I wrote:

The problem is that "can we cover the bigger rectangle with small rectangles such that no two rectangles overlap and no gap opens up"

Is it a correct usage of such that? Could I use "so that"? How about "in a way that"?

Is open ups the correct verb choice for gap here?

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    Another issue: With The problem is that the clausal complement of that must be a declarative; it cannot be a question. You would need to say The problem is, can we ....? such that is fine and idiomatic in this context, or so that. – Tᴚoɯɐuo Nov 8 '18 at 13:50
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    Yes - such that, so that, and in a way that (and in such a way that) are all perfectly good alternatives for your context. Or just plain so without that. It's just a stylistic choice, where it's a bit pointless to suggest any particular versions are "better" or "worse" than any others. Note that it's opens up (open ups is garbage English). – FumbleFingers Nov 8 '18 at 14:58
  • @FumbleFingers thank you, you mean ... so without two rectangles overlap or a gap opens up? – Ahmad Nov 8 '18 at 15:00
  • Certainly not! You can't just arbitrarily change no in the original to without in your rephrasing. The problem is Can we cover the bigger rectangle with small rectangles so no two rectangles overlap and no gap opens up? Note the punctuation (required because the "problem" is framed as a question). – FumbleFingers Nov 8 '18 at 16:10
  • @fumber thanks, LOL, I thought you meant so without is another alternative. Sorry! – Ahmad Nov 8 '18 at 17:02

Such that is a natural and appropriate term to use here. So that is also acceptable, and less formal. In a way that is not grammatical, though in such a way that is - and is essentially what such that means. You could also use in a way that means, though that comes across as much more informal.

"No gap opens up" will be understood, but is not idiomatic, and some might argue as to whether it is semantically correct. "There are no gaps" would be more natural.

You also need to lose the that in "The problem is that". That quotation marks have you covered. If you didn't have the quotation marks, you'd want something like "the problem is figure out if we can cover the bigger rectangle..."

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