Melody

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Melody  Feb 11, 2022
 #3
avatar+118724 
+13

 

1 2 3 4 5 6 7 8  FOUR IN A ROW
W W W L            $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$
      L W W W W
L W W W W L           $${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$
  L W W W W L  
    L W W W W L

 

 

1 2 3 4 5 6 7 8                          FIVE IN A ROW
W W W W  L          $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$
     L W W W W W
L W W W W W  L       $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$      
  L W W W W W  L

 

 

1 2 3 4 5 6 7 8                       SIX IN A ROW
W W W W  W  L        $${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$
   L  W W W W W W
L W W W W W  W  L     $${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

1 2 3 4 5 6 7 8    SEVEN IN A ROW
W W W W W W W L  
L W W W W W W W  

 

 

I TRIED TO PUT ALL THIS INTO A TABLE BUT THE TABLE DID NOT COPE VERY WELL   

 

FOUR IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{32}}}{{\mathtt{243}}}} = {\mathtt{0.131\: \!687\: \!242\: \!798\: \!353\: \!9}}$$

 

$${\mathtt{3}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{16}}}{{\mathtt{243}}}} = {\mathtt{0.065\: \!843\: \!621\: \!399\: \!177}}$$

 

FIVE IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{64}}}{{\mathtt{729}}}} = {\mathtt{0.087\: \!791\: \!495\: \!198\: \!902\: \!6}}$$

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}} = {\mathtt{0.029\: \!263\: \!831\: \!732\: \!967\: \!5}}$$

 

SIX IN A ROW

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}} = {\mathtt{0.058\: \!527\: \!663\: \!465\: \!935\: \!1}}$$

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}^{{\mathtt{2}}} = {\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}} = {\mathtt{0.009\: \!754\: \!610\: \!577\: \!655\: \!8}}$$

 

SEVEN IN A ROW  

 

$${\mathtt{2}}{\mathtt{\,\times\,}}{\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{7}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right) = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

EIGHT IN A ROW

 

$${\left({\frac{{\mathtt{2}}}{{\mathtt{3}}}}\right)}^{{\mathtt{8}}} = {\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\mathtt{0.039\: \!018\: \!442\: \!310\: \!623\: \!4}}$$

 

 

$${\frac{{\mathtt{32}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{16}}}{{\mathtt{243}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{729}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{128}}}{{\mathtt{2\,187}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{64}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{256}}}{{\mathtt{6\,561}}}} = {\frac{{\mathtt{112}}}{{\mathtt{243}}}} = {\mathtt{0.460\: \!905\: \!349\: \!794\: \!238\: \!7}}$$

.
May 17, 2015
 #1
avatar+118724 
+5
May 17, 2015
 #3
avatar+118724 
+5

Hi MathsGod1

 

 

 

 

 

Here is a video you should watch and absorb :)

 

https://www.khanacademy.org/math/geometry/right_triangles_topic/pyth_theor/v/pythagorean-theorem

 

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Oh I almost forgot.  Numbers that work for the pythagorean Theorem are called pythagorean triads.

May 17, 2015