E noʻonoʻo i ʻelua poʻe \(A\) a me \(B\) ʻaʻole like ka lā hānau, kahi o \(A\) i ʻoi aku ma mua o \(B\) . Hōʻike: ʻElua mau pūʻulu makahiki \(a,b \in \mathbb{N}\) e pili ana i kēia: \(2\cdot a = b\) . Hoʻonoho mua mākou i \(d \in \mathbb{R}^+\) i ka ʻokoʻa makahiki ma waena o \(A\) a me \(B\) i ka hānau ʻana o \(A\) me \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) Ke noʻonoʻo nei mākou i ka manawa kūpono \(x \in \mathbb{R}^+\) ma hope o ka hānau ʻana o \(A\) me \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
I kēia manawa i ka manawa, e like me ka wehewehe ʻana, \(a = \lfloor x \rfloor \) a me \(b = \lfloor x+d \rfloor\) . Hoʻoholo mākou i kēia manawa \(x\) no nā mea paʻa:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
Hihia 1st: \(0 \leq x_1 + d_1 < 1\):
A laila $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
ʻO kēia ka manaʻo ʻo $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ a me $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ ka $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ makahiki mua a $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .
2 hihia: \( 1 \leq x_1 + d_1 < 2 \):
A laila $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
ʻO kēia ka manaʻo ʻo $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ a me $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ ka ʻelua mau makahiki i makemake ʻia.
Ma nā huaʻōlelo koʻikoʻi, ʻo ia hoʻi, no ka laʻana: Inā hānau kou makuahine iā ʻoe i ka makahiki \(20\) , ʻelua pololei ʻo ia i kou mau makahiki ma \(40\) a me \(42\) . He mea hoihoi nō hoʻi ka hihia o ka nui o kou makuahine i ka \(n\) mau makahiki: Maanei ʻoe i hoʻonoho ai \(n \in \mathbb{N}\) me ka ʻole a loaʻa iā \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . E hana ana kēia inā a inā wale nō ka ʻokoʻa o ka makahiki integer \( \lfloor d \rfloor = d_0 \) he helu helu o \(n-1\) . No ka laʻana, ma ka hihia i luna, ʻo kou makuahine he \(24\) makahiki \(6\) mau makahiki o kou mau makahiki.