Puzzle and Solution: Making 30 with Numbers 1 to 9

In the world of puzzles and number games, there are countless challenges that can test one's mathematical skills and creativity. One such intriguing puzzle involves using the numbers 1 through 9 with plus or minus signs to achieve a sum of 30. While it may seem impossible at first, it's a problem that can be approached systematically to find a solution.

The Challenge

The question at hand is:

You have the numbers 1 2 3 4 5 6 7 8 and 9. How would you add plus or minus signs to make 30?

The challenge lies in the right utilization of the plus and minus signs to manipulate the sum of the given digits correctly. Let's explore how this puzzle can be solved through a step-by-step approach.

Initial Approach

One possible solution to this problem can be found by brute force, which typically involves trial and error, and can be time-consuming. Here is one solution:

1 2 3 4 5 6 7 8 - 9 30

Let's break this down:

The sum of 1 2 3 4 5 6 7 8 is 36. Subtracting 9 from 36 gives 27, which is not equal to 30.

Correct Solution

While the initial attempt did not work, a correct solution can be found by carefully adjusting the signs. Here’s a more accurate representation:

1 2 3 4 5 6 7 8 - 9 30

However, this is still incorrect. Let's explore a different approach.

Step-by-Step Solution

To find the correct solution, we can use a method where all the signs are initially set to plus, resulting in the sum of 123456789, which equals 45. We then need to adjust the signs to reduce this sum to 30. It's important to note that changing a sign from plus to minus affects the total sum by twice the value of that digit. For example, changing 2 to -2 subtracts 4 from the total sum.

Systematic Approach

Start with the sum of 1 2 3 4 5 6 7 8 9, which is 45. Subtract 15 to reach 30. Since 15 is an odd number and cannot be the sum of two identical digits, we need to find another combination. Realize that the solution involves subtracting 6 twice, as follows: 1 2 3 4 5 6 7 8 9 - 6 - 6 30

Explanation of Solution

The correct solution is:

1 2 3 4 5 6 7 8 9 - 6 30

This can be verified step by step:

Add all the numbers: 1 2 3 4 5 6 7 8 9 45. Subtract 18 (6 12) from 45: 45 - 18 27. Subtract the remaining 6: 27 - 6 30.

Alternative Approach

Another way to approach this is by setting all the signs to plus initially:

1 2 3 4 5 6 7 8 9 45. Subtract 15 to reach 30. The following combination works:

1 2 3 4 5 6 7 8 9 _______
 9  8  7  6  -  3 × 4  2 × 1 5   30
A. 9 8 7 6  30 30
B. 3 × 4  12 12
C. 2 × 1 5 12
D. 30 — 3 × 4  12 2 × 1 5  12  30
E. 30 — 12  12  30

Following the provided steps:

9 8 7 6 30 30 3 × 4 12 12 2 × 1 5 12 30 — 3 × 4 12 2 × 1 5 12 30 30 — 12 12 30

This approach confirms that subtracting the appropriate values correctly brings the total to 30. Thus, a solution to the puzzle is achieved.

For a precise solution, the arrangement should be: