Whiskey, Vodka, Rum, Tequila.

Now, the answer to last week’s Friday Challenge.

First, let’s recap:  We’ve got 12 identical-looking steel balls (yes we do), exactly one of which is either slightly heavier or else slightly lighter than the other 11 identical balls.  We’ve got only a balance scale and our wits, and we need to find out which ball is odd and whether it is lighter or heavier than the rest in only 4 uses of the scale.

Here’s how:

First, divide your balls into 4 groups of equal size.  We’ll name our four groups Whiskey, Vodka, Rum, and Tequila.  The odd ball must be in one of those 4 groups, so let’s examine the 4 different possibilities, and your method of addressing each.  Remember that we don’t know when we begin any of these scenarios where the odd ball is; we’ll begin methodically, as follows:

If the Odd Ball Were In the Tequila Group:

Weighing 1:

Weigh the entire Whiskey group against the Vodka group.  If they weigh the same, you know that those six balls are all standard, and that the odd ball is in either Rum or Tequila.  Let’s find out which group it’s in:

Weighing 2:

Keep Whiskey on the scale, and weigh it against the Rum group.  If they weigh the same, the odd ball must be in the Tequila group.  Let’s now find out if it’s heavier or lighter:

Weighing 3:

Keep Whiskey on the scale, and weigh it against the Tequila group.  Now you know whether the odd ball in the Tequila group is heavier or lighter than the others.  Now let’s find out which ball it is:

Weighing 4:

Weigh one randomly selected ball from the Tequila group against another.  If the two balls weigh the same, you’ll know the Tequila ball you didn’t weigh is the odd one (and you already know from Weighing 3 whether it is light or heavy).  If they’re different, then obviously you know that the odd ball is the one that is lighter or heavier (again, as determined in Weighing 3).  Done.

If the Odd Ball Were In the Rum Group:

Weighing 1:

Weigh the entire Whiskey group against the Vodka group.  If they weigh the same, you know that those six balls are all standard, and that the odd ball is in either Rum or Tequila.  Let’s find out:

Weighing 2:

Keep Whiskey on the scale, and weigh it against the Rum group.  If they are different, then Rum contains the odd ball, and you now know whether it is heavier or lighter than the standard balls in the Whiskey group.  Now, let’s find out which ball is odd:

Weighing 3:

Weigh one randomly selected ball from the Rum group against another.  If the two balls weigh the same, you’ll know the Rum ball you didn’t weigh is the odd one (and you already know from Weighing 2 whether it is light or heavy).  If they’re different, then obviously you know that the odd ball is the one that is lighter or heavier (again, as determined in Weighing 2).  Done.

If the Odd Ball Were In the Vodka Group:

Weighing 1:

Weigh the entire Whiskey group against the Vodka group.  If they are different, you know that the odd ball is in either Whiskey or Vodka (and that all six balls in Rum and Tequila are standard), but not yet whether the odd ball is heavy or light.  Let’s find out:

Weighing 2:

Keep Whiskey on the scale, and weigh it against the Rum group.  If they weigh the same, the odd ball must be in the Vodka group, and you know from Weighing 1 whether the odd Vodka group was heavier or lighter than the standard Whiskey group.

Weighing 3:

Weigh one randomly selected ball from the Vodka group against another.  If the two balls weigh the same, you’ll know the Vodka ball you didn’t weigh is the odd one (and you already know from Weighing 2 whether it is light or heavy).  If they’re different, then obviously you know that the odd ball is the one that is lighter or heavier (again, as determined in Weighing 2).  Done.

If the Odd Ball Were In the Whiskey Group:

Weighing 1:

Weigh the entire Whiskey group against the Vodka group.  If they are different, you know that the odd ball is in either Whiskey or Vodka (and that all six balls in Rum and Tequila are standard), but not yet whether the odd ball is heavy or light.  Let’s find out:

Weighing 2:

Keep Whiskey on the scale, and weigh it against the Rum group.  If they are different, the odd ball must be in the Whiskey group, and you know from Weighing 1 whether the odd Whiskey group was heavier or lighter than the standard Vodka group.

Weighing 3:

Weigh one randomly selected ball from the Whiskey group against another.  If the two balls weigh the same, you’ll know the Whiskey ball you didn’t weigh is the odd one (and you already know from Weighing 2 whether it is light or heavy).  If they’re different, then obviously you know that the odd ball is the one that is lighter or heavier (again, as determined in Weighing 2).  Done.