— Burt Kaliski Jr. (@modulomathy) June 27, 2012
For simplicity, let’s start with the assumption that “from the digits” means that the only numbers input to the equation are the single digits 1, 7 and 9 — not numbers consisting of two or more digits such as 17. Furthermore, let’s assume that parentheses are not allowed. Neither requirement is stated in the problem, but we should at least try this simple case first.
Without parentheses, multiplication will have priority over addition, so our equation will have the form
a + b + c + d + e
where one or two of the values a, b, c, d, e include a multiplication operation. These one or two values can each be 7, 9, 49, 63, or 81; anything larger (e.g., 7*7*7 = 343) would be too large for the target sum of 179.
If there’s only one multiplication result, then given its maximum of 81, the remaining four values would have to add to 98, which is too much for four single digits.
So there must be two multiplication results. The remaining three values will add to at most 27, so the sum of the two multiplication results will need to be at least 152. The only possibility in this case is for both multiplication results to be 81, producing a sum of 162.
The remainder of 17 can be covered by three single digits as 1 + 7 + 9.
Answer: 179 = 9*9 + 9*9 + 1 + 7 + 9