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%I #46 Sep 29 2021 18:57:38
%S 3,3,3,2,2,2,3,3,3,3,3,1,2,4,3,4,3,3,3,4,2,5,2,3,3,3
%N Minimum number of mathematical symbols and operators in common use required to write n using exactly three 4's. This version allows only addition, subtraction, multiplication, division, square root, exponentiation, factorial, subfactorial, decimal point, overline (an infinitely repeated digit), parentheses, as well as concatenation.
%C My web page gives examples of Four Nines up to 774,840,978 = (9-to-the-9) + (9-to-the-9); tablified a Four Pi's; and cites Knuth's Conjecture:"Representing Numbers Using Only One 4", Donald Knuth, [Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp. 308-310]. Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function. For example: Knuth shows how to make the number 64 using only one 4: |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt (4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _|. - _Jonathan Vos Post_, May 25 2014
%C In 1912, W. W. Rouse Ball calculated all the integers from 1 to 90 using the permissible symbols. - _Arkadiusz Wesolowski_, Sep 27 2021
%H W. W. Rouse Ball, <a href="http://www.jstor.org/stable/3603066">Four fours: Some arithmetic puzzles</a>, Math. Gazette No. 98, May 1912.
%H Donald Knuth, <a href="http://www.jstor.org/stable/2689238">Representing Numbers Using Only One 4</a>, Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp. 308-310.
%H Magic Dragon Multimedia, <a href="http://magicdragon.com/EmeraldCity/Nonfiction/four9s.html">The Four Nines Puzzle</a>.
%e n.................expression...............a(n)
%e 0................4 - sqrt(4*4)................3
%e 1...................(4/4)^4...................3
%e 2..................(4 + 4)/4..................3
%e 3...................4 - 4/4...................2
%e 4..................4 + 4 - 4..................2
%e 5...................4 + 4/4...................2
%e 6...............4 + 4 - sqrt(4)...............3
%e 7..................4*4 - !4...................3
%e 8.................sqrt(4*4*4).................3
%e 9.................!4 - 4 + 4..................3
%e 10..............4 + 4 + sqrt(4)...............3
%e 11...................44/4.....................1
%e 12.................4 + 4 + 4..................2
%e 13..............!4 + sqrt(4*4)................4
%e 14...............4*4 - sqrt(4)................3
%e 15.............!4 + 4 + sqrt(4)...............4
%e 16................4! - 4 - 4..................3
%e 17................!4 + 4 + 4..................3
%e 18...............4*4 + sqrt(4)................3
%e 19.................!4 + 4/.4..................4
%e 20..................4*4 + 4...................2
%e 21..............!4 + 4*sqrt(!4)...............5
%e 22................44/sqrt(4)..................2
%e 23.................4! - 4/4...................3
%e 24................4! - 4 + 4..................3
%e 25.................4! + 4/4...................3
%Y Cf. A242540.
%K nonn,more
%O 0,1
%A _Arkadiusz Wesolowski_, May 15 2014
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