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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 :)

$${Wonderful}! \\ $$$$\left.{old}\:{is}\:{gold}\::\right) \\ $$

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