Langford Timeline of Activity

The Most Recent 20+ years on a Timeline — Contains an 8 or 9 year inactivity gap!

1999 - Team Groth Solves L(2,19) in two months!

Rick Groth writes to tell how L(2,19) was computed.

2002 - Mike Godfrey determines L(2,20) with new Algebraic Method

In February 2002, Mike Godfrey, at the University of Manchester Institute of Science and Technology, UK, came up with an algebraic method of determining L(2,n) that was way faster than the classic search algorithm.

Using this new method, Godfrey and his colleague Ron van Bruchem cracked L(2,20), V(2,20), and V(2,21). The fellows also verified all previous results for L(2,n) and V(2,n). (See the tables below.)

Th Godfrey Method does not enumerate solutions, but rather extracts the number of solutions by repeated evaluations of a generating function.. then dividing the result by a very large number! Here is Godfrey's letter of explanation. No plans to publish this method have been set.

A Zip archive of Godfrey's FORTRAN source code is available for download HERE.

2004 - L(4,24) Solutions enumerated

All three L(4,24) solutions were computed by Richard Noble, terminating on May 28, 2004, after 2 years elapsed time. [Noble's Solutions]

2004 - L(2,23) Computed

In April, 2004, a grid computing experiment at Université de Reims Champagne-Ardenne used Godfrey's Method on a mix of 30 Intel and Sun machines, one with 24 processors. Five people worked on this project. See Michaël Krajecki's letter of explanation. There are nearly 4 quadrillion solutions to n=23.

2005 - L(2,24) Computed

Michaël Krajecki, Christophe Jaillet, Alain Bui obtained L(2,24) in April 2005 after a three month computation with 12 to 15 processors. There are nearly 47 quadrillion solutions!

References:

2006-2014

Nothing new? Better check my records!

2015 - L(2,27) Computed!

Christophe Jaillet wrote: We had a large experiment using best-effort distributed computing, and got L(2,27) with an average amount of 181 CPU-GPU machines during 2 days (1 day and 20 hours).

The team solved L(2,27) in late April 2015... reporting 111,683,606,778,027,803,456 solutions! However, this turned out to be erroneous. They had calculated the correct number, but fouled up and displayed this incorrect result. See News at top -

2015 - L(2,27) Confirmed, L(2,28) Computed!

July 2015 — Team Assarpour-Liu computed: 

    L(2,27) =   111,683,611,098,764,903,232
    L(2,28) = 1,607,383,260,609,382,393,152

April 2015 — Team Krajecki computed L(2,27) and sent results via email. When Team Assarpour-Liu executed their method for 27 and 28, the results for 27 was different than Team Krajecki's number, so, it was up to one or both of the parties to resolve the discrepancy.

The Team Assarpour-Liu paper might be downloaded from [arXiv.org]. We have a May 2017 PDF of The Team Assarpour-Liu paper [HERE].

2016 - Conflict explained!

November. Team Krajecki writes to say they had arrived at the same number as Team Assarpour-Liu (2015), but their program didn't display it correctly. You can find the details about their algorithm and the computation time in the paper on [SuperFri].

The above numbers have been added to the Table of Solutions.

2017 - Planar solutions up to 28 Confirmed

Boris Dimitrov reports that he has confirmed Knuth's results for planar solutions up to 28 pairs. See Planar Solutions on the main page.

2018 - P(2,31) and P(2,32) computed

Rory Molinari reports on Planar Langford pairings: P(2,31) and P(2,32). See [LINK] for his full report, and musings on end-run planarity solution..

2019 - Colombian Variant

A variation from Colombia — having to do with a 'Sentinel Pair' that allows only one number from each pair between them! We call it The Colombian Variant.

2020 - Tanton's Chairs

Also, check out a circular version of Langford's Problem that derives from a set of students sitting in a circle of chairs! [Tanton's Chairs]

2020 - Langford Quilts

Check out these beautiful Langford Quilts....

2021 - Asymptotic formula estimating L(2,n)

I've been informed of a paper (in preparation) that gives an asymptotic formula estimating the number of Langford and the Skolem sequences! The paper goes over the formula, and demonstrates its performance with a table of estimated vs actual values and the percentage difference. This Link appears on the Home Page. SO no further info on this page at this time (FYI - Raw URL here 4/17/22 https://eprint.arxitics.com/articles/langford.pdf)

2022 - ?

???