Twelve Marbles Solutions

Duncan Chiu (dchiu@idt.net)
Sat, 26 Dec 1998 17:55:41 -0500


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Hi Everyone,

I hope you all had a nice Christmas celebration.

I am sending you the solutions to Which One is The Odd Ball in two ways. The
three attachments are in Microsoft Word 7.0. One of them is a table. If you
could not open them, draw diagrams as you read the solutions that follow. Please

share them with your students if you think they are elegant and worthwhile.

BTW, thirteen marbles can be done in the same manner, except you cannot tell if
the odd ball is heavier or lighter than the rest.

Have fun and a Happy New Year!

duncan

Solution
First weighing

Divide 12 marbles into three groups of four and place two groups on the balance.

Either the two groups are the same
in weight (see column A) or they are different (see columns B and C.)

Column A
Second weighing

If the two groups are the same, take three of these eight equal-weight marbles
(Gs) and weigh them with three of the
remaining four marbles (numbered 1, 2, 3, 4.) The result is either they are
equal to, heavier than, or lighter than each
other:

Third weighing

a) If the three Gs are the same as the other three, weigh the remaining 4th
marble with a G. You can now tell if the
4th marble is heavier or lighter than the rest.

b) If the three Gs are heavier than the other three, you know the odd ball is
a light one on the other side of the
balance. Weigh any two of these three marbles.
Case i: They are equal to each other. That means the third one is the odd one
and is lighter than the rest.
Case ii: They are different. In this case the odd one is on the lighter side of

the balance.

c) If the three Gs are lighter than the other three, you know the odd ball is a

heavy one on the other side of the
balance. Weigh any two of these three marbles.
Case i: They are equal to each other. That means the third one is the odd one
and is heavier than the rest.
Case ii: They are different. In this case the odd one is on the heavier side of

the balance.

Column B

Second weighing

If the two groups are different in weight (1, 2, 3, 4 < > 5, 6, 7, 8,) take
three of the remaining four, which are equal
in weight (Gs), and replace any three (2, 3, 4) on one side of the balance.
Replace three marbles on the other side of
the balance (5, 6, 7) with the displacement (2, 3, 4.) The two outcomes are
either they are equal (1, G, G, G = 2,
3, 4, 8) or unequal to each other (1, G, G, G < > 2, 3, 4, 8.)

Third weighing

a) If the second weighing shows equal weight, the "odd ball" must be one of the
displaced three marbles (5, 6, 7.)

Case i: Weigh any two of these three marbles. If they are equal to each other,

then the weight of the third is
determined by the position of the balance on the first weighing.
Case ii: If they are different, then the odd one is also decided by the
position in the first weighing. (The
position of the first weighing actually decides the odd one's weight at the
end.)

b) If the second weighing shows unequal weights, then the balance is either of
the same or opposite position as the
first weighing.

Case i: Same position indicates that the odd ball is one of the 4th marbles not

replaced on either side of the
balance (1 or 8). Compare one of them with a G. The result determines the weight

of the odd ball.
Case ii: Opposite position indicates the odd ball is one of the three marbles
displaced by 3 Gs or the three
switched to the other side of the balance (2, 3, 4.) The odd ball can be found
by weighing two of these three
marbles.

Solution Two
by
G. Pocherstnik & L. Zhang

First weighing

Same as Solution One

Column A
Second weighing

Same as Solution One

Third weighing

Same as Solution One

Column B

Second weighing

If the two groups are different in weight (1, 2, 3, 4 < > 5, 6, 7, 8,) take
three of the remaining four, which are equal
in weight (Gs), and replace any three (5, 6, 7) on one side of the balance.
Switch the 4th marble (8) with any one (1)
on the other side of the balance: (8, 2, 3, 4 = or < > G, G, G, 1)

Third weighing

Solutions are similar to Solution One.

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