From an old book I found the following question.
Use the digits $1,2,3,4,5,6,7,8,9$ and the operations $"+,-,×,÷"$ with $( )$ for construct the result $100.$
During the computations the order of $123456789$ cannot be broken.
For example $$1+2+3+4+5+6+7+(8×9)=100$$ They have given only this example. I have found that $$1×(-2+3+4)+(5×6)+(7×8)+9=100$$ $$(1×2×3)-(4×5)+(6×7)+(8×9)=100$$
It is easy to prove that number of ways of constructing $100$ is finite. But find all possible constructing ways seems like a hard task.
Here I am curious about what are the positive integers constructable using only the previous operations and above sequence. For example $1$ and $10$ are constructable, because $$1+2-3-4+5+6-7-8+9=1$$ $$1×(2+3+4+5-6-7+9)=10.$$
is the largest constructable number (in the above sense).
We cannot construct all the numbers between $1$ and $362881.$ For an example we cannot construct
How many constructable numbers in between $1$ and $362881$?
Without computing can we find all of constructable numbers?
For a given integer $n$ between $1$ and $362881,$ can we determine it is constructable or not?
There are similar questions about the $123456789$ sequence. But they did not answer for my questions.