TED, you're just quibbling because you can't solve it (gg). As for me, I'm
a sucker for a logic problem. This one is rather straight forward to solve
in your head, but hard to put in writing. Here's what I would do:
The problem
Given that you have twelve metal balls of identical size, one of which is
heavier or lighter than the others, explain how, using a balance scale only
three times, you can determine which of the balls differs in weight from the
others and whether it is lighter or heavier.
1. Put aside 4 of the 12 balls and weigh the remaining 8, 4 on each side.
1.1 If the scales are EVEN, the "odd" ball is part of the group set aside;
Put an "X" on all 8 balls ON the scale as, by definition, they must
weigh the same. Proceed to step 2.
1.2 If the scales are UNEVEN, the "odd" ball is on the scale; Label the
4 balls OFF the scale with an "X", as, by definition, they must weigh
the same. Proceed to step 3.
2. Put 3 of the "X" balls on the left side of the scale and any 3 of the
remaining 4 on the right
2.1 If the scales are EQUAL, the remaining ball with no "X" is the odd one.
Weigh it against any other to see if it is heavier or lighter. Finished.
2.2 If the scales are UNEQUAL, the odd ball is on the right side and heavier
if the right side is heavier and lighter if the right side is lighter.
Label the odd ball out with an "X"; put one of the 3 remaining balls
aside, and weigh the other 2 against each other.
2.2.1 If they do NOT weigh the same, the odd ball is on the scale and weighs
as determined in step 2.2. Finished.
2.2.2 If they DO weigh the same, the odd ball is the remaining ball off the
scale and weighs as determined in step 2.2. Finished.
3. Remember which side was heaviest. Then, label the 4 balls on the left
side of the scale with "A" and the 4 balls on the right with "B". Set
aside 2 "A"s and 1 "B" and then exchange 1 "A" for 1 "B" on the scale,
add an "X" for the other removed "A" on the left, and weigh again.
3.1 If the scales TILT the SAME as in step 3, then proceed to step 4.
3.2 If the scales TILT the OPPOSITE as in step 3, then proceed to step 5.
3.3 If the scales DO NOT TILT AT ALL, the proceed to step 6.
4. The odd ball is one of the 3 balls that was NOT moved (the "A" on the left
or one of the 2 "B"s on the right). Put the "A" ball aside, then put a
"B" on each side of the scale and weigh.
4.1 If the scales are EQUAL, then the odd ball is the "A" ball that was put
aside in step 4. It weighs what was determined in step 3. Finished.
4.2 If the scales are UNEQUAL, the odd ball is the same weight as the right
side of the scale in step 3 (i.e. the heavier one if the right side was
heavier; the lighter one if the right side was lighter). Finished.
5. The odd ball is one of the 2 moved balls (the "B" on the left or the "A"
on the right). Remove all but these 2 balls. Replace the "B" ball on the
left with 1 "X" ball and weigh again.
5.1 If the 2 balls are EQUAL, then the odd ball is the "B" ball that was
just replaced on the left. It is heavier or lighter depending on the
outcome of the right side of the scale in step 3. Finished.
5.2 If the 2 balls are UNEQUAL, then the odd ball is the "A" ball on the
right. It is heavier or lighter depending on this weighing. Finished.
6. The odd ball is one of the 3 balls that was removed (2 "A"s from the left
and 1 "B" from the right. Put the "B" ball aside, then put an "A" on each
side of the scale and weigh.
6.1 If the scales are EQUAL, then the odd ball is the "B" ball that was put
aside in step 6. It weighs what was determined in step 3. Finished.
6.2 If the scales are UNEQUAL, the odd ball is the same weight as the left
side of the scale in step 3 (i.e. the heavier one if the left side was
heavier; the lighter one if the left side was lighter). Finished.
- Jeff |
