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So there's a small game where when we see a four digit number somewhere in public, my friends and I would try to form the value of ten using these four digits with only basic operations.
E.g.
If we see a four-digit number: $5829$

a possible way is to do:
$$9 + 5 - 8 \div 2 = 10$$

When I say basic I mean operations only include "$+,\div,-,\times"$ and the digits can be used in any order as done above.

Are there any sufficient conditions to guarantee that a string of numbers can attain a value of $10$? How many solutions are there?

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  • $\begingroup$ Are you looking for formal solutions of a number like $\overline{xyzt}$ or how many different ways we can get $10$? $\endgroup$ – Deniz Tuna Yalçın Oct 6 '17 at 10:58
  • $\begingroup$ are parentheses allowed ? if so you can make one number into possibly 6 different answers. $\endgroup$ – user451844 Oct 6 '17 at 11:04
  • $\begingroup$ oh and with rearrangement of the digits, like in the example, there's only about 375 unique 4 digit numbers . $\endgroup$ – user451844 Oct 6 '17 at 11:10
  • $\begingroup$ Must all digits be used ? Otherwise "$8+2$" would be an easier way in the example. $\endgroup$ – Peter Oct 6 '17 at 11:20
  • $\begingroup$ are leading 0's allowed etc. there's tons that haven't been verified. $\endgroup$ – user451844 Oct 6 '17 at 11:22
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There are $715$ ways to select $4$ digits when repetitions are allowed. I made a program to run through all combinations of digits and operators and the result was that there are $439$ digit selections which have at least one solution. The complete list is a bit big to post here, so I'm just including a snippet from the start and one from the end.

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