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Find the number of equilateral triangles in the grid.

 

 Dec 18, 2019
 #1
avatar+2551 
+1

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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+1

 

and don't forget the upside-down triangles

Guest Dec 18, 2019
 #3
avatar+2551 
+1

yes! remember that! very important.

CalculatorUser  Dec 18, 2019
 #4
avatar+23809 
+1

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

 

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

 

laugh

 Dec 18, 2019

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