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The phrase "other idea" is correct and usable in written English.
You can use it when you're talking about a particular idea and want to contrast it with another one. For example, "I had one idea for how to solve the problem, but then I came up with another idea that worked even better."
Exact(59)
Remember my other idea?
The other idea is generic cigarette packaging.
"I'd got fixated on the other idea.
John Cage's "4'33" ": that is silence, a whole other idea.
"Well, along with it, you have this other idea.
Whatever they said, surely this other idea would be faster.
The other idea, which Columbia followed, was to require applicants to take an aptitude test.
JUDGE: We say that because we don't have any other idea about them.
But, the assistant responded, you had so many questions about my other idea.
The other idea is that you are exchanging one form of pain for another".
Similar(1)
Other ideas surface regularly.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com