### Brain Teaser

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For those of you who like this sort of stuff,

You have nine cubes all the same size; one is slightly heavier, the rest weigh the same. With a balance scale, find the heavier cube in two tries.

This next one is a little more difficult. You have twelve cubes; one is either slightly heavier or slightly lighter than the rest. In three tries of the scale, you must find the odd cube and tell if it is heavy or light. Hint: you need the same information you had in the previous challenge going in the third try.

-- Maria (anon@ymous.com), May 09, 2000

The first one is trivial. Divide the 9 cubes into 3 groups of 3 each. Compare 2 groups on the scale. If one is heavier, it contains the heavier cube. If they're the same, the one you didn't weigh contains the heavier cube.

Next, take two of the three from the group known to contain the heavy cube and compare them. If either is heavier, you have it. If they're the same, the one you didn't weigh is the heavy one.

By extension, on the final weighing you must have reduced the candidate cubes to no more than 3, AND you must know the direction of difference (heavier or lighter).

So far, the second teaser has me stumped. I can reduce the candidates to 4 and know the direction every time, or I can reduce them to 3 and know the direction 75% of the time, but not 100%.

-- Flint (flintc@mindspring.com), May 09, 2000.

Does this mean you would like another hint, Flint? You're on the right track, you've grouped four.

-- Maria (anon@ymous.com), May 09, 2000.

Flint,

Hey that was not trivial. Give yourself some credit. Otherwise I'm a dunce.

-- (lar@indy.net), May 09, 2000.

2nd one:

Weigh 4 cubes on each side. If they weigh the same, odd one is in other 4.

If same, put 3 of other 4 on scale with one known cube from previous weighing. if these weigh the same, remaining cube is the odd one, compare it to a known to see what it is.

If different, keeping in mind which cube is of known weight, switch one unknown cube from one side of the balance to the side of the balance with the known cube, as well as replacing the unknown cube from the known side with and known cube to be put on the unknown side of the balance. if no change in balance direction, the unchanged cube is odd, and is either heavy or light depending on whether the known side is up or down. If the balance changes direction, the cube you switched is odd, and heavy or light respectively depending on how the balance changes. If the balance is now equal, you removed the odd cube, which is heavy or light depending on how the balance was in step 2.

Deep breath.

If original balance was different, (keep track of cubes) remove one cube from one side and two cubes from the other, as well as switching one cube from one side to the other, replace the removed cubes with knowns and weigh. If balance is now equal, a removed cube is odd.

Take the three removed cubes, and remove one of the cubes from the side you took two off and replace with a known cube. Now switch the unknown cube that was by itself to the side with the other unknown cube (that originally had 2). If the unknown was the cube by itself, the balance will change direction, if it was the cube you removed, the balance will be equal, and if no change, the odd cube is the one you didn't touch, in all cases heavy or light depending on the balance.

If the balance is still lopsided but in a different direction after removing the three cubes, one of the two switched cubes is odd. remove one of them and replace with a known and weigh. if equal, it was the other cube, and if unequal, that's it.

OTOH, if balance is unequal in the same direction, either one of the two unmoved cubes on the one side or the other is odd, so remove one cube from the side with two and switch the one cube by its lonesome, replacing the blanks with known cubes. Again, if balanced, the removed one was odd, if the same lopside, the unmoved is odd, and if changed, the moved is odd. If it's heavy or light is determined by the balance.

Brain teaser is right!

Frank

-- Someone (ChimingIn@twocents.cam), May 09, 2000.