Odd man out

This is called a Logical elimination puzzle.

Assume that a weighing is a metal ball ..... this is done in order to resolve confusion between a weighing (ball) and a weighing (action) in the following explanation

We know when balance scale goes up or down, all the balls participating in that weighing must have one odd-man out among them.

We will consider only left pan of the balance As heavy (>), light (<) or equal (=) compared to the right pan in a weighing.

So total combinations of >,<,= with repetitions in 3 weighings attempts are 3^3 = 27.

Explanation: 9 combinations for > in first place as follows


so such 9 9 combinations for < and = each in first place. Exactly 27.

But if we do not keep same ball or a set of balls on same side of the balance then we can eliminate 3 repetitions i.e. >>>,=== & <<< from 27 outcomes So total outcomes 24.

These 24 outcomes have 2 sets of 12 each which are counter combinations of each other i.e. >=< is the counter combination of <=> (where > is replaced by < and vice versa keeping = intact)

so we will arrange 12 balls (named from A to L) each per pair of a counter combination as follows




Thus A is arranged for a combination ><= & <>= (see the vertical column for A ).



Thus for
weighing 1: compare > balls which are A,B,C,D with < balls which are E,F,G,H
weighing 2: compare > balls which are G,H,I,K with < balls which are A,E,F,J
weighing 3: compare > balls which are B, E, H, I with < balls which are C, G, K, L

The results of these three weighings as observed on the left of the balance are noted. If the outcome is >, <, = we conclude that ball A is heavy & if =, =, < ball L is heavy.

If we obtain an outcome that appears in the lower array, we conclude that the corresponding ball is light. Thus for <, >, = ball A is light (counter combination), & =, =, > means ball L is light.

Bingo!

Of course how stupid of me not to see it straight away

this is a puzzle that was asked of me about 4 years ago and i worked out the answer in about 6 hours. the explaination given by vinsan looks rather complicated and i dare say its right but if you cannot decypher vinson's rather complex solution, then i can provide a much easier to understand solution

Please do debnpete.

odd man out
Don1, sorry about the delay in replying. did i say i had an easier to understand solution? oops! no, i jest. my solution is more graphical than vinsan's, but it wont allow me to put it on here only can type. the main difference between my solution and vinsan's is that mine depends on the result of the first weigh to determine which balls are used on the second weigh, and again for the third weigh. vinsan has obviously used higher maths to give the answer, whereas i have used lateral thinking and hard work. vinsan's is the smart way (weigh?) so to speak. i could email my answer to you to show it is different, but i can also explain vinsan's way more clearly. here goes.

number the balls 1 to 12

get your weighing scales
observe the left pan and note what happens to it
if it goes down then its heavy, write (H)
if it goes up then its light, write (L)
if it stays the same then its equal, write (E)

on the left pan place 1,2,3,4
on the right pan place 5,6,7,8
write down what happens to the left pan (H,L orE)

on the left pan now place 7,8,9,11
on the right pan now place 1,5,6,10
write down what happens to the left pan (H,L or E)

on the left pan now place 2,5,8,9
on the right pan now place 3,7,11,12
write down what happens to the left pan (H,L or E)

you should now have three answers, a combination of H,L,or E (heavy, light or equal) written down in order of weighing

here are the answers for each combination

H,L,E. 1 is heavy
L,H,E. 1 is light

H,E,H. 2 is heavy
L,E,L. 2 is light

H,E,L. 3 is heavy
L,E,H. 3 is light

H,E,E. 4 is heavy
L,E,E. 4 is light

L,L,H. 5 is heavy
H,H,L. 5 is light

L,L,E. 6 is heavy
H,H,E. 6 is light

L,H,L. 7 is heavy
H,L,H. 7 is light

L,H,H. 8 is heavy
H,L,L. 8 is light

E,H,H. 9 is heavy
E,L,L. 9 is light

E,L,E. 10 is heavy
E,H,E. 10 is light

E,H,L. 11 is heavy
E,L,H. 11 is light

E,E,L. 12 is heavy
E,E,H. 12 is light

phew!! i hope you understand this
like i said its a smart mathematical solution though i'm proud of the way i worked out the answer, but then i would be wouldn't i ?

by the way i live just up the road in Rotherham, South Yorkshire.

There's a much easier way to explain it
i dont understand the metal ball thing, but i do know that there's an easier way. you weigh all 12 (6 on each side), whichever is heavier you weigh that pile (3 on each side). whichever side is heavier, take 1 away from the stack and remember which it was. then weigh the remaining 2, (1 on each side) and you have your answer. if the 2 sides are equal, then the 3rd one you took away from the is your heaviest weight

Re: There's a much easier way to explain it

The odd ball may be heavier OR lighter - you don't know which ahead of time.

Re: There's a much easier way to explain it


whoops...i feel stupid now

You shouldn't. You just solved a different problem.