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## Math puzzle: Find the odd ball out

A friend on chessbomb.com asked a very interesting puzzle. Here it is and my attempt to solve it:

There are 15 balls, one of them has an odd weight (either light or heavy) and all the rest have equal weight. We know one of the 14 balls with the correct weight (good ball). Using 3 weighings on a beam balance we need to figure the odd ball out. We do not need to know if the odd ball is heavy or light.

My attempt at the solution

Let us suppose that the known good ball is {1}

- If {1,2,3,4,5} = {6,7,8,9,10} then odd ball is {11,12,13,14,15}
- If {1,11} = {12,13} then odd ball is {14,15 }
- If {1} = {14} then odd ball is {15}
- If {1} > {14} then light odd ball is {14}
- If {1} < {14} then heavy odd ball is {14}

- If {1,11} > {12,13} then odd ball is {11} (heavy) or (12,13} (light)
- If {12} = {13} then heavy odd ball is {11}
- If {12} > {13} then light odd ball is {13}
- If {12} < {13} then light odd ball is {12}

- If {1,11} < {12,13} then odd ball is {11} (light) or (12,13} (heavy)
- If {12} = {13} then light odd ball is {11}
- If {12} > {13} then heavy odd ball is {12}
- If {12} < {13} then heavy odd ball is {13}

- If {1,11} = {12,13} then odd ball is {14,15 }
- If {1,2,3,4,5} > {6,7,8,9,10} then odd ball is {2,3,4,5,6,7,8,9,10}
- If {2,3,6} = {4,5,7} then light odd ball is {8,9,10}
- If {8} = {9} then light odd ball is {10}
- If {8} > {9} then light odd ball is {9}
- If {8} < {9} then light odd ball is {8}

- If {2,3,6} > {4,5,7} then {2,3} is heavy odd ball or {7} is light odd ball
- If {2} = {3} then light odd ball is {7}
- If {2} > {3} then heavy odd ball is {2}
- If {2} < {3} then heavy odd ball is {3}

- If {2,3,6} < {4,5,7} then {4,5} is heavy odd ball or {6} is light odd ball
- If {4} = {5} then light odd ball is {6}
- If {4} > {5} then heavy odd ball is {4}
- If {4} < {5} then heavy odd ball is {5}

- If {2,3,6} = {4,5,7} then light odd ball is {8,9,10}
- If {1,2,3,4,5} < {6,7,8,9,10} then ball is {1,2,3,4,5,6,7,8,9,10}
- If {2,3,6} = {4,5,7} then heavy odd ball is {8,9,10}
- If {8} = {9} then heavy odd ball is {10}
- If {8} > {9} then heavy odd ball is {8}
- If {8} < {9} then heavy odd ball is {9}

- If {2,3,6} > {4,5,7} then {4,5} is light odd ball or {6} is heavy odd ball
- If {4} = {5} then heavy odd ball is {6}
- If {4} > {5} then light odd ball is {5}
- If {4} < {5} then light odd ball is {4}

- If {2,3,6} < {4,5,7} then {2,3} is light odd ball or {7} is heavy odd ball
- If {2} = {3} then heavy odd ball is {7}
- If {2} > {3} then light odd ball is {3}
- If {2} < {3} then light odd ball is {2}

- If {2,3,6} = {4,5,7} then heavy odd ball is {8,9,10}

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