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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?

enter image description here

Mazdak
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4 Answers4

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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.

Indrayudh Roy
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  • @bluebird7: this answer looks good to me. – TonyK Jul 07 '14 at 17:06
  • see the latest answer ! – Mazdak Jul 07 '14 at 17:11
  • @bluebird7: 'the latest answer': Do we have to work out which one that is, based on the time of you comment? I'd rather you just told us which answer you mean. – TonyK Jul 07 '14 at 17:24
  • @bluebird7 It's not false but it's not an answer, really, more like a hint. He's telling you that summing the sides overcounts the shaded balls (doublecounts them, to be more specific). – Lisa Jul 07 '14 at 21:22
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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:

$$X_1+X_2+X_3=9$$

rlartiga
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1

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.

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

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

Anunoy Chakraborty
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