What is the sum of the numbers in the shaded circles?

Question

Each of the 6 circles contains a different counting number. The sum of all 6 numbers is 21.The sum of the 3 numbers along each side of the triangle is shown in the diagram. so What is the sum of the numbers in the shaded circles?

Note that obviously the answer is easy. But what kind of ways you suggest for this problem?

Answer 1

Let $X_1,X_2,X_3$ the shaded corners. $Y_1,Y_2,Y_3$ the white ones.

$$X_1+Y_1+X_2=8$$ $$X_2+Y_2+X_3=8$$ $$X_1+Y_3+X_3=14$$ $$X_1+X_2+X_3+Y_1+Y_2+Y_3=21$$

Adding the three first equations:

$$2X_1+2X_2+2X_3+Y_1+Y_2+Y_3=30$$

And substract this equation with the fourth one:

$$X_1+X_2+X_3=9$$

Answer 2

We have $$8+8+14=({\rm sum \space of \space white \space balls})+2({\rm sum \space of \space shaded \space balls}).$$ You can take it from here.

Answer 3

9.

The numbers in the triangle are:

5

2 6

1 4 3

Two of the sides sum to 8 and one side sums to 14, and the whole thing sums to 21 as specified.

Answer 4

$$2\text{ (Sum of shaded balls)}+\text{sum of unshaded balls}=8+8+14=30 \tag1$$ A/Q $$\text{(sum of shaded balls)}+\text{sum of unshaded balls}=21 \tag2$$

Subtracting (1) & (2), Sum of shaded balls=9.