Makemakika ma ka pāʻani Dobble

I ke ahiahi ʻohana hope loa, ua lawe ʻia ka pāʻani Dobble (ma ka Harry Potter Edition) i ka papaʻaina e nā keiki. Ma hope o ka pōʻai nalowale 5 (me ka ʻike ʻole ʻia o kaʻu kāleka me ke kāleka pāʻani) ua haʻi ʻia iaʻu, me koʻu kahaha, hiki i kēlā me kēia mea pāʻani ke loaʻa i kahi paʻi i kēlā me kēia puni. Akā ʻo koʻu manaʻoʻiʻo ʻaʻole i ʻike ʻia me nā ʻūhā nalowale hou - ʻoi aku ka wikiwiki o nā keiki.


No ke kumu e nānā pono ai i ka pāʻani mai ka manaʻo makemakika. ʻO ka mea mua ke kumu o ka pāʻani: He pāʻani kāleka maʻalahi ʻo Dobble me \(55\) mau kāleka pōʻai, e hōʻike ana kēlā me kēia me ʻewalu mau hōʻailona like ʻole. Hana ʻia nā kāleka a pau, waiho wale i ke kāleka hope loa i waenakonu o ka papaʻaina. I kēia manawa pono nā mea pāʻani a pau e hoʻohālikelike i nā hōʻailona ma ke kāleka me nā hōʻailona ma kā lākou kāleka kiʻekiʻe o kēia manawa. Inā loaʻa i ka mea pāʻani ka hōʻailona like ma nā kāleka ʻelua, hiki iā ia ke kau i kāna kāleka ma ka waihona ma o ka wikiwiki loa i ka inoa ʻana i ka hōʻailona. ʻO ka mea pāʻani e hoʻolei i kā lākou kāleka a pau e lanakila mua.

Pehea e hiki ai ke loa'a \(55\) nā kāleka i hana 'ia ma ke 'ano i loa'a i nā kāleka 2 i ho'okahi hō'ailona like? He aha ka helu liʻiliʻi o ia mau hōʻailona pono e hoʻohana ʻia? He aha ka helu kiʻekiʻe o ia mau kāleka?

ʻO ka mea mua, kūkulu mākou i kēia mau kāleka me ka hoʻohana ʻana i nā ʻanuʻu kūpono (ʻo nā kāleka a pau i kūkulu ʻia ma hope o ka waiwai i hoʻokaʻawale ʻia lākou ma ka ʻaoʻao piʻi): Pono ka kāleka mua he 8 mau hōʻailona like ʻole, ʻo ia hoʻi.:

$$\left(\begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \\ 8 \end{array}\right)$$

I kēia manawa, kūkulu mākou i nā kāleka ma ke ʻano i loaʻa iā lākou hoʻokahi hōʻailona like me ka kāleka mua:

$$\left(\begin{array}{c} 1 \\ x_{1.2} \\ x_{1.3} \\ x_{1.4} \\ x_{1.5} \\ x_{1.6} \\ x_{1.7} \\ x_{1.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{2.2} \\ x_{2.3} \\ x_{2.4} \\ x_{2.5} \\ x_{2.6} \\ x_{2.7} \\ x_{2.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{3.2} \\ x_{3.3} \\ x_{3.4} \\ x_{3.5} \\ x_{3.6} \\ x_{3.7} \\ x_{3.8} \end{array}\right), \ldots, \left(\begin{array}{c} 1 \\ x_{k.2} \\ x_{k.3} \\ x_{k.4} \\ x_{k.5} \\ x_{k.6} \\ x_{k.7} \\ x_{k.8} \end{array}\right)$$

Hiki ke kūkulu ʻia kekahi helu o ia mau kāleka ma ʻaneʻi (e hoʻopiha wale ʻoe i nā wahi i ka piʻi ʻana, e hoʻomaka me \(9\) ). ʻAʻole hoihoi kēia hihia liʻiliʻi, akā naʻe, no ka mea makemake mākou i kahi hoʻonohonoho me ka helu liʻiliʻi o nā hōʻailona (a me ka nui o nā kāleka). I kēia manawa mākou e noʻonoʻo ai i ka lua o ka hōʻailona \( x_{l.2} \) o kēlā me kēia kāleka, kahi e pili pono ai kēia: \( x_{1.2} \neq x_{2.2} \neq x_{3.2} \neq \ldots \neq x_{k.2} \) . No laila, ua hoʻokomo mākou i \( k \) hōʻailona hou. Akā i kēia manawa \( k \leq 8-1 = 7 \) , no ka mea, ʻaʻohe o nā hōʻailona \( 7 \) \( x_{1.2},\, x_{1.3},\, x_{1.4},\, x_{1.5},\, x_{1.6},\, x_{1.7},\, x_{1.8} \) (o ke kāleka hema loa) hiki ke hoʻohālikelike i ka hōʻailona ʻelua o kēlā me kēia kāleka ʻē aʻe (inā ʻaʻole, ʻelua mau hōʻailona like. ).

