Tensor product of fields

From formulasearchengine
Jump to navigation Jump to search

In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.

Perfect magic cubes of order one are trivial; cubes of orders two to four can be proven not to exist, and cubes of orders five and six were first discovered by Walter Trump and Christian Boyer on November 13 and September 1, 2003, respectively. A perfect magic cube of order seven was given by A. H. Frost in 1866, and on March 11, 1875, an article was published in the Cincinnati Commercial newspaper on the discovery of a perfect magic cube of order 8 by Gustavus Frankenstein. Perfect magic cubes of orders nine and eleven have also been constructed. The first perfect cube of order 10 has been constructed in 1988. (Li Wen, China)

An alternative definition

In recent years, an alternative definition for the perfect magic cube was proposed by John R. Hendricks. It is based on the fact that a pandiagonal magic square has traditionally been called 'perfect', because all possible lines sum correctly. This is not the case with the above definition for the cube. See Nasik magic hypercube for an unambiguous alternative term

This same reasoning may be applied to hypercubes of any dimension. Simply stated; if all possible lines of m cells (m = order) sum correctly, the hypercube is perfect. All lower dimension hypercubes contained in this hypercube will then also be perfect. This is not the case with the original definition, which does not require that the planar and diagonal squares be a pandiagonal magic cube.

The original definition is applicable only to magic cubes, not tesseracts, dimension 5 cubes, etc.

Example: A perfect magic cube of order 8 has 244 correct lines by the old definition, but 832 correct lines by this new definition.

Order 8 is the smallest possible perfect magic cube. None can exist for double odd orders.

Gabriel Arnoux constructed an order 17 perfect magic cube in 1887. F.A.P.Barnard published order 8 and order 11 perfect cubes in 1888.

By the modern (Hendricks) definition, there are actually six classes of magic cube; simple magic cube, pantriagonal magic cube, diagonal magic cube, pantriagdiag magic cube, pandiagonal magic cube, and perfect magic cube.

Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik! C. Planck (1905) redefined Nasik to mean magic hypercubes of any order or dimension in which all possible lines summed correctly.

i.e. Nasik is an alternative, and unambiguous term for the perfect class of any dimension of magic hypercube.

First known Perfect Pandiagonaal Semi-magisch Magic Cube

Thomas Krijgsman, 1982 March, 21 number 5 / link: http://www.pythagoras.nu/pyth/nummer.php?id=253

-Row 1 (4x4)- - - - - -Row 2 (4x4)- - - - - -Row 3 (4x4)- - - - - - Row 4 (4x4)

[32]-[05]-[52]-[41] = [10]-[35]-[22]-[63] = [49]-[28]-[45]-[08] = [39]-[62]-[11]-[18]

[03]-[42]-[31]-[54] = [37]-[64]-[09]-[20] = [30]-[07]-[50]-[43] = [60]-[17]-[40]-[13]

[61]-[24]-[33]-[12] = [27]-[02]-[55]-[46] = [36]-[57]-[16]-[21] = [06]-[47]-[26]-[51]

[34]-[59]-[14]-[23] = [56]-[29]-[44]-[01] = [15]-[38]-[19]-[58] = [25]-[04]-[53]-[48]

3D solution in my head, fill the numbers on graph paper, that all.

Walter Trump and Christian Boyer, 2003-11-13

This cube consists of all numbers from 1 to 125. The sum of the 5 numbers in each of the 25 rows, 25 columns, 25 pillars, 30 diagonals and 4 triagonals (space diagonals) equals the magic constant 315.

1° level - 2° level

3° level - 4° level

5° level -

See also

References

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    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
  • Planck, C., The Theory of Paths Nasik, Printed for private circulation, A.J. Lawrence, Printer, Rugby,(England), 1905
  • H.D, Heinz & J.R. Hendricks, Magic Square Lexicon: Illustrated, hdh, 2000, 0-9687985-0-0

External links