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Question Number 141087 by greg_ed last updated on 15/May/21

hi, masters ! look at this thing carefully : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ................. find the line and the column where the number 795471 will appear !

$$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{masters}}\:! \\ $$$$\boldsymbol{\mathrm{look}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{thing}}\:\boldsymbol{\mathrm{carefully}}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\mathrm{3}\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\mathrm{6}\:\mathrm{7}\:\mathrm{8}\:\mathrm{9} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{10}\:\mathrm{11}\:\mathrm{12}\:\mathrm{13}\:\mathrm{14}\:\mathrm{15}\:\mathrm{16} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{17}\:\mathrm{18}\:\mathrm{19}\:\mathrm{20}\:................. \\ $$$$\:\:\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{line}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{column}}\:\boldsymbol{\mathrm{where}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{number}}\:\mathrm{795471}\:\boldsymbol{\mathrm{will}}\:\boldsymbol{\mathrm{appear}}\:! \\ $$

Answered by talminator2856791 last updated on 08/Sep/21

⌈(√(795471)) ⌉ = ⌈ 891.89181 ⌉ = 892 892 is the line 795471 − 891^2 = 1590 1590th column counting from most left

$$\: \\ $$$$\:\lceil\sqrt{\mathrm{795471}}\:\rceil\:=\:\lceil\:\mathrm{891}.\mathrm{89181}\:\rceil\:=\:\mathrm{892} \\ $$$$\:\mathrm{892}\:\mathrm{is}\:\mathrm{the}\:\mathrm{line} \\ $$$$\:\mathrm{795471}\:−\:\mathrm{891}^{\mathrm{2}} \:=\:\mathrm{1590} \\ $$$$\:\mathrm{1590th}\:\mathrm{column}\:\mathrm{counting}\:\mathrm{from}\:\mathrm{most}\:\mathrm{left}\: \\ $$$$\: \\ $$

Answered by qaz last updated on 15/May/21

1,3,7,13,21,... 1...1......1 2 4 6 8 2...3......3 2 2 2 3...7......5 4...13....7 ............ n...a_n ....Tn S_n = ((n),(1) )+2 ((n),(2) )+2 ((n),(3) ) a_n =S_n −S_(n−1) =n^2 −n+1 795471=n^2 −n+1 ⇒l=n≈892 If n=892,number 794773 is at [l:892,c:892] 795471−794773=698 So 795471 is at [l:892,c:1590]

$$\:\:\mathrm{1},\mathrm{3},\mathrm{7},\mathrm{13},\mathrm{21},...\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}...\mathrm{1}......\mathrm{1} \\ $$$$\:\:\:\:\mathrm{2}\:\:\mathrm{4}\:\:\mathrm{6}\:\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}...\mathrm{3}......\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\mathrm{2}\:\:\:\mathrm{2}\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}...\mathrm{7}......\mathrm{5} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}...\mathrm{13}....\mathrm{7} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:............ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{n}...{a}_{{n}} ....{Tn} \\ $$$${S}_{{n}} =\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}+\mathrm{2}\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}+\mathrm{2}\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}\: \\ $$$${a}_{{n}} ={S}_{{n}} −{S}_{{n}−\mathrm{1}} ={n}^{\mathrm{2}} −{n}+\mathrm{1} \\ $$$$\mathrm{795471}={n}^{\mathrm{2}} −{n}+\mathrm{1}\:\:\:\:\Rightarrow{l}={n}\approx\mathrm{892} \\ $$$${If}\:{n}=\mathrm{892},{number}\:\mathrm{794773}\:{is}\:{at}\:\left[{l}:\mathrm{892},{c}:\mathrm{892}\right] \\ $$$$\mathrm{795471}−\mathrm{794773}=\mathrm{698} \\ $$$${So}\:\mathrm{795471}\:{is}\:{at}\:\left[{l}:\mathrm{892},{c}:\mathrm{1590}\right] \\ $$

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