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# weighing puzzle 12 balls of equal size and weight except for 1

## Question

You have 12 balls of equal size and weight ,except one ball is either lighter or heavier than the others. Using an analytical balance only 3 times ,identify the odd ball and weather it is hvy or light

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Problem

Spoiler

12 balls, one is heavy or light (24 cases.) 24 <= 33 so we're good.

Spoiler

Reduce the cases to <= 32 in number.
Number the balls and compare { 1 2 3 4 } with { 5 6 7 8 }.

• If they balance, one of { 9 10 11 12 } is heavy or light (8 cases.)
• If they don't balance one of the lighter group is light or one of the heavier group is heavy (8 cases.)

Balance

Spoiler

Reduce the cases to <= 3 in number.
Compare { 10 11 12 } with { 1 2 3 }.

• If they balance, { 9 } is heavy or light (2 cases.) Compare { 9 } with any other ball to decide.
• Otherwise one of { 10 11 12 } is known heavy or light (3 cases.) Compare { 10 } with { 11 } to decide.

Don't balance

Spoiler

Reduce the cases to <= 3 in number.
Set aside { 4 7 8 } and compare { 1 2 5 } with { 3 6 12 }.

If { 1 2 3 4 } was heavier, then

Spoiler
• If { 1 2 5 } is heavier, 1 or 2 is heavy or 6 is light. Compare 1 with 2 to decide.
• If they balance, 4 is heavy or 7 or 8 is light. Compare 7 with 8 to decide.
• If ( 1 2 5 ) is lighter, 3 is heavy or 5 is light Compare 3 with 12 to decide

If { 5 6 7 8 } was heavier, then

Spoiler
• if { 1 2 5 } is heavier, 5 is heavy or 3 is light. Compare 3 with 12 to decide.
• If they balance, 4 is light or 7 or 8 is heavy. Compare 7 with 8 to decide.
• If { 1 2 5 } is lighter, 1 or 2 is light or 6 is heavy. Compare 1 with 2 to decide.

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Put 6 balls on either side. Divide the lighter 6 into two groups of 3 and put them on the scale. take the lighter 3. put one on each side of the scale. if they are balanced, the other one weighs lighter. if they don't, we'll know anyway which one is lighter from the balance.

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AAAA BBBB - CCDE

1= bad ball in CCDE

CC DA - E

11= E is bad, weigh against A to discover whether H or L

11 Left Heavy: relabel as HH LA

Weigh H vs H, if equal, L is bad. If not, heavier H is bad

12 Left Light: relabel as LL HA

weigh L vs L, if equal, H is bad. If unequal lighter L is bad

1<> relabel to HHHH LLLL (or LLLL HHHH)

12 weigh HHL vs HLC - aside HLL

12= L vs L = choose H, if not, choose lighter L

12 Left side heavy: Either left HH is heavy or right L is light, H vs H: = choose L, else heavier H

12 Left side light: either left L is light or right H is heavy, weigh L against a good ball, balance means H is bad, imbalance says L

Edited by CaptainEd
Elephant font, Removed
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Maybe this is clear:

Weighing #1: weigh 4 vs 4, leave 4 aside, call them AAAA BBBB -CCDE

There are two outcomes: Balanced vs Unbalanced.

The easy case is Balanced. The A’s and Bs are good, and one of CCDE is bad.

For weighing #2: weigh CC vs DA, leave E aside.

If this is balanced, then E is bad, third weighing is A vs E to determine Heavy or Light.

If the  Left side goes down, either a C is heavy, or D is Light. For Weighing 3, weigh the Cs against each other. If they balance, D is bad. If not, the C that went down is Heavy.

If Left side went Up, then the same argument applies: either a C is Light or D is heavy.

Now the harder case: AAAA and BBBB don’t balance. Relabel them as HHHH  and LLLL. We don’t know whether an H is Heavy or an L is Light, but we know that no H is Light and no L is Heavy. And the CCDE are all Good; call them G.

Weighing 2: HHL vs HLG, leave HLL aside

Three possibilities:

A) balanced: HLL is bad: weigh L vs L,  the lighter is bad, if neither, the H is bad and Heavy

B) If HHL goes down, then either the HH in the left is bad or the L in the right is bad.: weigh H vs H; the heavier is bad. If neither, L is bad and light.

C) If HHL goes up, then either the L in left pan is Light or the H in right pan is Heavy. Weigh H vs G, if they balance, the L is bad; else the H is bad.

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