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# help

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Find the number of equilateral triangles in the grid. Dec 18, 2019

#1
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We classify the types of equ-triangles starting with biggest to smallest.

The biggest is obviously the equ-triangle with a side length of 5, the smallest equ-triangle has a side length of 1.

Count the number of equ-triangles for each class. The first one has already been filled out for you.

 Side Length Number of $$\Delta$$ 5 1 4 ? 3 ? 2 ? 1 ?

Now once you complete the table, just find the total up the number of equ-triangles for the table.

Calcuser not logged in

Dec 18, 2019
#2
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and don't forget the upside-down triangles

Guest Dec 18, 2019
#3
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yes! remember that! very important.

CalculatorUser  Dec 18, 2019
#4
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Find the number of equilateral triangles in the grid ($$n=5$$).

$$a(n) = \Big\lfloor \dfrac{n(n+2)(2n+1)}{8} \Big \rfloor$$

Source:  http://oeis.org/A002717

$$\begin{array}{|rcll|} \hline a(5) &=& \Big\lfloor \dfrac{5(5+2)(2*5+1)}{8} \Big\rfloor \\ &=& \Big\lfloor \dfrac{5*7*11}{8} \Big\rfloor \\ &=& \lfloor 48.125 \rfloor \\ &=& \mathbf{48} \\ \hline \end{array}$$ Dec 18, 2019