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Reachable Numbers

Problem 259

Published on Saturday, 10th October 2009, 01:00 pm; Solved by 995; Difficulty rating: 70%

A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:

  • Uses the digits 1 through 9, in that order and exactly once each.
  • Any successive digits can be concatenated (for example, using the digits 2, 3 and 4 we obtain the number 234).
  • Only the four usual binary arithmetic operations (addition, subtraction, multiplication and division) are allowed.
  • Each operation can be used any number of times, or not at all.
  • Unary minus is not allowed.
  • Any number of (possibly nested) parentheses may be used to define the order of operations.

For example, 42 is reachable, since (1/23) * ((4*5)-6) * (78-9) = 42.

What is the sum of all positive reachable integers?