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gautam89

## Main Question or Discussion Point

Snooker balls Problems – An Approach to Elegant Solutions:

Problem Definition: In a set S of snooker balls, all of the same standard weight, there is one ball of non-standard weight.

Using a Scale Balance, find the non-standard ball with minimum number of trials- Provide a formula and a proof.

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Proposition 1. Set S contains One non-standard ball of known relative weight -Identify non-standard :

Solution: For Quantity = 3 (to power n), or less;

Min weighing required = n.

Proposition 2: Set S contains One non-standard ball- of unknown relative weight; find the ball and identify its relative (heavier or lighter) weight ?:

Solution: For Qty = ( 3(power n) + 3(n-1) + 3 (n-2) … 3(2) + 3(1) ) or less;

Min weighing required = n +1.

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An approach to proofs of solutions:

Definition 1: (Ternary partition)

Divide the set S into three subsets A, B, and C, where

1. Qty(S) = Qt(A)+Qty(B) +Qty(C), and

2. Qty(A) = Qty(B) >= 1/3 x Qty(S), and

3. Qty(C) = Qty(A) - i; where i = 0,1 or 2.

Proof For Proposition 1: Weigh A with B. This will identify which of the subsets A, B, or C has the non-standard (with known relative weight) ball. – Recursively use n-1 weighings to find the non-standard ball (of known relative weight).

Definition 2 : (“Weight type”)

1. A ball is of “ heavier weight type” when it can NOT be lighter ball.

2. A ball is of “ lighter weight type” when it can NOT be heavier ball.

3. A subset of balls is called of a specific “weight type” if all the balls in that subset is of that “weight type”

Proof For proposition 2:

1. Divide the set S into its Ternary subsets A, B, C (Definition 1), and weigh set A & B.

2. If A and B are Not equal, then using the “weight type” concept and Proposition 1, one can “cross-weigh” subsets of A, B and C n times, to find the non-standard ball from A or B, and identify if it is heavier or lighter ball

3. If subsets A and B are equal then the non-standard ball is in C, and again one can use “cross weighings” and Proposition 1, n times to find and identify the relative weight of the non-standard ball in subset C.

What about problem when there are TWO non-standard balls (of equal weights)...

Gautam Pandya

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Problem Definition: In a set S of snooker balls, all of the same standard weight, there is one ball of non-standard weight.

Using a Scale Balance, find the non-standard ball with minimum number of trials- Provide a formula and a proof.

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

Proposition 1. Set S contains One non-standard ball of known relative weight -Identify non-standard :

Solution: For Quantity = 3 (to power n), or less;

Min weighing required = n.

Proposition 2: Set S contains One non-standard ball- of unknown relative weight; find the ball and identify its relative (heavier or lighter) weight ?:

Solution: For Qty = ( 3(power n) + 3(n-1) + 3 (n-2) … 3(2) + 3(1) ) or less;

Min weighing required = n +1.

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

An approach to proofs of solutions:

Definition 1: (Ternary partition)

Divide the set S into three subsets A, B, and C, where

1. Qty(S) = Qt(A)+Qty(B) +Qty(C), and

2. Qty(A) = Qty(B) >= 1/3 x Qty(S), and

3. Qty(C) = Qty(A) - i; where i = 0,1 or 2.

Proof For Proposition 1: Weigh A with B. This will identify which of the subsets A, B, or C has the non-standard (with known relative weight) ball. – Recursively use n-1 weighings to find the non-standard ball (of known relative weight).

Definition 2 : (“Weight type”)

1. A ball is of “ heavier weight type” when it can NOT be lighter ball.

2. A ball is of “ lighter weight type” when it can NOT be heavier ball.

3. A subset of balls is called of a specific “weight type” if all the balls in that subset is of that “weight type”

Proof For proposition 2:

1. Divide the set S into its Ternary subsets A, B, C (Definition 1), and weigh set A & B.

2. If A and B are Not equal, then using the “weight type” concept and Proposition 1, one can “cross-weigh” subsets of A, B and C n times, to find the non-standard ball from A or B, and identify if it is heavier or lighter ball

3. If subsets A and B are equal then the non-standard ball is in C, and again one can use “cross weighings” and Proposition 1, n times to find and identify the relative weight of the non-standard ball in subset C.

What about problem when there are TWO non-standard balls (of equal weights)...

Gautam Pandya

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