Marble-Weighing Problem
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You have 12 marbles. They all weigh the same, except one. You don't know if that one is heavier or lighter. You have a balance scale. You can perform up to a maximum of three weighings to find out which marble has the different weight, and if it is heavier or lighter than the others.

Avoid this common, but false, "solution:" If you weigh the two groups of six and one group descends, it may be because there's a LIGHTER marble in the other group. Do not assume that the group descending must contain a heavy marble. Your goal is to find the odd marble, and to determine whether it's lighter or heavier than the rest.

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Here is the answer:

General Scenario One:

You weigh two groups of four. If there's an imbalance, you take three from the elevated group (it can be from the heavier group, but let's keep it simple) and put them in the lower group - replacing three of the lower ones in that group. You take three from the unweighed group and put them in the elevated tray.

If the trays now keep the same imbalance, then you know the marble in the lower tray that was not replaced, and the marble in the elevated tray that was not replaced, together contain a heavy or light marble. If that's the case, you weigh one of these marbles against a normal one to see if it is heavy, normal, or light. Keeping in mind which tray (elevated or lower) you took it from, you can tell from that weighing the final status of the marble (or its counterpart) straightforwardly enough.

If the two groups become balanced, then you know the three you took from the lower tray has a heavy marble. If that's the case, see RULE OF THREE below.

If the balance reverses, then you know the three marbles transferred from the elevated to the lower tray contains a light marble. If that's the case, see the RULE OF THREE below.

General Scenario Two:

You weigh two groups of four. If there's a balance, then you know the four unweighed marbles contain a heavy or light one. You replace three of the normal marbles in one of the trays with three of the unweighed marbles. If the tray with the new replacements sinks, you know the three replacements contain a heavy marble. If it becomes elevated, then you know they contain the light marble. If either's the case, go to RULE OF THREE below. If the trays remain even, then you know the last unweighed marble is heavy or light. If that's the case, just weigh it with a normal marble to determine its status.

RULE OF THREE:

Once you determine a specific quality (heavy or light) for a group of 3, a single weighing of one marble against the other (leaving one out) is made. If the two marbles are even, then the one left out has the quality (heavy or light) determined for that group. If the two marbles are uneven, then the elevated marble is light (if light was determined for that group) or the heavier marble is heavy (if heavy was determined for that group).


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