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This question already has an answer here:

how i can find mathematical way to know the number of triangles are in this photo?

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I sure that the solution like sequence or series but how to find it ?

if I added new line so what is the number of triangles become?

thanks for all

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marked as duplicate by Brian M. Scott, Zev Chonoles, Jared, Micah, Amzoti Jun 13 '13 at 23:37

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ @BrianM.Scott :sorry I dont know so delete my question or what ? $\endgroup$ – mhd.math Jun 13 '13 at 23:17
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    $\begingroup$ It’s okay to leave it. It’s likely to be closed, but that doesn’t hurt anything, and you might get an answer that you like better than the ones at the other question (though they were pretty comprehensive). $\endgroup$ – Brian M. Scott Jun 13 '13 at 23:21
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Any triangle either has a top most vertex, or a bottom most vertex.

First, count triangles with a top most vertex. Starting from the vertex at the top we have $6$ possible triangles. The two vertices in the second row each have $5$ possible triangles, and so on.

The pattern continues:

$$1\cdot6+2\cdot5+3\cdot4+4\cdot3+5\cdot2+6\cdot1=56$$

Next, count triangles with a bottom most vertex. Starting with the vertices at the bottom, there are $1+2+3+2+1=9$ such triangles. In the second row from the bottom, there are $1+2+2+1=6$ such triangles. Continuing we get $1+2+1=4$ and $1+1=2$ for the next two rows, finishing with $1$ near the top. This gives us $22$ triangles with a lower most vertex.

Altogether, we have $78$ triangles.

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  • $\begingroup$ Are the individual triangles counted in? $\endgroup$ – SJuan76 Jun 13 '13 at 23:33
  • $\begingroup$ @SJuan76: They are. See if you can follow my reasoning to see that all of the individual triangles are included in the count. $\endgroup$ – Jared Jun 13 '13 at 23:35
  • $\begingroup$ Ok, I see, of the top vertex one of them is with a "height" of 1, and the same with the other descending, account for the "up pointing" triangles, and the same for the "vertex down" triangles for the reversed triangles. thanks $\endgroup$ – SJuan76 Jun 13 '13 at 23:38

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