Question

1. the digits from 1 to 9 should be placed in the circles so that the numbers on each side add up to 21. find the sum of the numbers in the vertices.

Explanation:

Step1: Define sum of all digits

The sum of digits from 1 to 9 is given by the formula $\sum_{n = 1}^{9}n=\frac{9\times(9 + 1)}{2}=45$.

Step2: Analyze the sum of the three - side totals

Since the sum of the numbers on each side of the triangle should add up to 21, and there are 3 sides, the sum of the three - side totals is $21\times3 = 63$. But when we add up the three - side totals, the numbers at the vertices are counted twice.

Step3: Find the sum of the vertices

Let the sum of the numbers at the vertices be $x$. Then $45+x=63$, so $x=63 - 45=18$.

Step4: Try combinations for vertices

The possible combinations of three distinct numbers from 1 to 9 that sum to 18 are: (3, 6, 9), (4, 5, 9), (4, 6, 8), (5, 6, 7).

Step5: Fill in the remaining numbers

After choosing a combination for the vertices, we can fill in the remaining numbers to meet the side - sum requirement. For example, if the vertices are 3, 6, 9:

• For the side with 3 and 6, the middle number is $21-(3 + 6)=12$, and we can use numbers from the remaining set to complete the sides.

After trial - and - error, one possible arrangement is: vertices are 3, 7, 8.
The completed triangle:
Top - left vertex: 3, middle number on the left - hand side: 4, bottom - left vertex: 7
Bottom - left vertex: 7, middle number on the bottom side: 5, bottom - right vertex: 9
Bottom - right vertex: 9, middle number on the right - hand side: 2, top - right vertex: 8

The sum of the numbers at the vertices is 18.

Answer:

18