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

im in the 9th grade and this was given to me it took me a few hours to get but there was in fact a simpler way to find it then the way you did it…

Your solution might find the odd bag out but does it know if its heavier or lighter?

Email me it and if it is simpler I’ll post it here with your name on it.

Thanks for posting.

blm