Editing Kepler conjecture (section)

==Twentieth century==
The next step toward a solution was taken by [[László Fejes Tóth]]. {{harvtxt|Fejes Tóth|1953}} showed that the problem of determining the maximum density of all arrangements (regular and irregular) could be reduced to a [[finite set|finite]] (but very large) number of calculations. This meant that a [[proof by exhaustion]] was, in principle, possible. As Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem.

Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. English mathematician [[Claude Ambrose Rogers]] (see {{harvtxt|Rogers|1958}}) established an upper bound value of about 78%, and subsequent efforts by other mathematicians reduced this value slightly, but this was still much larger than the cubic close packing density of about  74%.

In 1990, [[Wu-Yi Hsiang]] claimed to have proven the Kepler conjecture. The proof was praised by ''Encyclopædia Britannica'' and ''Science'' and Hsiang was also honored at joint meetings of AMS-MAA.<ref>{{cite journal | doi=10.1007/BF03024356 | volume=16 | issue=3 | date=June 1994 | title=The Status of the Kepler Conjecture | first=Thomas C. | last=Hales | s2cid=123375854 |author-link=Thomas Callister Hales | journal=[[The Mathematical Intelligencer]] | pages=47–58}}</ref> {{harvs|txt|first=Wu-Yi|last=Hsiang|year1=1993|year2=2001}}  claimed to prove the Kepler conjecture using geometric methods. However [[Gábor Fejes Tóth]] (the son of László Fejes Tóth) stated in his review of the paper "As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs."  
{{harvtxt|Hales|1994}} gave a detailed criticism of Hsiang's work, to which {{harvtxt|Hsiang|1995}} responded. The current consensus is that Hsiang's proof is incomplete.<ref>{{cite book |first=Simon |last=Singh |author-link=Simon Singh |title=Fermat's Last Theorem |location=New York |publisher=Walker |year=1997 |isbn=978-0-80271-331-5 |url-access=registration |url=https://archive.org/details/fermatsenigmaepi00sing_0 }}</ref>