Question

I need 23, 28, and 40 desperately

Take the digits 5, 4, 3, 2 and 1, in that order. Using those digits and the four arithmetic signs — plus, minus, times and divided by — you can get 1 with the sequence 5 - 4 + 3 - 2 - 1. You can get 2 with the sequence (5 - 4 + 3 - 2) x 1.

The question is ... how many numbers from 1 to 40 can you get using the digits 5, 4, 3, 2, and 1 in that order along with the four arithmetic signs?

You can group digits with parentheses, as in the example. There are no tricks to this, though. It's a straightforward puzzle. How many numbers from 1 to 40 can you get — and, specifically, what number or numbers can you not get?

Answer 1

9514 1404 393

Answer:

• (5 +4 +3)×2 -1 = 23
• 5 +(4×3×2) -1 = 28
• 5×(4 +3 +2 -1) = 40

Explanation:

The above expressions give the values you requested. I was able to form all integers 1-40 except 39.

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Additional comment

I used a computer program to evaluate different sequences of operations. Attached is one result for each integer 1-40 except 39. The notation here is postfix, but is not difficult to translate to infix with parentheses. Starting with the left-most operation (op), the pattern (a, b, op) → (a op b) is used. For example, the listing for 38 is 543+*21++, so this becomes 5*(4+3)+(2+1) = 38.