This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

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a)
Step 1: Determine the magic sum for a triangle using numbers to . The sum of numbers to is . In a magic triangle, each vertex number is part of two sides, and each middle-side number is part of one side. If is the magic sum for each side, and is the sum of the numbers at the vertices, then . To make divisible by , must be divisible by . The smallest possible sum for three distinct numbers from to (for vertices) is . If , then , so .
Step 2: Arrange the numbers to to achieve the magic sum. Place the numbers at the vertices. The remaining numbers are . These will be placed in the middle of the sides. For a side with vertices and , the middle number must be . For a side with vertices and , the middle number must be . For a side with vertices and , the middle number must be .
Step 3: Show the arrangement. The magic sum is . The arrangement is as follows: • Place at the top vertex. • Place at the bottom-left vertex. • Place at the bottom-right vertex. • Place on the side between and . • Place on the side between and . • Place on the side between and .
The sum of numbers on each side
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Determine the magic sum for a triangle using numbers 1 to 6. The sum of numbers 1 to 6 is 1+2+3+4+5+6 = 21.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.