### 4 integers that can be combined with + or - to make the numbers 1 through 40 - SOLUTION

There are four positive integers, *a*, *b*, *c*, *d*, which can be combined using plus or minus to get all the integers from 1 to 40 inclusive. These combinations of *a*, *b*, *c*, and *d* can involve any number of *a*, *b*, *c*, and *d*, but none of these four integers can be used more than once. That is, some possible combinations are:

However, these types of combinations are not allowed:

What are the four integers?

#### Solution

Much like the consecutive powers of 2 (from 2^0 to 2^n) can be combined in different
ways with addition to get all possible integers from 1 to (2^(n+1) - 1),
the consecutive powers of 3 (from 3^0 to 3^n) can be combined using addition and
subtraction to get all the possible integers from 1 to (3^0 + 3^1 + ...
+ 3^n). The four integers are 1, 3, 9, and 27.

Ken Ho's home page