if aunt Maye is correct, and if we trust her scale for the first three measurements, how can we weigh the twelve die to find the fixed die

Question

Your hosting n old time game night. Old uncle hogswalla brought twelve of his "shooting dice" ,but can't remember which one is the cheater. Luckily, Aunt Maye brought her old balance scale and suggest that you use it to figure out which die is fixed.Her theory is the fixed die will be heavier or lighter. The only catch is her scale is only good for about three readings. After that, its fairly unreliable.

Answer 1

SOLVABLE! What is often missed with using a balance scale, is that there are three possible outcomes. In Computer Science, we represent this as a node in a tree structure with three branches. At the second and third weighing levels of the tree, the nodes also have three branches. Thus after one weighing, 3 results; after 2 weighings, 3²=9 results; after 3 weighings, 3³=27 results. This is enough to determine one of 24 possibilities (which of 12 dice and whether heavier or lighter).

Since this is a very standard problem, I copied the text of a solution as follows:

PROBLEM: You have 12 dice identical in size and appearance but 1 is an odd weight (could be either light or heavy).

You have a set of balance scales which will give 3 possible readings:

Left = Right,
Left > Right, or
Left < Right
(i.e. Left and Right have equal weight, Left is Heavier, or Left is Lighter).

You have only 3 chances to weigh the dice in any combination using the scales.

GOAL: Determine which die is the odd one and whether it is heavier or lighter than the rest.

How do you do it?

SOLUTION:

Number the dice 1, 2, 3, ... 10, 11, 12

Start off with them in 3 groups: [1, 2, 3 and 4], [5, 6, 7 and 8] and [9,10,11 and 12]

Weigh 1, 2, 3 and 4 vs 5, 6, 7 and 8 with 3 possible outcomes:

1. If they balance then 9,10,11,12 have the odd die, so weigh 6,7,8 vs 9,10,11 with 3 possible outcomes:

1a
If 6,7,8 vs 9,10,11 balances, 12 is the odd die. Weigh it against any other die to determine if heavy or light.

1b
If 9,10,11 is heavy then they contain a heavy die. Weigh 9 vs 10, if balanced then 11 is the odd heavy die, else the heavier of 9 or 10 is the odd heavy die.

1c
If 9,10,11 is light then they contain a light die. Weigh 9 vs 10, if balanced then 11 is the odd light die, else the lighter of 9 or 10 is the odd light die.


2. If 5,6,7,8 > 1,2,3,4 then either 5,6,7,8 contains a heavy die or 1,2,3,4 contains a light die so weigh 1,2,5 vs 3,6,12 with 3 possible outcomes:

2a
If 1,2,5 vs 3,6,12 balances, then either 4 is the odd light die or 7 or 8 is the odd heavy die. Weigh 7 vs 8, if they balance then 4 is the odd light die, or the heaviest of 7 vs 8 is the odd heavy die.

2b
If 3,6,12 is heavy then either 6 is the odd heavy die or 1 or 2 is the odd light die. Weigh 1 vs 2, if balanced then 6 is the odd heavy die, or the lightest of 1 vs 2 is the odd light die.

2c
If 3,6,12 is light then either 3 is light or 5 is heavy. Weigh 3 against any other die, if balanced then 5 is the odd heavy die else 3 is the odd light die.
 

3. If 1,2,3,4 > 5,6,7,8 then either 1,2,3,4 contains a heavy die or 5,6,7,8 contains a light die so weigh 5,6,1 vs 7,2,12 with 3 possible outcomes:

3a
If 5,6,1 vs 7,2,12 balances, then either 8 is the odd light die or 3 or 4 is the odd heavy die. Weigh 3 vs 4, if they balance then 8 is the odd light die,  or the heaviest of 3 vs 4 is the odd heavy die.

3b
If 7,2,12 is heavy then either 2 is the odd heavy die or 5 or 6 is the odd light die. Weigh 5 vs 6, if balanced then 2 is the odd heavy die, or the lightest of 5 vs 6 is the odd light die.

3c
If 7,2,12 is light then either 7 is light or 1 is heavy. Weigh 7 against any other die, if balanced then 1 is the odd heavy die else 7 is the odd light die.