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#1 2020-02-12 10:06:02

Mama Donna
Member
Registered: 2020-02-12
Posts: 3

Show me how to fit 195 triangles into a 45 inch octagon

I need help with this and I'm hoping one of you "math-wizards" can do this.  Extremely important and urgent!  Thank you ahead of time.
Sincerely, Donna

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#2 2020-02-12 22:33:35

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,774

Re: Show me how to fit 195 triangles into a 45 inch octagon

hi Donna,

Welcome to the forum.

If I'm understanding the problem correctly, then what difference does it make how big the octagon is?  If you made an octagon of a certain size, counted the triangles, then shrunk it to half size, it would still have those many triangles.

I started with a regular octagon:

jA2pGeF.gif

There are, if I've counted correctly, 65 points.  I noticed that 195 = 65 x 3, so that's hopeful.  If you picked a point (65 choices) then another different point (64 choices), then another (63 choices) then you could make a total of 65 x 64 x 63 triangles.  Except you'd be counting every one several times (eg ABC, ACB, BAC, BCA, CAB, CBA) so divide by six.  That's still way to many.  You'd also be counting three points that actually don't make a triangle but rather a straight line.  There's a lot of them.

But you didn't say it's a regular octagon.  If not then that opens up more possibilities.  I've arrowed a point where three diagonals cross at a single point.  If you adjust the points at the edge of the octagon just slightly then that three way crossing becomes another triangle (see insert).

So how to proceed from here?  When I get a problem that looks this hard I try simpler cases first to see if I can establish what to do.

Suppose the shape was just a single triangle.  How many triangles? Easy. Just the one.

What about a quadrilateral and a pentagon?

W3L2bt3.gif

I can easily count the triangles in the quadrilateral … 8, 4 small ones and another 4 that are made from two small triangles.

Is there a way to get this answer by reasoning it out?  There are 5 points so if I try to make all the triangles I can; that's 5 choices for vertex 1, followed by 4 choices for vertex 2 followed by 3 for vertex 3, divide by the repeated triangle number explained above, that's 6 so I get (5x4x3)/6 = 10.  Whoops, not 8.  But there are two diagonals with 3 points on them.  A 'triangle' made from those three points doesn't count as it's a straight line not a triangle.  Two diagonals like that so subtract 2 … 10 -2 = 8 and I've got the right answer.

For the pentagon the number to subtract for 'straight line' triangles is much larger because each diagonal has four points, let's call them A,B,C and D.  'Triangles' ABC, ABD, ACD and BCD are all just straight lines so you'd need to subtract 4 for every diagonal, 5 so subtract 20 altogether.

General method.  Draw the polygon.  Adjust the points so that there are no 3 or more way crossings of lines. Count the number of points, N.  Calculate Nx(N-1)x(N-2)/6.   Count how many diagonals there are and how many ways of picking any 3 points on a diagonal to subtract the straight line cases.

At this point I'm going to pause and await your reply.  I need to know if I'm interpreting the question correctly and whether we're meant to be considering a regular octagon.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#3 2020-02-13 05:17:06

Mama Donna
Member
Registered: 2020-02-12
Posts: 3

Re: Show me how to fit 195 triangles into a 45 inch octagon

Thank you, Bob, for undertaking my request. 
     If you enter a "star of David" in the very center of the octagon, and go from there to create the rest of the 195 (189) triangles to fit into the octagon (a regular octagon) , can you show me how to do this?  ALL triangles!!  No other shapes but triangles!  I'm not "testing" you, rather I'm asking for your help.  This is very important.
            Thank you,
                    Donna

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#4 2020-02-13 07:13:01

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,774

Re: Show me how to fit 195 triangles into a 45 inch octagon

Hi Donna,

Sorry but I'm really not getting this.  An octagon has 8 sides.  A star of David has 6 points.  So how to put these two together?  What size is each?

A diagram would really help.  Or a longer description such as (1) Draw a regular octagon ABCDEFGH.  (2) Make a star of David by ???? please continue …

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#5 2020-02-13 16:11:52

Mama Donna
Member
Registered: 2020-02-12
Posts: 3

Re: Show me how to fit 195 triangles into a 45 inch octagon

The red icon in the corner of your post....the one turning round and round....that is in the center of the larger octagon.  Except for the center of the star,  all the spaces are made up of triangles...12 I believe.  Now expand outwards making more triangles.  Expand more, until you have 195 triangles in all.  Does this explanation help?  I will try to draw this and send via email.  I think if the triangles are 3" each, they should fit into the 45" octagon.
   Thanks again! smile
     Donna

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