12 bags of gold math problem

I solved this eventually many months ago when my boss gave it to us in the office to ponder. Took a long time but can still remember how to do it.

The problem goes likes this.

You have 12 bags of gold that are identical in every way – except that one bag is different to the others in weight. It can be heavier or lighter.

The only method you have for finding which one is different is a set of scales. Easy enough you say, but you can only use the set of scales 3 times.

Go on, have a go.

The solution is in below.

Blm

12 bags of gold numbered 1->12

Step 1
Compare 1 2 3 4 to 5 6 7 8 (WEIGH 1)
If same then odd bag in 9 10 11 12 – goto step 2
If 1 2 3 4 heavier than 5 6 7 8 – goto step 6
If 1 2 3 4 lighter than 5 6 7 8 – goto step 10

Step 2
Compare 9 10 11 to 1 2 3 (WEIGH 2)
If same the 12 is different – goto step 3
If 9 10 11 heavier than 1 2 3 then we know the odd one out is heavier and in 9 10 11 – goto Step 4
If 9 10 11 lighter than 1 2 3 then we know the odd one out is lighter and in 9 10 11 – goto Step 5

Step 3
We know 12 is the odd one out – just need to find if its lighter or heavier
Compare 12 to 1 (WEIGH 3)
If 12 lighter then 12 is odd one out and lighter
If 12 heavier then 12 is odd one out and heavier

Step 4
Compare 9 to 10 (WEIGH 3)
if 9 heavier then 9 odd one out and heavier
if 10 heavier then 10 odd one out and heavier
if same then 11 odd one out and heavier

Step 5
Compare 9 to 10 (WEIGH 3)
if 9 lighter then 9 odd one out and lighter
if 10 lighter then 10 odd one out and lighter
if same then 11 odd one out and lighter

Step 6
Time to swap some stuff about
Some new notation here
+ = possibly heavier,
– = possibly lighter,
= = known standard
Compare 1+ 5- 9= to 2+ 6- 7- (WEIGH 2)

If 1 5 9 heavier – goto 7
If 2 6 7 heavier – goto 8
If same – goto 9

Step 7
We know that the balance beteen Step 1 and Step 6 stayed the same therefore the odd bag has not moved so it must be 1+ 6- or 7-
Compare 6- to 7- (WEIGH 3)
if 6 is lighter then its odd one out and lighter
if 7 is lighter then its odd one out and lighter
if same then 1 odd one out and heavier

Step 8
The balance between Step 1 and Step 6 has changed therefore the odd bag has moved so it must be 5- or 2+
Compare 2+ to 9= (WEIGH 3)
If 2 heavier then 2 is odd one and heavier
if same then 5 is odd one and lighter

Step 9
Odd one out must be in 3+ 4+ or 8-
Compare 3+ to 4+ (WEIGH 3)
if same then 8 is odd and lighter
if 4 heavier then 4 is odd and heavier
if 3 heavier then 3 is odd and heavier

Step 10
Like step 6 but other way round
Compare 1- 5+ 9= to 2- 6+ 7+ (WEIGH 2)

if 2 6 7 heavier – goto Step 11
if 1 5 9 heavier – goto Step 12
if same – goto Step 13

Step 11
Balance has stayed the same therefor the odd bag has not moved ant it must be in 1- 6+ 7+
Compare 6+ to 7+ (WEIGH 3)
If 6 is heavier then 6 is odd and heavier
if 7 is heavier then 7 is odd and heavier
if same then 1 is odd and lighter

Step 12
Balance had move there must be in a moved bag ie – 5+ or 2-
Compare 5+ and 9= (WEIGH 3)
if 5 is heavier then 5 is odd and heavier
if same then 2 is odd and lighter

Step 13
Odd one out must be in -3 -4 or 8+
Compare -3 to -4 (WEIGH 3)
if same then 8 is odd and heavier
if 3 is lighter then 3 is odd and lighter
if 4 is lighter then 4 is odd and lighter

Dadaaaaaaaa