus de visualisation complémentaire du registre numérique »
 This Wikipedia entry.

Rick Mabry,
"A Subset of the Plane Having Constant Linear Shades,"
a talk at the
Twentysecond Summer Symposium on Real Analysis, UCSB,
Santa Barbara, CA, June 23–27, 1998. [See also the
Conference Report
in the Real Analysis Exchange, Vol. 24(1), 1998/99, pp. 35–38.]

Rick Mabry and Paul Deiermann,
"Adding Like Sines",
Mathematics Magazine, 71, no. 2 (April 1998),
p.130. Reprinted on p. 54 of Proofs Without Words II: More Exercises in Visual Thinking by Roger B. Nelsen, MAA 2000.

Rick Mabry,
"Shades of the Cauchy functional equation,"
a talk at the
Special Session on Real Analysis, 931st Meeting of the
AMS, Louisville, KY, March 21, 1998.

Rick Mabry,
"Shading the plane,"
a talk at the 75th Annual LouisianaMississippi Section Meeting of
the MAA,
New Orleans, LA, Feb 26, 1998.

Rick Mabry,
Stan Wagon, and
Doris Schattschneider,
Automating Escher's Combinatorial Patterns",
Mathematica in Education and Research,
5, no. 4 (1996), 38–52.

Paul Deiermann and Rick Mabry,
Problem 1457, "The center of a sliced pizza",
Mathematics Magazine, 68, no. 4 (1995), 312–315.

Rick Mabry,
"Bipartite Graphs and the Four Color Theorem",
Bulletin of the Institute for Combinatorics and its
Applications, 14 (1995), 119–122. (See also Frank Harary's rejoinder.)

Paul Deiermann and Rick Mabry,
"Throwing Another Fallacy out the Window (Using Minimum Energy)" (FFF
#81),
College Journal of Mathematics,
25, no. 5 (1994), 434.

Paul Deiermann and Rick Mabry,
"Generalizing an Approach to Radius of Curvature" (FFF #77),
College Journal of Mathematics,
25, no. 4 (1994), 309–310.

Rick Mabry,
"D^{**} functions and a
wellknown 'translation invariant' lemma,"
a talk at the Ninth Annual MiniConference in Real Analysis,
Auburn University, Auburn, AL, March 20, 1993.

Rick Mabry,
"Families of Shadings of the Real Line,"
a talk at the Eighth Annual MiniConference in Real
Analysis,
Auburn University, Auburn, AL, March 21, 1992.

R.D. Mabry,
"Sets which are welldistributed and invariant
relative to all
isometry invariant total extensions of Lebesgue measure",
Real Analysis Exchange,
16, no. 2 (1991–1992), 425–459.

A.G. Kartsatos and R. Mabry,
"Controlling the space with preassigned responses,"
Journal of Optimization Theory and Applications,
54, no. 3 (1987), 517–540.

A.G. Kartsatos and R. Mabry,
On the solvability in Hilbert space of certain nonlinear operator
equations depending on parameters,
Journal of Mathematical Analysis and Applications,
120, no.2 (1986), 670–678.
Other Stuff
Jim Hoffman and I played with
symmetric Venn diagrams
for awhile, trying to find one for N=11. ("Nonsimple" ones have recently been discovered by Peter Hamburger, et al.)
(See the
Symmetric Venn Diagrams page, part of the
Venn Diagrams Survey maintained by Frank Ruskey, in the
Electronic Journal of Combinatorics.)
We didn't succeed, but had a lot of fun. A prettier version of
my graphic ("ice cream cone curves")
mentioned in the above page can be found
HERE. A really ugly version (actually the same one) is
HERE.
And then there is
THIS!
While we were playing around, we
convexified a 7symmetric Venn diagram
using the
Geometer's Sketchpad.
(Click
Here for a GSP file. A reasonable printout takes 6 pages. It
needs to
be pretty big to even begin to see what is going on because each of
the 7
pieces is a convex polygon with 37 sides.) Not all 7symmetric Venn
diagrams
can be convexified. For such an example, see Figure 5 in Branko Grunbaum's article, "Venn Diagrams II,"
Geombinatorics, 2, no. 2 (Oct. 1992), 25–32.