*Originally posted by sarathian*

** I can only say that even 39 is not the highest number of marbles for 4 weighings. The highest number of marbles with 4 times use of the balance, is 40.
**

As Plumber has already hinted , there is a soli ...[text shortened]... be chosen at the 4 different stages of the BALANCING act.

Let me put forth my steps. Am I missing something? Where ?

Let us have 4 boxes named U, E, H & L , for keeping the marbles for the following different categories.

(i)U-box - undecided; the odd marble may be in this group and it is not known whether it is heavier or lighter.

(ii)E-box- for marbles with weight equal to the norm; The odd marble is decidedly not in this group.

(iii)H-box- for marbles which may contain the odd marble if it is heavier.

(iv)L-box- for marbles which may contain the odd marble if it is lighter .

In the case of 39 marbles the initially contents of the boxes are-

U- 39, E -0, H- 0, L-0.

The contents of these boxes will change after each stage of balancing as a consequence of the inference of that weighing.

First WEIGHING: (13 U marbles on LHS) <-vs-> (13 U marbles on RHS).

Case( C1)LHS=RHS =>U-13, E-26, H-0, L-0.

(In this case the odd marble can be decided in next 3 weighings as in the 13-marbles puzzle).

Case(C2)LHS.NE.RHS =>U-0, E-13, H-13, L-13.

Second WEIGHING: ( 9H+6L on LHS)<-vs->(3H+12E onRHS).

Case(C21)LHS=LHS=>U-0, E-31, H-1, L-7.

Third WEIGHING under C21: (3L on LHS)<-vs->(3L on RHS).

Case(C211)LHS=RHS=>U-0, E-37, H-1, L-1. In one further weighing the odd one out of the H-1 & L-1 can be decided.

Case(C212)LHS.NE.RHS=>U-0, E-35, H-0, L-4.In this case too, one additional weighing locates the odd one.

Case (C22)LHS>RHS=>U-0, E-30, H-9, L-0.

Third WEIGHING under C22: (3H on LHS)<-vs->(3H on RHS).

Case(C221)LHS=RHS=>U-0, E-36, H-3, L-0(In next 1 weighing the odd one gets decided).

Case(C222)LHS.NE.RHS=>One of the 3 in the heavier side contains the odd one and can be decided in next weighing.

Case(C23)LHS<RHS=>U-0, E-30, H-3, L-6.

Third weighing under C23: (3L on LHS)<-vs->(3L on RHS).

Case (C231)LHS=RHS=>U-0, E-36, H-3, L-0. And last 1 weighing locates the ghost.

Case(C232)LHS.NE.RHS=>One of the three on the lighter side contains the culprit and can be nabbed in 1 last weighing.

This decides and locates the demonic odd marble out of 39. If there are 40 marbles containing an odd one , then after the first stage of weighing , in case C1 you will have U-14, E-26, L-0, H-0. And with 14 undecided category marbles, you cannot decide the odd one out in 3 weighings, as has been shown above by one poster above(TheMaster37).

Am I missing something?

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