The 2001 Puzzle

The powerful wizard Bumblebee gives Henrietta Porter one magical stone along with 11 ordinary stones. The stones look and feel identical, but the magical stone's weight is slightly different from the other eleven stones, which all weigh the same. Bumblebee also provides Henrietta with a balance scale, and tells her that the power of the magical stone will be hers if she can use the scale just three times to determine which of the stones is magical. Henrietta must also determine if the magical stone is heavier or lighter than the ordinary stones.

Solution by Chris Worley  (1997, B.S.)

1)      Take eight of the stones and place 4 on each side of the balance scale.  Arrange the stones in each group of 4 (including the group of 4 not on the scale) as if they were at the ends of a +, so that the stones can be identified.  The stone at the left will be #1, the top #2, the right #3, and the bottom #4.  The scale will either be balanced or unbalanced.  If it is balanced, go to step 2, if it is unbalanced, go to step 6.

2)      The scale is balanced, meaning the magical stone is among the 4 stones not on the scale.  Call the group of stones on the left side of the scale group A, the stones on the right side of the scale group B, and the stones off the scale group C.  Remove stones 2, 3, and 4 from group A and replace with stones 2, 3, and 4 from Group C.  The scale will now remain balanced, will tilt down on the left side, or will tilt down on the right side.  If the scale remains balanced go to step 3.  If it tilts left, go to step 4.  If it tilts right, go to step 5.

3)      The scale has remained balanced, meaning the only stone to have not been placed on the scale, stone C1, is the magical stone.  Clear the scale and be sure to keep the magical stone isolated.  Place stone C1 on the left side of the scale and any of the other 11 stones on the right side.  If the scale tilts down on the left side, the magical stone is heavier than the other stones, but if the scale tilts right, the magical stone is lighter.  Go to step 10.

4)      The scale has tilted left, meaning the magical stone is among stones C2, C3, and C4, and is heavier than the other stones. Clear the scale, keeping stones C2, C3, and C4 isolated from the other stones.  Place stone C2 on the left side of the scale, and stone C3 on the right side.  If the scale is balanced, then stone C4 is the magical stone.  If the scale tilts down on the left, then stone C2 is the magical stone, and if it tilts right, stone C3 is the magical stone.  Go to step 10.

5)      The scale has tilted right, meaning the magical stone is among stones C2, C3, and C4, and is lighter than the other stones.  Clear the scale, keeping stones C2, C3, and C4 isolated from the other stones.  Place stone C2 on the left side of the scale, and stone C3 on the right side.  If the scale is balanced, then stone C4 is the magical stone.  If the scale tilts down on the left, then stone C3 is the magical stone, and if it tilts right, stone C2 is the magical stone.  Go to step 10.

6)      The scale is unbalanced, meaning the magical stone is among the 8 stones on the scale.  Position the scale so that the heavy side is on the left.  Call the group of stones that is heavier group A, the group of stones that is lighter group B, and the group of stones not on the scale group C.  Switch stones A1, A3, and A4 with stones C1, C3, and C4.  Switch stones A2 and B2.  Stones C1, B2, C3, and C4 should now be on the left side, and stones B1, A2, B3, and B4 should now be on the right, with stones A1, C2, A3, and A4 off the scale.  The scale could now be balanced, or unbalanced in either direction.  If the scale is balanced, go to step 7.  If the scale is tilting down on the left side, go to step 8.  If the scale is tilting down on the right side, go to step 9.

7)      The scale is now balanced, which means the magical stone was removed from the scale and is among stones A1, A3, and A4.  The magical stone is also heavier than the other stones since the scale tilted down to the side these stones were on when they were on the scale.  Clear the scale, being sure to keep track of stones A1, A3, and A4.  Place stone A1 on the left side of the scale, and stone A3 on the right side.  If the scale is balanced, stone A4 is the magical stone.  If the scale tilts down to the left, A1 is the magical stone.  If it tilts to the right, then A3 is the magical stone.  Go to step 10.

8)      The scale is still tilting to the left side with 4 different stones on that side than the first try.  This means that the magical stone is on the right side and lighter than the other stones.  Since stone A2 was not on the right side on the first use, the magical stone must be among stones B1, B3, and B4.  Clear the scale, being sure to keep track of stones B1, B3, and B4.  Place stone B1 on the left side of the scale, and stone B3 on the right side.  If the scale is balanced, then stone B4 is the magical stone.  If the scale tilts down to the left, B3 is the magical stone.  If it tilts down to the right, B1 is the magical stone.  Go to step 10.

9)      The scale has switched and is now tilting down on the right side.  This means that the magical stone switched sides of the scale, and must be either stone A2 or stone B2.  Clear the scale, keeping track of stones A2, B2, and C2 (any of the normal stones can be used in place of C2).  Place stone A2 on the left of the scale, and stone C2 on the right of the scale.  If the scale is balanced, then stone B2 is the magical stone, and the magical stone is lighter than the others, since the side B2 was on was always lighter than the other side.  If the scale tilts, it will tilt left, and stone A2 is the magical stone and is heavier than the other stones.  The scale can not tilt right because stone A2 was always on the heavy side of the scale, and must be either heavier or the same weight as all the other stones.  Go to step 10.

10)  The magical stone is in hand, the comparative weight is known, and the wizard’s puzzle has been solved.  Henrietta can now enjoy the power of the magical stone.