Ua loaʻa iā mākou ka nui o kēia mau kāleka hou 7:

$$\left(\begin{array}{c} 1 \\ x_{1.2} \\ x_{1.3} \\ x_{1.4} \\ x_{1.5} \\ x_{1.6} \\ x_{1.7} \\ x_{1.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{2.2} \\ x_{2.3} \\ x_{2.4} \\ x_{2.5} \\ x_{2.6} \\ x_{2.7} \\ x_{2.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{3.2} \\ x_{3.3} \\ x_{3.4} \\ x_{3.5} \\ x_{3.6} \\ x_{3.7} \\ x_{3.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{4.2} \\ x_{4.3} \\ x_{4.4} \\ x_{4.5} \\ x_{4.6} \\ x_{4.7} \\ x_{4.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{5.2} \\ x_{5.3} \\ x_{5.4} \\ x_{5.5} \\ x_{5.6} \\ x_{5.7} \\ x_{5.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{6.2} \\ x_{6.3} \\ x_{6.4} \\ x_{6.5} \\ x_{6.6} \\ x_{6.7} \\ x_{6.8} \end{array}\right), \left(\begin{array}{c} 1 \\ x_{7.2} \\ x_{7.3} \\ x_{7.4} \\ x_{7.5} \\ x_{7.6} \\ x_{7.7} \\ x_{7.8} \end{array}\right)$$

Me ka hoʻopaʻapaʻa like mākou e kūkulu nei i nā palapala 'āina \(7\) aʻe (ʻo ka mua o kēia mau palapala 'āina e hui pū me kā mākou palapala hoʻomaka, ʻaʻole me \(1\) , inā ʻaʻole me ka \(7\) ma mua. loaʻa nā palapala 'āina ):

$$\left(\begin{array}{c} 2 \\ x_{8.2} \\ x_{8.3} \\ x_{8.4} \\ x_{8.5} \\ x_{8.6} \\ x_{8.7} \\ x_{8.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{9.2} \\ x_{9.3} \\ x_{9.4} \\ x_{9.5} \\ x_{9.6} \\ x_{9.7} \\ x_{9.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{10.2} \\ x_{10.3} \\ x_{10.4} \\ x_{10.5} \\ x_{10.6} \\ x_{10.7} \\ x_{10.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{11.2} \\ x_{11.3} \\ x_{11.4} \\ x_{11.5} \\ x_{11.6} \\ x_{11.7} \\ x_{11.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{12.2} \\ x_{12.3} \\ x_{12.4} \\ x_{12.5} \\ x_{12.6} \\ x_{12.7} \\ x_{12.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{13.2} \\ x_{13.3} \\ x_{13.4} \\ x_{13.5} \\ x_{13.6} \\ x_{13.7} \\ x_{13.8} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{14.2} \\ x_{14.3} \\ x_{14.4} \\ x_{14.5} \\ x_{14.6} \\ x_{14.7} \\ x_{14.8} \end{array}\right)$$

Hiki ke hoʻomau ʻia kēia hoʻopaʻapaʻa no nā kāleka \(7\) aʻe; Huina \(8-2 = 6\) manawa hou aku. Ua like nā kāleka \(7\) hope loa:

$$\left(\begin{array}{c} 8 \\ x_{50.2} \\ x_{50.3} \\ x_{50.4} \\ x_{50.5} \\ x_{50.6} \\ x_{50.7} \\ x_{50.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{51.2} \\ x_{51.3} \\ x_{51.4} \\ x_{51.5} \\ x_{51.6} \\ x_{51.7} \\ x_{51.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{52.2} \\ x_{52.3} \\ x_{52.4} \\ x_{52.5} \\ x_{52.6} \\ x_{52.7} \\ x_{52.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{53.2} \\ x_{53.3} \\ x_{53.4} \\ x_{53.5} \\ x_{53.6} \\ x_{53.7} \\ x_{53.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{54.2} \\ x_{54.3} \\ x_{54.4} \\ x_{54.5} \\ x_{54.6} \\ x_{54.7} \\ x_{54.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{55.2} \\ x_{55.3} \\ x_{55.4} \\ x_{55.5} \\ x_{55.6} \\ x_{55.7} \\ x_{55.8} \end{array}\right), \left(\begin{array}{c} 8 \\ x_{56.2} \\ x_{56.3} \\ x_{56.4} \\ x_{56.5} \\ x_{56.6} \\ x_{56.7} \\ x_{56.8} \end{array}\right)$$

Inā ʻoe e hoʻohui i kahi kāleka ʻē aʻe $$\left(\begin{array}{c} 9 \\ x_{57.2} \\ x_{57.3} \\ x_{57.4} \\ x_{57.5} \\ x_{57.6} \\ x_{57.7} \\ x_{57.8} \end{array}\right)$$ e hāʻule no ka mea ʻaʻole kaʻana kēia kāleka i kahi hōʻailona me ke kāleka hoʻomaka. Ua kūkulu mākou i ka palena o \(1 + 8 \cdot 7 = 57\) palapala 'āina. ʻO kā mākou pahuhopu i kēia manawa, ʻo ia ke kūkulu ʻana i nā mea he nui.

No ka hana ʻana i kēia, nānā mākou i nā kāleka hou 7 mua loa i loaʻa a hiki i ka hopena e pono mākou i nā hōʻailona hou \(7 \cdot 7\) ma aneʻi (ʻaʻohe kāleka i loaʻa i kahi hōʻailona ʻelua a ʻaʻole pono e ʻike ʻia kēlā me kēia hōʻailona e hāʻawi ʻia. ʻelua, ʻoiai ʻo \(1\) ua pālua ʻia):

$$\left(\begin{array}{c} 1 \\ 9 \\ 10 \\ 11 \\ 12 \\ 13 \\ 14 \\ 15 \end{array}\right), \left(\begin{array}{c} 1 \\ 16 \\ 17 \\ 18 \\ 19 \\ 20 \\ 21 \\ 22 \end{array}\right), \left(\begin{array}{c} 1 \\ 23 \\ 24 \\ 25 \\ 26 \\ 27 \\ 28 \\ 29 \end{array}\right), \left(\begin{array}{c} 1 \\ 30 \\ 31 \\ 32 \\ 33 \\ 34 \\ 35 \\ 36 \end{array}\right), \left(\begin{array}{c} 1 \\ 37 \\ 38 \\ 39 \\ 40 \\ 41 \\ 42 \\ 43 \end{array}\right), \left(\begin{array}{c} 1 \\ 44 \\ 45 \\ 46 \\ 47 \\ 48 \\ 49 \\ 50 \end{array}\right), \left(\begin{array}{c} 1 \\ 51 \\ 52 \\ 53 \\ 54 \\ 55 \\ 56 \\ 57 \end{array}\right)$$

No laila pono mākou i nā hōʻailona liʻiliʻi \(8 + (7 \cdot 7) = 57\) (e like me ka nui o nā hōʻailona e like me nā kāleka!). Ke hoʻāʻo nei mākou e loaʻa i kēia helu a e ʻimi i kahi lula kūkulu no nā mea ʻē aʻe a pau. No ka hana ʻana i kēia, kūkulu mākou i kahi dobble liʻiliʻi liʻiliʻi i loaʻa nā hōʻailona \(3\) no kēlā me kēia kāleka a loaʻa iā ia ma ke kāleka hoʻomaka.

$$\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right)$$

a me nā kāleka ʻē aʻe

$$\left(\begin{array}{c} 1 \\ 4 \\ 5 \end{array}\right), \left(\begin{array}{c} 1 \\ 6 \\ 7 \end{array}\right)$$

$$\left(\begin{array}{c} 2 \\ x_{3.2} \\ x_{3.3} \end{array}\right), \left(\begin{array}{c} 2 \\ x_{4.2} \\ x_{4.3} \end{array}\right)$$

$$\left(\begin{array}{c} 3 \\ x_{5.2} \\ x_{5.3} \end{array}\right), \left(\begin{array}{c} 3 \\ x_{6.2} \\ x_{6.3} \end{array}\right)$$

me ka huina o \(1 + 3 \cdot 2 = 7\) kāleka a me \( 3 + (2 \cdot 2) = 7\) hōʻailona. Me kahi hoʻāʻo liʻiliʻi a me ka hewa (a me ka hoʻohana ʻana i nā hōʻailona i hāʻawi ʻia) loaʻa iā ʻoe ka dobble aʻe:

$$\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right)$$

$$\left(\begin{array}{c} 1 \\ 4 \\ 5 \end{array}\right), \left(\begin{array}{c} 1 \\ 6 \\ 7 \end{array}\right)$$

$$\left(\begin{array}{c} 2 \\ 4 \\ 6 \end{array}\right), \left(\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right)$$

$$\left(\begin{array}{c} 3 \\ 4 \\ 7 \end{array}\right), \left(\begin{array}{c} 3 \\ 5 \\ 6 \end{array}\right)$$

Hiki ke ʻike ʻia kēia me ka systematically? No ka hana ʻana i kēia, hoʻokomo mākou i nā hōʻailona hou i hāʻawi ʻia \(4, 5, 6, 7\) i loko o kahi matrix square.:

$$\begin{array}{ccc} 4 & & 5 \\ & & \\ 6 & & 7\end{array}$$

I kēia manawa, noʻonoʻo mākou no nā kāleka mua ʻelua (e hoʻomaka me nā hōʻailona hoʻomaka \ \(4\) a me \(5\) ) nā laina hoʻohui kū pololei i nā hōʻailona haʻahaʻa \(6\) a me \(7\):

$$\begin{array}{ccc} 4 & & 5 \\ \vdots & & \vdots \\ 6 & & 7\end{array}$$

No ka mea, ʻaʻole e ʻokoʻa kēia mau laina, loaʻa iā mākou (ma ka hoʻolālā ʻana i nā hōʻailona ma nā laina hoʻohui laina a me ka laina) nā kāleka kūpono loa.:

$$\left(\begin{array}{c} 2 \\ 4 \\ 6 \end{array}\right), \left(\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right)$$

ʻO ka hope, manaʻo mākou e hoʻohui i nā laina me kahi pali ʻē aʻe (i kēia hihia me ka slope \(1\) ):

$$\begin{array}{ccccc} & 4 & & 5 & \\ \ddots & & \ddots & & \ddots \\ & 6 & & 7 &\end{array}$$

Haʻalele ka laina hoʻohui ʻelua (ma waena o \(5\) a me \(6\) ) i ka matrix ma ka ʻaoʻao ʻākau a komo hou ma ka ʻaoʻao hema. Ma ke koho akamai ʻana i ka pali, hōʻoia mākou ma ka ʻaoʻao hoʻokahi ʻaʻole i hui nā laina hoʻohui me kekahi, akā ʻaʻole i hui nā laina hoʻohui mua (vertical). Ke alakaʻi nei kēia manaʻo hoʻolālā i ke ʻano hoʻolālā ma lalo nei:

ʻO kahi pālua me \(k \in \mathbb{N} \, | \, (k-1) \text{ prim} \) he \(1+(k \cdot (k-1)) = k^2-k+1 = k + (k-1)(k-1)\) kāleka a me nā hōʻailona. No ka palapala 'āina \(K_x\) me \(x \in \mathbb{N}\) a me \(0 \leq x \leq (k-1) \cdot k\) pili.:

$$K_x = \left(\begin{array}{c} f(x,1) \\ f(x,2) \\ \vdots \\ f(x,k) \end{array}\right), \,\, m = \left\lfloor \frac{x-1}{k-1} \right\rfloor + 1,$$

$$f(x,y) = \left\{\begin{array}{ll} y & \text{falls } x = 0 \\ \lfloor \frac{x-1}{k-1} \rfloor + 1, &\text{sonst falls } y = 1 \\ (k+1) + (k-1)(x-1) + (y-2), & \text{sonst falls } 0 < x < k \\ \left( \left((m-1)(k-1)+x\right)-1+ \left( (m-2)(y-2) \right) \right) \% (k-1) &\text{sonst} \\ + (k+1) + (k-1)(y-2)&\end{array}\right.$$

Aia nā \((k-1)\cdot k + 1 = k + (k-1)(k-1)\) o kēia mau kāleka. I kēia manawa koe wale nō e hōʻike:

$$ \forall x_1 < x_2 \in \{ 1, \ldots, k+(k-1)(k-1) \} \, \exists \, ! \, y_1, y_2 \in \{ 1, \ldots, k \}: f(x_1, y_1) = f(x_2, y_2) $$

  • Hihia 1st: \( x_1 = 0 \)
    • Hihia 1a: \( 0 < x_2 < k \)
      • No \(y_1 = 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = f(0, 1) = 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \lfloor \frac{x_2-1}{k-1} \rfloor + 1 = 1\) .
      • No \(y_1 \neq 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = f(0, y_1) = y_1 \neq 1\)
        \(f(x_2, y_2) = f(x_2, y_2) = \lfloor \frac{x_2-1}{k-1} \rfloor + 1 = 1\)
      • No \(y_1 = 1\) a me \(y_2 \neq 1\) :
        \(f(x_1, y_1) = f(0, 1) = 1\)
        \(f(x_2, y_2) = f(x_2, y_2) = (k+1) + (k-1)(x-1) + (y-2) =\)
        \((k+1)(x-1) + (k-1) + y \geq (k+1)(x-1)+y > 1\)
      • No ka \(y_1 \neq 1\) a me \(y_2 \neq 1\) :
        \(f(x_1, y_1) = f(0, y_1) = y_1 \leq k\)
        \(f(x_2, y_2) = f(x_2, y_2) = (k+1) + (k-1)(x-1) + (y-2) > k\)
    • Hihia 1b: \( x_2 \geq k \)
      • No \(y_1 = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\) a me \(y_2 = 1\) iā mākou:
        \(f(x_1, y_1) = f(0, \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\)
      • No ka \(y_1 \neq \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\) a me \(y_2 = 1\) penei:
        \(f(x_1, y_1) = f(0, y_1) = y_1 \neq \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1\)
      • No ka \(y_2 \neq 1\) :
        \(f(x_1, y_1) = f(0, y_1) = y_1 \leq k\)
        \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1)\)
        \(+ (k+1) + (k-1)(y_2-2) \geq (k+1)+(k-1)(y_2-2) > k \)
  • 2 hihia: \( 0 < x_1 < k \)
    • Hihia 2a: \( 0 < x_2 < k \)
      • No \(y_1 = 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = f(x_1, 1) = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 = 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1 = 1\)
      • No \(y_1 \neq 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = f(x_1, y_1) = (k+1)+(k-1)(x_1-1)+(y_1-2) > 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1 = 1\)
      • No \(y_1 = 1\) a me \(y_2 \neq 1\) :
        \(f(x_1, y_1) = f(x_1, 1) = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 = 1\)
        \(f(x_2, y_2) = f(x_2, y_2) = (k+1)+(k-1)(x_2-1)+(y_2-2) > 1\)
      • No ka \(y_1 \neq 1\) a me \(y_2 \neq 1\) :
        \(f(x_1, y_1) = (k+1)+(k-1)(x_1-1)+(y_1-2) \leq\)
        \((k+1)+(k-1)(x_1-1)+(k-2)\)
        \(f(x_2, y_2) = (k+1)+(k-1)(x_2-1)+(y_2-2) \geq\)
        \((k+1)+(k-1)((x_1+1)-1)+(y_2-2) =\)
        \((k+1)+(k-1)(x_1-1) + (k-1) + (y_2-2) \geq\)
        \((k+1)+(k-1)(x_1-1) + (k-1) + (2-2) \geq\)
        \((k+1)+(k-1)(x_1-1) + (k-1) > (k+1)+(k-1)(x_1-1) + (k-2)\)
    • Hihia 2b: \( x_2 \geq k \)
      • No \(y_1 = 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = f(x_1, 1) = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 = 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1 \geq \left\lfloor \frac{k-1}{k-1} \right\rfloor + 1 = 2 > 1\)
      • No \(y_1 = 1\) a me \(y_2 \neq 1\) :
        \(f(x_1, y_1) = f(x_1, 1) = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 = 1\)
        \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1)\)
        \(+ (k+1) + (k-1)(y_2-2) \geq (k+1) + (k-1)(y_2-2) > 1\)
      • No \(y_1 \neq 1\) a me \(y_2 = 1\) :
        \(f(x_1, y_1) = \left( \left((m_1-1)(k-1)+x_1\right)-1+ \left( (m_1-2)(y_1-2) \right) \right) \% (k-1)\)
        \(+ (k+1) + (k-1)(y_1-2) \geq (k+1) + (k-1)(y_1-2) > 1\)
        \(f(x_2, y_2) = f(x_2, 1) = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor + 1 = 1\)
      • No ka mea \(y_1 \neq 1\) a \(y_2 \neq 1\) ʻo ia:
        \((k+1) + (k-1)(x_1-1) + (y_1-2) =\)
        \(\left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1)\)
        \(+ (k+1) + (k-1)(y-2)\)
        \(\Leftrightarrow y_1 = (k-1)y_2 - (k-1)(x_1+1) +\)
        \(\left( 2 + \left( \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1) \right) \right) \)
        No ka mea \(y_2 = x_1+1\) me \( 2 \leq y_2 \leq k\) ʻo ia
        \(y_1 = 2 + \left( \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1) \right)\) me \( 2 \leq y_1 \leq k\).
        Hoʻokahi wale nō hoʻonā maʻaneʻi \( (y_1, y_2) \).
        No ka mea, koho mākou \(y^*_2=y_2-1\) e like me ka waiwai, he \(y^*_1 = y_1-(k-1) < 2\).
        Eia hou, no ka \(y^*_2*=y_2+1\) alaila \(y^*_1 = y_1+(k-1) > k\).
  • 3. Ka hihia: \( x_1 \geq k \)
    • Hihia 3a: \( x_2 \geq k \)
      • Hihia 3a': \(m_1 = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor +1 = \left\lfloor \frac{x_2-1}{k-1} \right\rfloor +1 = m_2\)
        • No \(y_1 = 1\) a me \(y_2 = 1\) :
          \(f(x_1, y_1) = f(x_1, 1) = m_1\)
          \(f(x_2, y_2) = f(x_2, 1) = m_2 = m_1\)
        • No \(y_1 = 1\) a me \(y_2 \neq 1\) :
          \(f(x_1, y_1) = f(x_1, 1) = m_1 = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 \leq \left\lfloor \frac{((k-1) \cdot k)-1}{k-1} \right\rfloor + 1 =\)
          \(\left\lfloor k - \frac{1}{k-1} \right\rfloor + 1 = (k - 1) + 1 = k\)
          \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_2-2) \geq\)
          \((k+1) + (k-1)(y_2-2) \geq (k+1) > k\)
        • No \(y_1 \neq 1\) a me \(y_2 = 1\) :
          E nana \(y_1 = 1\) a me \(y_2 \neq 1\) .
        • No ka mea \(y_1 \neq 1\) a \(y_2 \neq 1\) ʻo ia:
          \(f(x_1, y_1) = \left( \left((m_1-1)(k-1)+x_1\right)-1+ \left( (m_1-2)(y_1-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_1-2) = L_1 + (k+1) + (k-1)(y_1-2)\)
          \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_2-2) = L_2 + (k+1) + (k-1)(y_2-2)\)
          A laila \(f(x_1, y_1) = f(x_2, y_2) \Leftrightarrow\)
          \(L_1 + (k+1) + (k-1)(y_1-2) = L_2 + (k+1) + (k-1)(y_2-2) \Leftrightarrow\)
          \(L_1 + (k-1)(y_1-2) = L_2 + (k-1)(y_2-2) \Leftrightarrow\)
          \(L_1 - L_2 = (k-1)(y_2-y_1)\)
          No ka mea \(y_1 \neq y_2\) ʻo ia \(L_1-L_2 \leq (k-2 - 0) = k-2 < (k-1)(y_2-y_1)\).
          No ka mea \(y_1 = y_2\) ʻo ia \(L_1 - L_2 = 0 \Leftrightarrow L_1 = L_2\) a
          \(\left( \left((m_1-1)(k-1)+x_1\right)-1+ \left( (m_1-2)(y_1-2) \right) \right) \% (k-1) =\)
          \(\left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1) \Leftrightarrow\)
          \(x_1 = x_2 + (k-1)\cdot l\) kue i \(m_1 = m_2\).
      • Hihia 3a'': \(m_1 = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor +1 \neq \left\lfloor \frac{x_2-1}{k-1} \right\rfloor +1 = m_2\)
        • No \(y_1 = 1\) a me \(y_2 = 1\) :
          \(f(x_1, y_1) = f(x_1, 1) = m_1\)
          \(f(x_2, y_2) = f(x_2, 1) = m_2 \neq m_1\)
        • No \(y_1 = 1\) a me \(y_2 \neq 1\) :
          \(f(x_1, y_1) = f(x_1, 1) = m_1 = \left\lfloor \frac{x_1-1}{k-1} \right\rfloor + 1 \leq \left\lfloor \frac{((k-1) \cdot k)-1}{k-1} \right\rfloor + 1 =\)
          \(\left\lfloor k - \frac{1}{k-1} \right\rfloor + 1 = (k - 1) + 1 = k\)
          \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_2-2) \geq\)
          \((k+1) + (k-1)(y_2-2) \geq (k+1) > k\)
        • No \(y_1 \neq 1\) a me \(y_2 = 1\) :
          E nana \(y_1 = 1\) a me \(y_2 \neq 1\) .
        • No ka mea \(y_1 \neq 1\) a \(y_2 \neq 1\) ʻo ia:
          \(f(x_1, y_1) = \left( \left((m_1-1)(k-1)+x_1\right)-1+ \left( (m_1-2)(y_1-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_1-2) = L_1 + (k+1) + (k-1)(y_1-2)\)
          \(f(x_2, y_2) = \left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \%\)
          \((k-1) + (k+1) + (k-1)(y_2-2) = L_2 + (k+1) + (k-1)(y_2-2)\)
          A laila \(f(x_1, y_1) = f(x_2, y_2) \Leftrightarrow\)
          \(L_1 + (k+1) + (k-1)(y_1-2) = L_2 + (k+1) + (k-1)(y_2-2) \Leftrightarrow\)
          \(L_1 + (k-1)(y_1-2) = L_2 + (k-1)(y_2-2) \Leftrightarrow\)
          \(L_1 - L_2 = (k-1)(y_2-y_1)\)
          No ka mea \(y_1 \neq y_2\) ʻo ia \(L_1-L_2 \leq (k-2 - 0) = k-2 < (k-1)(y_2-y_1)\).
          No ka mea \(y_1 = y_2\) ʻo ia \(L_1 - L_2 = 0 \Leftrightarrow L_1 = L_2\) a
          \(\left( \left((m_1-1)(k-1)+x_1\right)-1+ \left( (m_1-2)(y_1-2) \right) \right) \% (k-1) =\)
          \(\left( \left((m_2-1)(k-1)+x_2\right)-1+ \left( (m_2-2)(y_2-2) \right) \right) \% (k-1) \Leftrightarrow\)
          \(y = \frac{(k-1)\cdot l + (3-k)(m_2 - m_1) + (x_1 - x_2)}{m_2 - m_1}\)
          Ma laila no \(2 \leq y \leq k\) mau a \(l \in \mathbb{N}_0\), no laila
          \(m_2 - m_1 \mid (k-1)\cdot l + (3-k)(m_2 - m_1) + (x_1 - x_2)\).
          Hōʻoia: ma laila \((k-1)\) ʻo ka prime, ʻo ia (no ka lemma o Bézout)
          \((k-1)\cdot l \equiv -\left( (3-k)(m_2-m_1) + (x_1-x_2) \right) \, \mod (m_2-m_1)\)
          hiki ke hoopauia, no ka mea \(\text{ggT}\left((k-1),(m_2-m_1)\right) = 1\) Māhele \(-\left( (3-k)(m_2-m_1) + (x_1-x_2) \right)\).
          A laila ʻo kēia wale nō ka hopena \(l_1\), no ka mea, hookahi
          \(l_2 = l_1 + (m_2-m_1)\) ʻo ia \( y_2 = y_1 + (k-1) > k\).

Hiki iā ʻoe ke ʻike i ka ʻike hope hoihoi e pili ana i ke kumuhana o ka dobble a me ka makemakika ma aneʻi a ma aneʻi . Ma ka palapala aʻe hiki iā ʻoe ke ʻike i ke ʻano i hōʻoia mua ʻia i ka hana: Hiki ke hana ʻia nā Dobbles (no \((k-1)\) prim) me ke kaomi ʻana i kahi pihi:

See the Pen DOBBLE CREATOR by David Vielhuber (@vielhuber) on CodePen.

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