How many triangles?

Printable View

• Jan 21st 2011, 01:20 PM
kepler
How many triangles?
Hi,

How many triangles are in this picture:

Attachment 20537

Kind regards,

Kepler
• Jan 21st 2011, 02:02 PM
undefined
Quote:

Originally Posted by kepler
Hi,

How many triangles are in this picture:

<image>

Kind regards,

Kepler

Nice problem. I might have counted wrong.

Spoiler:

There are 9 vertices/points of intersection, and consider sets of three of them. We have 3+1+2+2*C(4,3) = 14 sets that are collinear, where C(n,k) is binomial coefficient. Then consider sets of two of them. I count 6 that are not connected. Each of these 6 would have given 7 triangles, had they been connected. So I get C(9,3) - 14 - 6*7 = 28 total.
• Jan 21st 2011, 02:10 PM
kepler
Hi,

yep, that's my guess too.

Kind regards,

Kepler
• Jan 21st 2011, 03:45 PM
Archie Meade
Quote:

Originally Posted by kepler
Hi,

How many triangles are in this picture:

Attachment 20537

Kind regards,

Kepler

Counting sequentially......

If there are no internal lines, 1 triangle.
Draw the vertical line, 2 new triangles.
Draw the line from the left vertex to the right side, 5 new triangles.
Draw the line from the right vertex to the left side, 6 new triangles.
Draw the line from the left side to the mid-base, 7 new triangles.
Draw the line from the right side to the mid-base, 7 new triangles.

Total 28.
• Feb 12th 2011, 11:56 AM
beeliz
I found 34 triangles (Wink)
picture of them: Attachment 20770
• Feb 12th 2011, 12:51 PM
Archie Meade
Well done!

I miscounted.
My totals should have been....

No internal lines.....1 triangle
1 internal line........2 more
3 lines..................5 more
4 lines..................8 more
5 lines..................8 more
6 lines..................10 more
• Feb 12th 2011, 02:44 PM
tom@ballooncalculus
Quote:

Originally Posted by beeliz
I found 34 triangles (Wink)
picture of them: Attachment 20770

(Wait) 33 and 34 are duplicating your 4 and 5.

I agree with the total, though... (Rock) (Thinking)
• Feb 12th 2011, 04:06 PM
Archie Meade
Here's another counting process
• Feb 13th 2011, 01:09 PM
beeliz
Quote:

Originally Posted by tom@ballooncalculus
(Wait) 33 and 34 are duplicating your 4 and 5.

I agree with the total, though... (Rock) (Thinking)

whoooops. well I found 34 somewhere... (Thinking)

oh in the image Archie Meade posted, it's #2 in the fourth triangle and #1 in the sixth triangle that I missed in my picture. but I found them the first time I counted during a boring english class (Smile)

better picture: Attachment 20791 because I'm a perfectionist (Wink)