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Math Puzzle - Triangle
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Math Puzzle - Triangle











































Puzzle

using each number once, take the numbers 1-9 and put them in a triangle with 4 slots so that each side equals 17

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Feb 26, 2013
Puzzle
by: Staff


Answer

Part I

The Solution:

Math Integer Triangle Puzzle





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Finding the solution:

Use the following formula to find the number of combinations possible when selecting 4 numbers at a time (out of a group of 9 numbers)

Combinations Formula





----------------------------------------------------

Feb 26, 2013
Puzzle
by: Staff

----------------------------------------------------



Part II

The number of possible combinations = 126

These are:

Combinations without repetition (n=9, r=4)

{1,2,3,4} {1,2,3,5} {1,2,3,6} {1,2,3,7} {1,2,3,8} {1,2,3,9} {1,2,4,5} {1,2,4,6} {1,2,4,7} {1,2,4,8} {1,2,4,9} {1,2,5,6} {1,2,5,7} {1,2,5,8} {1,2,5,9} {1,2,6,7} {1,2,6,8} {1,2,6,9} {1,2,7,8} {1,2,7,9} {1,2,8,9} {1,3,4,5} {1,3,4,6} {1,3,4,7} {1,3,4,8} {1,3,4,9} {1,3,5,6} {1,3,5,7} {1,3,5,8} {1,3,5,9} {1,3,6,7} {1,3,6,8} {1,3,6,9} {1,3,7,8} {1,3,7,9} {1,3,8,9} {1,4,5,6} {1,4,5,7} {1,4,5,8} {1,4,5,9} {1,4,6,7} {1,4,6,8} {1,4,6,9} {1,4,7,8} {1,4,7,9} {1,4,8,9} {1,5,6,7} {1,5,6,8} {1,5,6,9} {1,5,7,8} {1,5,7,9} {1,5,8,9} {1,6,7,8} {1,6,7,9} {1,6,8,9} {1,7,8,9} {2,3,4,5} {2,3,4,6} {2,3,4,7} {2,3,4,8} {2,3,4,9} {2,3,5,6} {2,3,5,7} {2,3,5,8} {2,3,5,9} {2,3,6,7} {2,3,6,8} {2,3,6,9} {2,3,7,8} {2,3,7,9} {2,3,8,9} {2,4,5,6} {2,4,5,7} {2,4,5,8} {2,4,5,9} {2,4,6,7} {2,4,6,8} {2,4,6,9} {2,4,7,8} {2,4,7,9} {2,4,8,9} {2,5,6,7} {2,5,6,8} {2,5,6,9} {2,5,7,8} {2,5,7,9} {2,5,8,9} {2,6,7,8} {2,6,7,9} {2,6,8,9} {2,7,8,9} {3,4,5,6} {3,4,5,7} {3,4,5,8} {3,4,5,9} {3,4,6,7} {3,4,6,8} {3,4,6,9} {3,4,7,8} {3,4,7,9} {3,4,8,9} {3,5,6,7} {3,5,6,8} {3,5,6,9} {3,5,7,8} {3,5,7,9} {3,5,8,9} {3,6,7,8} {3,6,7,9} {3,6,8,9} {3,7,8,9} {4,5,6,7} {4,5,6,8} {4,5,6,9} {4,5,7,8} {4,5,7,9} {4,5,8,9} {4,6,7,8} {4,6,7,9} {4,6,8,9} {4,7,8,9} {5,6,7,8} {5,6,7,9} {5,6,8,9} {5,7,8,9} {6,7,8,9}

Given these combinations, identify those combinations where the sum of the integers = 17

There are only nine combinations which meet this criteria:

{1,2,5,9} {1,2,6,8} {1,3,4,9} {1,3,5,8} {1,3,6,7} {1,4,5,7} {2,3,4,8} {2,3,5,7} {2,4,5,6}

The numbers within each of the nine combinations can be arranged in different ways.

Use the following formula to find the number of permutations possible for each of the nine combinations.

Permutations Formula





----------------------------------------------------

Feb 26, 2013
Puzzle
by: Staff


----------------------------------------------------


Part III


These are:

{1,2,5,9}
{1,2,5,9} {1,2,9,5} {1,5,2,9} {1,5,9,2} {1,9,2,5} {1,9,5,2} {2,1,5,9} {2,1,9,5} {2,5,1,9} {2,5,9,1} {2,9,1,5} {2,9,5,1} {5,1,2,9} {5,1,9,2} {5,2,1,9} {5,2,9,1} {5,9,1,2} {5,9,2,1} {9,1,2,5} {9,1,5,2} {9,2,1,5} {9,2,5,1} {9,5,1,2} {9,5,2,1}


{1,2,6,8}
{1,2,6,8} {1,2,8,6} {1,6,2,8} {1,6,8,2} {1,8,2,6} {1,8,6,2} {2,1,6,8} {2,1,8,6} {2,6,1,8} {2,6,8,1} {2,8,1,6} {2,8,6,1} {6,1,2,8} {6,1,8,2} {6,2,1,8} {6,2,8,1} {6,8,1,2} {6,8,2,1} {8,1,2,6} {8,1,6,2} {8,2,1,6} {8,2,6,1} {8,6,1,2} {8,6,2,1}


{1,3,4,9}
{1,3,4,9} {1,3,9,4} {1,4,3,9} {1,4,9,3} {1,9,3,4} {1,9,4,3} {3,1,4,9} {3,1,9,4} {3,4,1,9} {3,4,9,1} {3,9,1,4} {3,9,4,1} {4,1,3,9} {4,1,9,3} {4,3,1,9} {4,3,9,1} {4,9,1,3} {4,9,3,1} {9,1,3,4} {9,1,4,3} {9,3,1,4} {9,3,4,1} {9,4,1,3} {9,4,3,1}



