Dhowrkii maalmood ee la soo dhaafay, waxaan ku baranayay StackExchange su'aasha soo socota ee ku saabsan isku dheelitirnaanta tirada. Waxaan raadineynaa qaaciido xiran oo loogu talagalay iskuduwaha walxaha \(n\) -th ee isku-dhafan isku-dhafan ee soo socda, kaas oo ka sii ballaaranaya asalka dibedda iyo ka sii sii durugsanaanta:
.. 9 10 11 12
23 8 1 2 13
22 7 0 3 14
21 6 5 4 15
20 19 18 17 16Marka hore waxaan u kala qaybinaynaa tirooyinka dabiiciga kooxaha soo socda:
$$G_1 = \{ 1,...,8 \}\\G_2 = \{ 9, ..., 24 \}\\G_3 = \{ 25, ... 48 \}\\...\\G_k = \left\{ (2k-1)^2, (2k+1)^2 - 1 \right\}$$
Waxaan wacnaa
$$a^2 = \left(\left \lfloor \left( \frac{\left \lfloor \sqrt{n} \right \rfloor + 1}{2} \right) \right \rfloor \cdot 2 - 1 \right)^2$$
nambarka bilowga ee koox.
Kooxda gudaheeda waxaa ku jira afar koox-hoosaad oo is le'eg \(G_1\) waa \(g_1 = \{ 1,2 \}, g_2 = \{ 3,4 \}, g_3 = \{ 5,6 \}, g_4 = \{7,8\}\) ). Haddii \(n\) hal koox-kooxeed una gudubto ku xigga, jihada dariiqa wareegga (saacad ahaan tusaalaheenna) sidoo kale isla waqtigaas ayaa la beddelayaa. Waxaan kuheleynaa jihada hada \( b \in \{0,1,2,3\} \) adoo uqaybinaya booska \(na^2\) koox gudaheeda tirada walxaha kujira koox hoosaad:
$$b = \left \lfloor \frac{ n - a^2 }{ \text{abs}\left( a \right) + 1 } \right \rfloor$$
Hadda waxaan go'aan ka gaari karnaa \(g_n\) ugu horreeya koox-hoosaad \(g_n\) tiro badan \(b\) lambarka \(a+1\) ee xubno ku dhex jira koox lambarka ugu horreeya \(a^2\):
$$c = a^2 + b \cdot (a+1)$$
Hadda waxaan ku go'aamin karnaa kormeer fudud:
$$x_{right} = \left(n - c - \frac{ a + 1 }{2}+1\right),\, y_{right} = \left(\frac{ a + 1 }{2}\right) \\ x_{bottom} = \left(\frac{ a + 1 }{2}\right),\, y_{bottom} = (-1) \cdot \left( n - c - \frac{ a + 1 }{2}+1\right) \\ x_{left} = (-1) \cdot \left(n - c - \frac{ a + 1 }{2}+1\right),\, y_{left} = (-1) \cdot \left(\frac{ a + 1 }{2}\right) \\ x_{top} = (-1) \cdot \left(\frac{ a + 1 }{2}\right),\, y_{top} = \left( n - c - \frac{ a + 1 }{2}+1\right)$$
Waxaan dooneynaa inaan ku matalno \(f(n)\) wakiil matalaad xiran iyo waliba iyadoo aan kiisaska la kala saarin. Waxaan u adeegsannaa shaqada Signum, waxaan ku sharaxay nidaamka halkan . Si aan u helno qaaciido saafi ah oo shaqeysa \(n\) , waan helnaa:
Caawinta SVG.js , waxaan ku arki karnaa muuqaalkan yar in wax walba runti shaqeeyaan:
See the Pen ulam spiral by David Vielhuber (@vielhuber) on CodePen.
Wareegtada waxaa sidoo kale loo yaqaanaa ' Ulam's muquuninta' waxayna leedahay xiriir xiiso leh oo ku saabsan tirooyinka ugu muhiimsan.