Logic Problem

[RATATTAK]

Just joined in the discussion about cakes https://www.pistonheads.com/gassing/topic.asp?h=0&... and that made me think of a logic problem that I solved many years ago ... so here is the problem which I hope the PH massive should enjoy solving :

You have 12 objects that all look and feel exactly the same; lets use matchsticks as an example. One of the matchsticks is slightly different to the 11 others in that it weighs more or could be less.

You have a chemical balance but no standard weights



You can only use the chemical balance three times to discover which matchstick is different and whether it's heavier or lighter taking into account all possibilities

Good luck if you care to have a go

Edited by RATATTAK on Thursday 11th January 21:08

Edited by RATATTAK on Thursday 11th January 21:36

[Fast and Spurious]

1st measure 6 per side.
One side heavier.
Next take the 6 in the heavier side, on the scales 3 per side. (2nd measure)
Then take the heavier side (3 matches), put 1 each in either side of the scales.
(3rd measure)
If one or the other side is heavier, that's the heavier one. If both sides equal, the match not on the scales is the heavier one.

[Fast and Spurious]

Oh hang on, heavier OR lighter... Need to think again!
Sorry too much Leffe

[anonymous-user]

Fast and Spurious said:

1st measure 6 per side.
One side heavier.
Next take the 6 in the heavier side, on the scales 3 per side. (2nd measure)
Then take the heavier side (3 matches), put 1 each in either side of the scales.
(3rd measure)
If one or the other side is heavier, that's the heavier one. If both sides equal, the match not on the scales is the heavier one.

Ah, but you don’t know whether the outlier is heavier or lighter. You’ve assumed it is heavier...

I remember coming across this puzzle as a kid. It’s kind of fun.

[RATATTAK]

Anyone got the first weighing yet ?

[V8LM]

First weighing is 4 on each pan.

[RATATTAK]

V8LM said:

First weighing is 4 on each pan.

Yep

[RATATTAK]

RATATTAK said:

V8LM said:

First weighing is 4 on each pan.

Yep

and let's say the worst case is that one side goes down (for the purposes of this exercise let's say the RHS goes down)

so the scales look like this:


^^^^ l vvvv

[anonymous-user]

Cannot solve because you can only guess if the odd match was supposed to be heavier or lighter if the odd one is allowed to be either heavier or lighter. No amount of weighing can deduce which is wrong. Even if you factor it down to two suspects an imbalance between two does not show that one is wrong and one is right, just that they are different.

[RATATTAK]

anonymous said:

[redacted]

it's solvable ... keep trying ... what would be your second weighing ?

[RATATTAK]

Clue

That first weighing tells you that it could be one of the four lighter ones on the LHS or one of the four heavier ones on the RHS but ...

more importantly, it tells you that it's NOT one of the four that were not on the scales, so they become 'standards'

so graphically you have:

^^^^ it could be one of these and if it is, it's got to be lighter

vvvv it could be one of these and if it is, it's got to be heavier

ssss all of these are "standards'

Edited by RATATTAK on Friday 12th January 21:49

[Phud]

Next two weighs are to find which of the vvvv or ^^^^ has the mismatched stick.

Then as you will have two sticks in each pan, remove one stick from each pan, this will tell you if the stick remains in a pan or in your hand, if mismatched stick is in the pan it will rise or fall, if pans are level then you will know form which pan you too the stick from.

[RATATTAK]

Phud said:

Next two weighs are to find which of the vvvv or ^^^^ has the mismatched stick.

Then as you will have two sticks in each pan, remove one stick from each pan, this will tell you if the stick remains in a pan or in your hand, if mismatched stick is in the pan it will rise or fall, if pans are level then you will know form which pan you too the stick from.

is this what you mean ?

second weighing

vv l vv

which will give you vv and ssssssssss

third weighing

v l v

which will give you v and ssssssssssss

but what if your second weighing balances ?

you've then got ^^^^ and ssssssss and you will need a third and a fourth weighing

so FAIL I'm afraid

Edited by RATATTAK on Friday 12th January 22:27

[brman]

Phud said:

Next two weighs are to find which of the vvvv or ^^^^ has the mismatched stick.

Then as you will have two sticks in each pan, remove one stick from each pan, this will tell you if the stick remains in a pan or in your hand, if mismatched stick is in the pan it will rise or fall, if pans are level then you will know form which pan you too the stick from.

say that again?

Haven't you just done 4 weighs? Which is the least I can manage.....

Edited by brman on Friday 12th January 22:10

[brman]

ash73 said:

Matches 123456ABCDEF, let's say 2 is lighter

123456 / ABCDEF, left side up so 1-6 must be lighter

123ABC / 456DEF, left side up so 1-3 must be lighter

1 / 2, right side up so 2 lighter (if level 3 would be lighter)

but what if D had been heavier? Both first weighs would give the same result, last weigh would give 1/2 the same so you would choose 3. Wrong result.....

Edited by brman on Friday 12th January 22:11

[RATATTAK]

ash73 said:

Matches 123456ABCDEF, let's say 2 is lighter (or E could be heavier)

123456 / ABCDEF, left side up so 1-6 must be lighter (or ABCDEF must be heavier)

123ABC / 456DEF, left side up so 1-3 must be lighter ( or DEF must be heavier)

1 / 2, right side up so 2 lighter (if level 3 would be lighter) (but what about DEF ?)

[Phud]

second weighing

vv/vv if nothing third weighing is ^^/^^ as you have already seen SSSS can be ignored

if you your second or third weighing one side dips then remove a single from each side so you would end up with v/v or ^/^ and then either v and v in a hand or ^ and ^

the result of you taking a single stick out will show you which one of the 4 was the odd one

[RATATTAK]

ash73 said:

True!

Maybe need to do three combinations, e.g.
123456 / ABCDEF
123ABC / 456DEF
135ACE / 246BDF

If left up, left up, right up 2 must be lighter
If left up, left up, left up D must be heavier
etc

Devil's advocate says:

How does that work if 3 is lighter or for that matter if F is heavier ?

[V8LM]

2nd weighing is take three from the left and put on the right, take three from the right and put on the table, and take three from the table on put on the left.

[RATATTAK]

Phud said:

second weighing

vv/vv if nothing third weighing is ^^/^^ as you have already seen SSSS can be ignored

if you your second or third weighing one side dips then remove a single from each side so you would end up with v/v or ^/^ and then either v and v in a hand or ^ and ^

the result of you taking a single stick out will show you which one of the 4 was the odd one

That's four weighings ... you can only have three

Clue:

Do NOT ignore your standards