{1,3,5,8}
{1,3,5,8} {1,3,8,5} {1,5,3,8} {1,5,8,3} {1,8,3,5} {1,8,5,3} {3,1,5,8} {3,1,8,5} {3,5,1,8} {3,5,8,1} {3,8,1,5} {3,8,5,1} {5,1,3,8} {5,1,8,3} {5,3,1,8} {5,3,8,1} {5,8,1,3} {5,8,3,1} {8,1,3,5} {8,1,5,3} {8,3,1,5} {8,3,5,1} {8,5,1,3} {8,5,3,1}



{1,3,6,7}
{1,3,6,7} {1,3,7,6} {1,6,3,7} {1,6,7,3} {1,7,3,6} {1,7,6,3} {3,1,6,7} {3,1,7,6} {3,6,1,7} {3,6,7,1} {3,7,1,6} {3,7,6,1} {6,1,3,7} {6,1,7,3} {6,3,1,7} {6,3,7,1} {6,7,1,3} {6,7,3,1} {7,1,3,6} {7,1,6,3} {7,3,1,6} {7,3,6,1} {7,6,1,3} {7,6,3,1}


{1,4,5,7}
{1,4,5,7} {1,4,7,5} {1,5,4,7} {1,5,7,4} {1,7,4,5} {1,7,5,4} {4,1,5,7} {4,1,7,5} {4,5,1,7} {4,5,7,1} {4,7,1,5} {4,7,5,1} {5,1,4,7} {5,1,7,4} {5,4,1,7} {5,4,7,1} {5,7,1,4} {5,7,4,1} {7,1,4,5} {7,1,5,4} {7,4,1,5} {7,4,5,1} {7,5,1,4} {7,5,4,1}


{2,3,4,8}
{2,3,4,8} {2,3,8,4} {2,4,3,8} {2,4,8,3} {2,8,3,4} {2,8,4,3} {3,2,4,8} {3,2,8,4} {3,4,2,8} {3,4,8,2} {3,8,2,4} {3,8,4,2} {4,2,3,8} {4,2,8,3} {4,3,2,8} {4,3,8,2} {4,8,2,3} {4,8,3,2} {8,2,3,4} {8,2,4,3} {8,3,2,4} {8,3,4,2} {8,4,2,3} {8,4,3,2}


{2,3,5,7}
{2,3,5,7} {2,3,7,5} {2,5,3,7} {2,5,7,3} {2,7,3,5} {2,7,5,3} {3,2,5,7} {3,2,7,5} {3,5,2,7} {3,5,7,2} {3,7,2,5} {3,7,5,2} {5,2,3,7} {5,2,7,3} {5,3,2,7} {5,3,7,2} {5,7,2,3} {5,7,3,2} {7,2,3,5} {7,2,5,3} {7,3,2,5} {7,3,5,2} {7,5,2,3} {7,5,3,2}


{2,4,5,6}
{2,4,5,6} {2,4,6,5} {2,5,4,6} {2,5,6,4} {2,6,4,5} {2,6,5,4} {4,2,5,6} {4,2,6,5} {4,5,2,6} {4,5,6,2} {4,6,2,5} {4,6,5,2} {5,2,4,6} {5,2,6,4} {5,4,2,6} {5,4,6,2} {5,6,2,4} {5,6,4,2} {6,2,4,5} {6,2,5,4} {6,4,2,5} {6,4,5,2} {6,5,2,4} {6,5,4,2}



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Feb 26, 2013
Puzzle
by: Staff


----------------------------------------------------


Part IV


Now, eliminate entries with duplicate end points to reduce the number of possibilities:

After the duplicate end points are eliminated, the number of possibilities is greatly reduced:



{1,2,5,9}
{1,2,5,9} {1,2,9,5} {1,5,9,2} {2,1,5,9} {2,1,9,5} {5,1,2,9}


{1,2,6,8}
{1,2,6,8} {1,2,8,6} {1,6,8,2} {2,1,6,8} {2,1,8,6} {6,1,2,8


{1,3,4,9}
{1,3,4,9} {1,3,9,4} {1,4,9,3} {3,1,4,9} {3,1,9,4}{4,1,3,9}


{1,3,5,8}
{1,3,5,8} {1,3,8,5} {1,5,8,3}{3,1,5,8} {3,1,8,5}{5,1,3,8}


{1,3,6,7}
{1,3,6,7} {1,3,7,6} {1,6,7,3} {3,1,6,7} {3,1,7,6} {6,1,3,7}


{1,4,5,7}
{1,4,5,7} {1,4,7,5} {1,5,7,4} {4,1,5,7} {4,1,7,5} {5,1,4,7}


{2,3,4,8}
{2,3,4,8} {2,3,8,4} {2,4,8,3} {3,2,4,8} {3,2,8,4} {4,2,3,8}


{2,3,5,7}
{2,3,5,7} {2,3,7,5} {2,5,7,3} {3,2,5,7} {3,2,7,5} {5,2,3,7}


{2,4,5,6}
{2,4,5,6} {2,4,6,5} {2,5,6,4} {4,2,5,6} {4,2,6,5} {5,2,4,6}


As a last step, match the various possibilities with one another until you get a match.

Refer to the top of this post for the final solution.





Thanks for writing.

Staff
www.solving-math-problems.com



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