# Arithmetic game with four numbers to get a value of $10$

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?

• Are you looking for formal solutions of a number like $\overline{xyzt}$ or how many different ways we can get $10$? – Deniz Tuna Yalçın Oct 6 '17 at 10:58
• are parentheses allowed ? if so you can make one number into possibly 6 different answers. – user451844 Oct 6 '17 at 11:04
• oh and with rearrangement of the digits, like in the example, there's only about 375 unique 4 digit numbers . – user451844 Oct 6 '17 at 11:10
• Must all digits be used ? Otherwise "$8+2$" would be an easier way in the example. – Peter Oct 6 '17 at 11:20
• are leading 0's allowed etc. there's tons that haven't been verified. – user451844 Oct 6 '17 at 11:22

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.