Menu Close

Question-66684




Question Number 66684 by Tinkutara@ last updated on 18/Aug/19
Answered by MJS last updated on 18/Aug/19
this is old...  you always say in words what you read and  write down in numbers  1  one “1” = 1 1  two “1s” = 2 1  one “2” and one “1” = 1 2 1 1  and so on  1 1 1 2 2 1  3 1 2 2 1 1  1 3 1 1 2 2 2 1  1 1 1 3 2 1 3 2 1 1  ...  you never get a “4”  you can proof if a given “number” is correct  or which had been the “input number” by  going back  uneven numbers digits can be input numbers  only, i.e.  a_n =13213 = one “3” and two “1s” and 3???    a_n =132131 is not correct, it must be 1351  a_n =1351  a_(n−1) =311111 but this isn′t correct too, it    must be 51 ⇒ neithef 132131 nor 1351 are  correct    a_n =231231  a_(n−1) =332111 is not correct...    but of course we can start with anything  1               2                      3                        x∉{1, 2, 3}  11            12                    13                      1x  21            1112               1113                  111x  1211        3112               3113                  311x  111221  132112           132113             13211x  312211  1113122112  1113122113   111312211x  ...    4321  14131211  11141113111221  31143113312211  132114132123112221  1113122114111312111213213211  ...  it has been explored by Goedel...
$$\mathrm{this}\:\mathrm{is}\:\mathrm{old}… \\ $$$$\mathrm{you}\:\mathrm{always}\:\mathrm{say}\:\mathrm{in}\:\mathrm{words}\:\mathrm{what}\:\mathrm{you}\:\mathrm{read}\:\mathrm{and} \\ $$$$\mathrm{write}\:\mathrm{down}\:\mathrm{in}\:\mathrm{numbers} \\ $$$$\mathrm{1} \\ $$$$\mathrm{one}\:“\mathrm{1}''\:=\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{two}\:“\mathrm{1s}''\:=\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{one}\:“\mathrm{2}''\:\mathrm{and}\:\mathrm{one}\:“\mathrm{1}''\:=\:\mathrm{1}\:\mathrm{2}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{so}\:\mathrm{on} \\ $$$$\mathrm{1}\:\mathrm{1}\:\mathrm{1}\:\mathrm{2}\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{3}\:\mathrm{1}\:\mathrm{2}\:\mathrm{2}\:\mathrm{1}\:\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{3}\:\mathrm{1}\:\mathrm{1}\:\mathrm{2}\:\mathrm{2}\:\mathrm{2}\:\mathrm{1} \\ $$$$\mathrm{1}\:\mathrm{1}\:\mathrm{1}\:\mathrm{3}\:\mathrm{2}\:\mathrm{1}\:\mathrm{3}\:\mathrm{2}\:\mathrm{1}\:\mathrm{1} \\ $$$$… \\ $$$$\mathrm{you}\:\mathrm{never}\:\mathrm{get}\:\mathrm{a}\:“\mathrm{4}'' \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{proof}\:\mathrm{if}\:\mathrm{a}\:\mathrm{given}\:“\mathrm{number}''\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{or}\:\mathrm{which}\:\mathrm{had}\:\mathrm{been}\:\mathrm{the}\:“\mathrm{input}\:\mathrm{number}''\:\mathrm{by} \\ $$$$\mathrm{going}\:\mathrm{back} \\ $$$$\mathrm{uneven}\:\mathrm{numbers}\:\mathrm{digits}\:\mathrm{can}\:\mathrm{be}\:\mathrm{input}\:\mathrm{numbers} \\ $$$$\mathrm{only},\:\mathrm{i}.\mathrm{e}. \\ $$$${a}_{{n}} =\mathrm{13213}\:=\:\mathrm{one}\:“\mathrm{3}''\:\mathrm{and}\:\mathrm{two}\:“\mathrm{1s}''\:\mathrm{and}\:\mathrm{3}??? \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{132131}\:\mathrm{is}\:\mathrm{not}\:\mathrm{correct},\:\mathrm{it}\:\mathrm{must}\:\mathrm{be}\:\mathrm{1351} \\ $$$${a}_{{n}} =\mathrm{1351} \\ $$$${a}_{{n}−\mathrm{1}} =\mathrm{311111}\:\mathrm{but}\:\mathrm{this}\:\mathrm{isn}'\mathrm{t}\:\mathrm{correct}\:\mathrm{too},\:\mathrm{it} \\ $$$$\:\:\mathrm{must}\:\mathrm{be}\:\mathrm{51}\:\Rightarrow\:\mathrm{neithef}\:\mathrm{132131}\:\mathrm{nor}\:\mathrm{1351}\:\mathrm{are} \\ $$$$\mathrm{correct} \\ $$$$ \\ $$$${a}_{{n}} =\mathrm{231231} \\ $$$${a}_{{n}−\mathrm{1}} =\mathrm{332111}\:\mathrm{is}\:\mathrm{not}\:\mathrm{correct}… \\ $$$$ \\ $$$$\mathrm{but}\:\mathrm{of}\:\mathrm{course}\:\mathrm{we}\:\mathrm{can}\:\mathrm{start}\:\mathrm{with}\:\mathrm{anything} \\ $$$$\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\notin\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\} \\ $$$$\mathrm{11}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{12}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{13}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}{x} \\ $$$$\mathrm{21}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1112}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1113}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{111}{x} \\ $$$$\mathrm{1211}\:\:\:\:\:\:\:\:\mathrm{3112}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3113}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{311}{x} \\ $$$$\mathrm{111221}\:\:\mathrm{132112}\:\:\:\:\:\:\:\:\:\:\:\mathrm{132113}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{13211}{x} \\ $$$$\mathrm{312211}\:\:\mathrm{1113122112}\:\:\mathrm{1113122113}\:\:\:\mathrm{111312211}{x} \\ $$$$… \\ $$$$ \\ $$$$\mathrm{4321} \\ $$$$\mathrm{14131211} \\ $$$$\mathrm{11141113111221} \\ $$$$\mathrm{31143113312211} \\ $$$$\mathrm{132114132123112221} \\ $$$$\mathrm{1113122114111312111213213211} \\ $$$$… \\ $$$$\mathrm{it}\:\mathrm{has}\:\mathrm{been}\:\mathrm{explored}\:\mathrm{by}\:\mathrm{Goedel}… \\ $$
Commented by Tinkutara@ last updated on 18/Aug/19
Wonderful!  old is gold :)
$${Wonderful}! \\ $$$$\left.{old}\:{is}\:{gold}\::\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *