I kekahi manawa, hoʻokahi nīnau wale nō ma ka papahana ahiahi (ma kēia hihia mai ka mea hoʻolaha hanohano ʻo Kai Pflaume) ua lawa ia e hoʻololi i kahi hopena nīnau pilikia ʻole i kahi pilikia hoʻonui liʻiliʻi. ʻO ia ka mea e hana nei ma "Who Knows What?" Nīnau kumu: Ua ʻike ʻia ke ʻano, ʻaʻole i kēia manawa ka pane - akā naʻe, ua hoʻoholo mua nā pilikia i nā hopena maikaʻi.
E lawe kākou i ʻelua mau hui \(A\) a me \(B\) . Ma mua o ka nīnau hope loa, ua lanakila ka hui \(A\) i ka nui \(x_a\) , a ua lanakila ka hui \(B\) i ka nui \(x_b\) . Ke noʻonoʻo nei mākou i ka hihia.
$$
x_a > x_b > 0.
$$
Ke piliwaiwai nei nā kime i nā helu holoʻokoʻa.
$$
1 \leq y_a \leq x_a,\qquad 1 \leq y_b \leq x_b.
$$
Inā pololei ka pane, hoʻohui ʻia ka nui i hoʻopaʻa ʻia; inā hewa ka pane, unuhi ʻia. No nā hopena ʻehā, ʻo kēia nā helu hope loa.:
$$
\begin{array}{c|c|c}
\text{Fall} & A & B\\
\hline
A \text{ richtig}, B \text{ richtig} & x_a+y_a & x_b+y_b\\
A \text{ richtig}, B \text{ falsch} & x_a+y_a & x_b-y_b\\
A \text{ falsch}, B \text{ richtig} & x_a-y_a & x_b+y_b\\
A \text{ falsch}, B \text{ falsch} & x_a-y_a & x_b-y_b
\end{array}
$$
Ma ka hōʻike, alakaʻi kahi paʻa i kahi nīnau kuhi. No laila, ʻike mau ʻia ia ma ka matrix ma ke ʻano he hihia kaʻawale "=". No nā waiwai pakeneka o ka hiki ke lanakila, helu mākou i nā lanakila pololei ma nā hihia liʻiliʻi ʻehā like. ʻO kahi ʻano e like me \(|A|B|A|A|\) e loaʻa ai ʻekolu lanakila pololei no ka Hui \(A\) a hoʻokahi no ka Hui \(B\) . ʻAʻole helu ʻia kahi paʻa ma ke ʻano he lanakila pololei no kekahi hui. He mea koʻikoʻi kēia helu pololei; ʻaʻole lawa ka helu wale ʻana i kahi cell holoʻokoʻa he "uliuli" a "ʻulaʻula" paha.
Aia i loko o kēia kumu hoʻohālike kahi kuhi: Hoʻomaopopo like mākou i nā hui pane ʻehā me he mea lā like ka hiki. No laila, ʻaʻole ia e pili ana i ka ʻike maikaʻi ʻana o ka Hui \(A\) a i ʻole ka Hui \(B\) i ka māhele, akā e pili ana i ke ʻano hana i hoʻohana ʻia ma mua o ka pane ʻana.
E
$$
d=x_a-x_b.
$$
A laila \(d>0\) ka pōmaikaʻi o ka hui \(A\) . ʻO ka nīnau i kēia manawa: He aha ka stake kūpono?
Hiki ke helu ʻia ka matrix piha o nā pili hiki ke hana ʻia me ka ikaika.:
Ka manaʻo o ka Hui A
E nānā mua mākou i ka wā e lanakila ai ka Hui \(A\) me nā pili paʻa \(y_a\) a me \(y_b\) .
ʻO ka hihia
$$
A \text{ richtig}, B \text{ falsch}
$$
Hele mau ia i ka Hui \(A\) , no ka mea
$$
x_a+y_a > x_b-y_b
$$
Pili pono kēia no ka mea \(x_a>x_b\) a me \(y_a,y_b>0\) .
No nā hihia ʻekolu ʻē aʻe, loaʻa iā mākou:
$$
\begin{array}{c|c}
\text{Fall} & A \text{ gewinnt genau dann}\\
\hline
A \text{ richtig}, B \text{ richtig} & x_a+y_a>x_b+y_b\\
A \text{ falsch}, B \text{ richtig} & x_a-y_a>x_b+y_b\\
A \text{ falsch}, B \text{ falsch} & x_a-y_a>x_b-y_b
\end{array}
$$
Me \(x_a=x_b+d\) lilo kēia:
$$
\begin{array}{c|c}
\text{Fall} & A \text{ gewinnt genau dann}\\
\hline
A \text{ richtig}, B \text{ richtig} & d+y_a>y_b\\
A \text{ falsch}, B \text{ richtig} & d-y_a>y_b\\
A \text{ falsch}, B \text{ falsch} & d-y_a>-y_b
\end{array}
$$
No laila:
$$
\begin{array}{c|c}
\text{Fall} & A \text{ gewinnt genau dann}\\
\hline
A \text{ richtig}, B \text{ richtig} & y_b<y_a+d\\
A \text{ falsch}, B \text{ richtig} & y_b<d-y_a\\
A \text{ falsch}, B \text{ falsch} & y_b>y_a-d
\end{array}
$$
He mea koʻikoʻi ke ʻano helu i kēia manawa. Ma mua, hiki ke hoʻowalewale ʻia kekahi e loiloi i kēlā me kēia cell ma muli wale nō o ka loaʻa ʻana o nā hihia \(A\) ma mua o \(B\) . Eia nō naʻe, he maʻalahi loa kēia no ka helu ʻana i ka hiki ke lanakila. ʻO nā hihia liʻiliʻi ʻehā he mau hanana like ia. No laila, ʻaʻole helu ʻia \(|A|B|A|A|\) lanakila hoʻokahi no \(A\) , akā, he ʻekolu mau hihia liʻiliʻi i lanakila no \(A\) .
No kahi stake paʻa \(y_a\) o ka hui \(A\) no laila ke hoʻohui nei mākou i nā komo pākahi \(A\) i nā hihia ʻehā ma luna o nā stake āpau \(y_b=1,2,\ldots,x_b\) .
ʻO ka hihia " pololei \(A\) , hewa \(B\) " e hele mau i ka hui \(A\) . Ua loaʻa mua kēia i nā hihia liʻiliʻi \(x_b\) i lanakila.
ʻO ka hopena o nā helu ma lalo nei no nā hihia ʻē aʻe ʻekolu.:
$$
\begin{aligned}
N_1(y_a)&=\min(x_b,d+y_a-1),\\
N_3(y_a)&=\min(x_b,\max(0,d-y_a-1)),\\
N_4(y_a)&=\begin{cases}
x_b, & y_a\leq d,\\
\max(0,x_b-y_a+d), & y_a>d.
\end{cases}
\end{aligned}
$$
ʻO ke ʻano kēia o ka helu o nā hihia liʻiliʻi i lanakila ʻia e ka Hui \(A\)
$$
N_A(y_a)=x_b+N_1(y_a)+N_3(y_a)+N_4(y_a).
$$
ʻO ka likelika kūpono o ka lanakila ʻana
$$
P_A(y_a)=\frac{N_A(y_a)}{4x_b}.
$$
ʻOiai ʻaʻole i hoʻokomo ʻia kahi paʻa ma ʻaneʻi, \(P_A\) ka likelika o ka lanakila pololei ʻana i ka nīnau haku (me ka ʻole o kahi nīnau kuhi).
Hoʻololi iki kēia helu pololei i ka optimum i hoʻohālikelike ʻia me ka hapa nui o nā cell. No ka Hui \(A\) ʻo nā wahi noi kūpono ma lalo nei ka hopena.:
$$
\boxed{
\begin{cases}
1\leq y_a\leq2, & x_b=1,\ d=2,\\
d\leq y_a\leq x_b-d+1, & 2d\leq x_b+1,\\
1\leq y_a\leq d, & 2d=x_b+2,\\
1\leq y_a\leq \max(1,x_b-d+1,d-x_b-1), & 2d>x_b+2.
\end{cases}
}
$$
ʻO nā pili āpau ma kēia wahi e hoʻonui i ka hiki ke lanakila o ka Hui \(A\) Inā makemake ʻoe e pili i ka nui loa ma waena o nā pili like maikaʻi, pono ʻoe e hoʻohana mau i ka lihi ʻākau o ka wahi.
He laʻana:
$$
x_a=30,\qquad x_b=22.
$$
A laila
$$
d=x_a-x_b=8.
$$
Aia ma laila
$$
2d=16\leq 23=x_b+1
$$
Hoʻopili ʻia ka laulā kūpono.
$$
8\leq y_a\leq 15.
$$
No laila, ʻo ka hoʻohana kūpono loa
$$
\boxed{y_a=15}.
$$
ʻO ke ʻano kahiko o ka noʻonoʻo ʻana i nā cell holoʻokoʻa e hōʻike ana i ka pae \(9\leq y_a\leq 14\) . Hōʻike pololei ka helu hihia hapa i nā waiwai palena ʻelua \(8\) a me \(15\) he kūpono hoʻi.
Ka manaʻo o ka Hui B
I kēia manawa, ke noʻonoʻo nei mākou i ke kūlana like mai ka manaʻo o ka hui ma hope \(B\) . Maanei hoʻi, ʻaʻole mākou e helu hou i nā cell holoʻokoʻa wale nō, akā, ʻo nā komo pākahi \(B\) i nā hihia liʻiliʻi ʻehā.
ʻO ka hihia
$$
A \text{ richtig}, B \text{ falsch}
$$
E eo mau ana ka Hui \(B\) Ma nā hihia ʻekolu i koe, loaʻa i ka Hui \(B\) ka helu o nā hihia i lanakila ʻia no kahi stake paʻa \(y_b\) ke hoʻohui ʻia ma luna o \(y_a=1,2,\ldots,x_a\):
$$
\begin{aligned}
M_1(y_b)&=\max(0,y_b-d-1),\\
M_3(y_b)&=x_a-\max(0,d-y_b),\\
M_4(y_b)&=\max(0,x_b-y_b).
\end{aligned}
$$
Pela no
$$
N_B(y_b)=M_1(y_b)+M_3(y_b)+M_4(y_b)
$$
a me ka hiki ke lanakila e pili ana
$$
P_B(y_b)=\frac{N_B(y_b)}{4x_a}.
$$
No \(y_b\leq d\) e hoʻomaʻalahi kēia i
$$
N_B(y_b)=2x_b.
$$
No \(y_b>d\) loaʻa wale nō kekahi
$$
N_B(y_b)=2x_b-1.
$$
ʻO kēia ka wahi kūpono loa no ka Hui \(B\)
$$
\boxed{
1\leq y_b\leq \min(d,x_b).
}
$$
He hoʻoponopono koʻikoʻi kēia i hoʻohālikelike ʻia me ke ʻano hana hapa nui o ke kelepona cruder: ʻAʻole pono ka bet kūpono no ka Hui \(B\) he \(1\) wale nō. No ka laʻana, inā ʻoi aku ka Hui \(A\) me \(d=8\) a hiki i ka Hui \(B\) ke pili ma ka nui \(22\) , a laila kūpono nā bet āpau no \(B\) e pili ana i ka hiki ke lanakila.:
$$
1\leq y_b\leq 8.
$$
Ua like nō ka manaʻo: ʻAʻole pono ka hui ma hope e pili i ke kiʻekiʻe nui ʻole. ʻOiai ʻo nā pili kiʻekiʻe loa e hoʻomaikaʻi i nā hiʻohiʻona pākahi, hoʻonui lākou i nā mea ʻē aʻe. I ka wā e ʻoi aku ai \(y_b\) ma mua o ka hemahema \(d\) , e nalowale ana ka hui \(B\) i kahi hiʻohiʻona holoʻokoʻa. No laila, pili ka hui alakaʻi ma ke ʻano, ma nā pili āpau a ka hoa paio, lanakila lākou i nā hiʻohiʻona pākahi he nui e like me ka hiki.
ʻAʻole pono ka mea hahai e pili pono i hoʻokahi euro, akā ma ka nui e like me ka hemahema. No laila, ʻo ka nīnau nui he laʻana maikaʻi ia o ka nui o ke kumumanaʻo pāʻani i loko o kahi lula nīnau maʻalahi: ʻo ka mea nui ʻaʻole wale ka cell e pau i ka uliuli a ʻulaʻula paha, akā ehia o nā hihia ʻehā i loko o kēlā cell i lanakila maoli.