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[[Image:Hammersley sofa animated.gif|right|280px|thumb|The Hammersley sofa has area 2.2074... but is not the largest solution]] | |||
The '''moving sofa problem''' was formulated by the Austrian-Canadian mathematician [[Leo Moser]] in 1966. The problem is a two-dimensional idealisation of real-life furniture moving problems, and asks for the rigid two-dimensional shape of largest area ''A'' that can be maneuvered through an L-shaped planar region with legs of unit width. The area ''A'' thus obtained is referred to as the ''sofa constant''. The exact value of the sofa constant is an [[Unsolved problems in mathematics|open problem]]. | |||
==Lower and upper bounds== | |||
As a semicircular disk of unit radius can pass through the corner, a lower bound for the sofa constant <math>\scriptstyle A\,\geq\,\pi/2\,\approx\, 1.570796327</math> is readily obtained. | |||
[[John Hammersley]] derived a considerably higher lower bound <math>\scriptstyle A\,\geq\,\pi/2 + 2/\pi\,\approx\,2.207416099</math> based on a [[handset]]-type shape consisting of two quarter-circles on either side of a 1 by 4/π rectangle from which a semicircle of radius <math>\scriptstyle 2/\pi\,</math> has been removed.<ref>{{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |authorlink3=Richard K. Guy |title=Unsolved Problems in Geometry |series=Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics |volume=II |editor-last=Hamos |editor-first=Paul R. |publisher=Springer-Verlag |year=1994 |isbn=978-0-387-97506-1 |url=http://www.springer.com/mathematics/geometry/book/978-0-387-97506-1 |accessdate=24 April 2013}}</ref><ref>[http://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm Moving Sofa Constant] by Steven Finch at MathSoft, includes a diagram of Gerver's sofa</ref> | |||
Gerver found a sofa that further increased the lower bound for the sofa constant to 2.219531669.<ref>{{cite journal |last=Gerver |first=Joseph L. |title=On Moving a Sofa Around a Corner |journal=Geometriae Dedicata |issn=0046-5755 |volume=42 |issue=3 |pages=267–283 |year=1992 |doi=10.1007/BF02414066}}</ref><ref>{{MathWorld|urlname=MovingSofaProblem|title=Moving sofa problem}}</ref> | |||
In a different direction, an easy argument by Hammersley shows that the sofa constant is at most <math>\scriptstyle 2\sqrt{2}\,\approx\, 2.8284</math>.<ref>{{cite journal |last=Wagner |first=Neal R. |title=The Sofa Problem |journal=The American Mathematical Monthly |volume=83 |issue=3 |year=1976 |pages=188–189 |doi=10.2307/2977022 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |jstor=2977022}}</ref><ref>{{cite book |last=Stewart |first=Ian |authorlink=Ian Stewart (mathematician) |title=Another Fine Math You've Got Me Into... |date=January 2004 |publisher=Dover Publications |location=Mineola, N.Y. |isbn=0486431819 |url=http://store.doverpublications.com/0486431819.html |accessdate=24 April 2013}}</ref> | |||
==See also== | |||
*[[Mountain climbing problem]] | |||
*[[Moser's worm problem]] | |||
==References== | |||
{{reflist}} | |||
[[Category:Discrete geometry]] | |||
[[Category:Unsolved problems in mathematics]] | |||
[[Category:Recreational mathematics]] | |||
Revision as of 16:48, 30 January 2014
The moving sofa problem was formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem is a two-dimensional idealisation of real-life furniture moving problems, and asks for the rigid two-dimensional shape of largest area A that can be maneuvered through an L-shaped planar region with legs of unit width. The area A thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem.
Lower and upper bounds
As a semicircular disk of unit radius can pass through the corner, a lower bound for the sofa constant is readily obtained.
John Hammersley derived a considerably higher lower bound based on a handset-type shape consisting of two quarter-circles on either side of a 1 by 4/π rectangle from which a semicircle of radius has been removed.[1][2]
Gerver found a sofa that further increased the lower bound for the sofa constant to 2.219531669.[3][4]
In a different direction, an easy argument by Hammersley shows that the sofa constant is at most .[5][6]
See also
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Moving Sofa Constant by Steven Finch at MathSoft, includes a diagram of Gerver's sofa
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534