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        riddles >> medium >> how many triangles can a human distinguish
        
(Message started by: TheoH54 on May 24^th, 2012, 7:16am)

Title: how many triangles can a human distinguish
Post by TheoH54 on May 24^th, 2012, 7:16am

Suppose the human eye can distinguish a difference of one degree between two angles (which is about right), how many different triangles are there to the human eye? In other words: how many different triangles are there with angles an integer number of degrees ? Of course ignoring flipping, rotation & resizing. ::)

Title: Re: how many triangles can a human distinguish
Post by towr on May 24^th, 2012, 11:10am

It's just a matter of counting the triples, a,b,c such that 1<=a<=b<=c<=178 and a+b+c=180

So, I think:
[hide] ~~sum(c=60..178, (179-c)-ceil((180-c)/2) + 1 ) = 3600~~[/hide]
[edit][hide]ah, wait, I forgot b <=c, so sum(c=60..89, c-ceil((180-c)/2) + 1 )+sum(c=90..178, (179-c)-ceil((180-c)/2) + 1 ) = 2700 [/hide][/edit]

Title: Re: how many triangles can a human distinguish
Post by pex on May 24^th, 2012, 11:53am

[hideb]The number of triples of positive integers (a,b,c) with a+b+c=180 is (180-1 choose 2-1) = 15931. In this total:
- one triangle, (60,60,60), has been counted once;
- 88 triangles, (1,1,178) ... (89,89,2) except (60,60,60), have been counted three times;
- all other triangles have been counted 3! = 6 times.
Thus, the number of triangles is the solution to 1*1 + 88*3 + (n-1-88)*6 = 15931, which matches towr's n = 2700.[/hideb]