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My daughter got this question and I cannot solve it - or even give her direction. It appears there in not enough information.

the number of equilateral triangles of side 1 into which an equilateral triangle of side n can be divided? ( n is a whole number)

@kovenlo what is her level? The answer would be $n^2$. But do you need a proof? — , Aug 29 '15 at 22:52
to the left is a picture of one equilateral trianangle and then another that is 4 equilaterals making one triangle and then another equil triangle with 9.... I guess that is the n2 1^2 = 1 2^2 = 4 3^2=9..... — kovenlo, Aug 29 '15 at 23:05

Peter Smith · Answer 1 · 2015-08-29T23:07:58.337

Your daughter might like to think about two ways of tackling this ...

Approach A

Draw a triangle of side 2, and fit four unit triangles into it.

Extend that picture to a triangle of side 3: how many new unit triangles of side one can you fit into the newly added strip (the trapezoid 2 units along the top, 3 along the bottom)? So how many triangle fit into the whole triangle?

Extend that picture to a triangle of side 4: how many new unit triangles of side one can you fit into the newly added strip this time? So how many in the whole triangle?

What's the pattern?

Approach B

Think about areas. What is area of a square of side 2 compared with the area of a unit square? What is area of an equilateral triangle of side 2 compared with the area of a unit triangle? Why can you know the ratio of the triangles without calculating the actual areas?

What is area of an equilateral triangle of side $n$ compared with the area of a unit triangle?

I guess we got lost in the terminology - the use of 1 side triangle meant a single triangle - so they wanted to know how many can fit into a n=th sided triangle - n2 - THANKS ALL — kovenlo, Aug 29 '15 at 23:18

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