Back in Granada

img_2014Back in Granada on my way to Alicante, staying once again at university residence Carmen de la Victoria, I revisited the site of a 13-year-old diary entry, dated June 15, 2003:

A dusty pedestrian road by the Alhambra is lined with engraved quotations. My favorite on Saturday was by Thoreau, translated into Spanish. Here are my own English version constructed from memory as I continued my walk, followed by the English original which I later found on the web, and finally my revision. Continue reading ‘Back in Granada’ »

Bridge in Buenos Aires

R24 Recoleta'16Got to fill in at last minute with Director Marcelo de Carlo at the Recoleta Bridge Club in Buenos Aires 3:30 pm Monday September 4, 2016. On the pictured Hand 2, East on my right opened 1S, I bid 2S showing hearts and a minor, West passed, North bid 3H, East 4D, South 4H, West passed! and so did everyone else. West led the Spade 4 and followed with the 6 as East played K and A. It was surprising that West would pass with three spades, but I decided that W having all three hearts was more likely than false carding with two spades, so I trumped low. To pass with three spades, West must have almost nothing, so East must have the diamond A and club K. I played three rounds of trump to enter dummy and immediately led a diamond, East erring by playing low. Now when I run the hearts, leaving AQ of clubs and J of diamonds in dummy, East falls to an endplay. Unfortunately I chickened out at the last minute and took the losing club finesse.

Fortunately on the last Hand 24 I competed to 5Sx with everyone else in game the other way for 15 IMPS to tie for first.

Working at Home Depot

A guest post by Donna Kalinowsky.

I teach mathematics as an adjunct instructor at Berkshire Community College (BCC) in Massachusetts, which is one of a handful of states that does not allow its employees to pay into Social Security. To get my last requisite credits toward Social Security, I took a part-time job at Home Depot. Home Depot has taught me a lot about gratitude and dignity.

I have met several people at Home Depot who are good, smart, hard working people. I go to them often when I have an issue I can’t solve on my own. Some are part time like me, working unpredictable hours at $11 an hour. I doubt even the few full-time people earn a whole lot more. (Just like BCC, Home Depot tries to minimize full timers in order to avoid paying benefits, though Home Depot does offer some benefits to part timers if they work there long enough.)

I work there by choice, for a special reason, but for these people. This is their life. One girl has no car, so she has to take the bus to work. But if she is scheduled on a day when  no bus service is available, she has to take a cab to work. If she’s scheduled for four hours  at $11 an hour and a taxi costs $20, then it’s almost like she’s working for taxi money. Continue reading ‘Working at Home Depot’ »

More Solutions to Fermat with Fractional Exponents

Nicola Marino has offered to post the following preprint:

Irrational Exponents in Fermat’s Last Theorem. The n > 2 case.

by Nicola Marino

In this paper we find explicit irrational exponents greater than 2 solving the Fermat equation, which we call Algebraically Derived Ratio of Irrational (or ADRI) numbers. We then further expand the class of ADRI numbers to cover all the solution exponents for any Fermat equation.

Continue reading

Euler’s Theorem by Induction on Vertices

In my Discrete Math class today, Ryan Patton asked to prove Euler’s circuit theorem by induction on the number of vertices. Alex Summers contributed an idea about pairing/short-cutting instead of removing the edges incident to a deleted vertex. The class came up with the following proof. Is this proof out there somewhere?

Theorem (Euler). A pseudograph has a circuit containing all edges and vertices if it is connected and every vertex has even degree.

Proof by induction on the number n of vertices.

Base case n=1. Just follow the loops in succession.

Now assume for n and prove for a pseudo-graph of n+1 vertices. Pick a vertex. Since degree even, you can pair the incident edges, and you can avoid pairing the two ends of a loop. Short-cut each pair to avoid the vertex and delete it. By induction, each component of the new pseudo-graph has the desired circuit. Now restore the vertex and undo the short cuts to obtain the desired circuit.

Relaxed Disk Packings

Relaxed Disk Packings

G151-300x220by John Berry, Matthew Dannenberg, Jason Liang, and Yengyi Zeng
2015 NSF “SMALL” undergraduate research Geometry Group

With update below by Iglesias-Ham (all already known).

Abstract. The classic result about the optimal hexagonal packing of unit disks in the plane has recently been partially generalized by Edelsbrunner et al. to allow but penalize overlap for the case of lattice packings. We attempt to remove the restriction to lattice packings. Continue reading ‘Relaxed Disk Packings’ »

Dale Hoffman on Community College Students

DaleOffice2010When speaking at the 2015 Washington Two-Year Mathematics Conference at Campbell’s Resort, Lake Chelan, I met many wonderful people, including Dale Hoffman of Bellevue College (formerly Bellevue Community College), who agreed to send me the following inspiring chronicle of some of his past students: Continue reading ‘Dale Hoffman on Community College Students’ »

Back To School at Age 35

Guest post by Lorna Larsen, whom I met at the 2015 Washington State Two-Year College Mathematics Conference. After years as a painter, Lorna returned to school and this fall takes over as the Math Learning Center Tutor Supervisor at Shoreline Community College. Her children often attended class with her and are now both interested in education, one working with developmentally disabled adults.

Spring of 2002 was the beginning of the season that would make or break me as a painting contractor. I’d decided two years earlier to put my experience to work for me, and in some ways it was paying off. I was finally starting to charge what I was worth. But both the kids were still small, and I was constantly torn between their needs and my company’s. The mountain of taxes, bids, payroll, invoices and all the other things that go with small business ownership was becoming an avalanche. Continue reading ‘Back To School at Age 35’ »

Medgar Evers uses TOC to Stem Attrition

Math enrollments rise from 7 to 65.

Guest post by Professor Umesh Nagarkatte, Medgar Evers College, CUNY

course-description-sub-links-siteboxIn  urban colleges, student attrition due to absenteeism and failure has been a common problem. Attrition happens because students get bogged down by academic and non-academic issues. In 2002, three faculty members found that the Theory of Constraints (TOC) and its logic-based Thinking Processes (TP) tools can address both absenteeism and failure. They began with two weeks training in TOC and TP at Goldratt Institute, New Haven, CT, and the chair got departmental agreement for a new approach for Spring 2002. Continue reading ‘Medgar Evers uses TOC to Stem Attrition’ »

New Optimal Pentagonal Tilings

Updated with new discoveries 31 January —11 February 2015; first published 27 May 2014. (Incidentally, new type of pentagonal tile discovered July 2015 by Casey Mann, Jennifer McLoud-Mann, and David Von Derauc.)

A joint paper [C1] with my SMALL undergraduate research Geometry Group found least-perimeter pentagonal unit-area tiles, Cairo and Prismatic:

They proved that mixtures of unit-area convex pentagonal tiles can do no better, but found many examples of Cairo-Prismatic tilings that do equally well [C1, C2]. Since their work nine more have been discovered.  Continue reading ‘New Optimal Pentagonal Tilings’ »