$8$ Stars, $8$ Bars and $8$ Pars

Number Theory Level 2

How many distinct possible ways are there to express $1000$ as the sum of all positive integers in decreasing order (but not strictly), using only $8$ 's?

Assumption: The sum can include terms that concatenate more than one $8$ . For instance, $88$ , $888$ and $8888$ .

Inspiration.

11

14

10

15

9

or less

16

or more

12

13

1 solution

Sathvik Acharya
Mar 20, 2021

We need to represent $1000$ as the sum of positive integers that only contain $8$ in their decimal expansion. This is equivalent to finding the number of non-negative integer solutions to the equation, $888x+88y+8z=1000\;\; \Leftrightarrow\;\; 111x+11y+z=125$ Since $x,y,z\ge 0$ , $\begin{aligned} x\le \left\lfloor \dfrac{125}{111}\right\rfloor &\implies x\in \{0,1\} \\ y\le \left\lfloor \dfrac{125}{11}\right\rfloor &\implies y\in \{0,1,2,\cdots, 10,11\} \\ z\le \left\lfloor \dfrac{125}{1}\right\rfloor &\implies z\in\{0,1,2,\cdots ,124,125\} \end{aligned}$ Case 1: $\;\;x=0 \implies 11y+z=125$

Note, $z=125-11y$ is a non-negative integer for all $y\in \{0,1,2,\cdots, 10,11\}$ . Hence, this case leads to $12$ possible solutions.

Case 2: $\;\;x=1 \implies 11y+z=14$

Note, $z=12-11y$ is a non-negative integer for all $y\in \{0,1\}$ . Hence, this case leads to $2$ possible solutions.

Therefore, there are $12+2=\boxed{14}$ possible solutions to the given equation.

List of all possible ways to represent $1000$ using only $8$ 's :

$\begin{aligned} 1000&=\underbrace{8+\cdots+8}_{\text{125}} \\ &=\underbrace{88}_{\text{1}}+\underbrace{8+\cdots+8}_{\text{114}} \\ &=\underbrace{88+88}_{\text{2}}+\underbrace{8+\cdots+8}_{\text{103}} \\ &=\underbrace{88+\cdots+88}_{\text{3}}+\underbrace{8+\cdots+8}_{\text{92}} \\ &=\underbrace{88+\cdots+88}_{\text{4}}+\underbrace{8+\cdots+8}_{\text{81}} \\ &=\underbrace{88+\cdots+88}_{\text{5}}+\underbrace{8+\cdots+8}_{\text{70}} \\ &=\underbrace{88+\cdots+88}_{\text{6}}+\underbrace{8+\cdots+8}_{\text{59}} \\ &=\underbrace{88+\cdots+88}_{\text{7}}+\underbrace{8+\cdots+8}_{\text{48}} \\ &=\underbrace{88+\cdots+88}_{\text{8}}+\underbrace{8+\cdots+8}_{\text{37}} \\ &=\underbrace{88+\cdots+88}_{\text{9}}+\underbrace{8+\cdots+8}_{\text{26}} \\ &=\underbrace{88+\cdots+88}_{\text{10}}+\underbrace{8+\cdots+8}_{\text{15}} \\ &=\underbrace{88+\cdots+88}_{\text{11}}+\underbrace{8+\cdots+8}_{\text{4}} \\ &=\underbrace {888}_{1}+\underbrace{8+\cdots+8}_{\text{14}} \\ &=\underbrace {888}_{1}+\underbrace{88}_{\text{1}}+\underbrace{8+\cdots+8}_{\text{3}} \\ \end{aligned}$

Sathvik Acharya - 2 months, 3 weeks ago

8 8 8 Stars, 8 8 8 Bars and 8 8 8 Pars

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$8$ Stars, $8$ Bars and $8$ Pars