### Notes regarding the exercise.

First of all, this is a **hard** problem. The purpose of this exercise is absolutely not to cause
everyone to solve the 12-ball problem, or worse, to show everyone the solution. The persistent, perhaps the obsessed,
will solve it eventually.
The exercise is rather about basic mathematical method, along with some beginning thoughts in combinatorics.

The problem is probably beyond many students until they do some preliminary work on smaller problems. The "profound observation" (thanks Dan K.) is the need to break up possible answers into relatively same-sized groups with each weighing. With that the whole problem may become accessible.

#### Downloads

OddBall, a simple Windows program, is available for download. (Written in VB6 quite a while ago, so no idea if it will still work on your machine. Test it.)

It allows you to select the number of balls (2 to 15), and whether the odd ball is light, heavy, or unknown. You drag balls onto the scale pans, then click the "Test" button to see the results. When you've solved the problem, drag the odd ball to the appropriate box for credit.

- OddBall.exe [44KB] - this is just the executable. If you have other VB6 programs on the system it may run correctly.
- OddBall.CAB [1.3MB] and setup.exe [1.6KB] - the CAB contains all necessary files. Download both pieces and run "setup.exe".

#### Analysis

Students must analyze the problem in terms of separate cases. They must decide what results are possible from a given test, then focus and draw conclusions from each individual result in turn. Finally, the conclusions must be brought together to assure that all possibilities are accounted for.

#### Sequential operations

The most important combinatoric result is the multiplicative effect of successive tests. While one test has three possible results (left, balanced, or right), two tests produce nine possible outcomes (left-left, left-balanced, etc.). The sequence, or combination, of tests has meaning. In fact, the balls chosen for the second test may depend on the results of the first.

Note: Although there are nine theoretical outcomes from two tests, students may recognize that four of these combinations will never actually occur.

If students can correctly determine the number of tests for a 100-ball or 1000-ball problem (with one heavy ball), that is much more important than solving the original 12-ball challenge. Likewise, the realization that the 12-ball problem presents 25 possible outcomes and is therefore potentially within reach of three tests is more important than the actual list of tests.

#### Models

Keeping track of the tests will require some organized record-keeping, especially for problems with more balls. A graphic decision tree is an excellent tool. An alternative would be a decision outline, as sketched below. Only the format is different.

Sample solution to five-ball problem (with the odd ball heavy)

- Weigh ball #1 (left) against ball #2 (right).

- If the left side is heavier, #1 is the odd ball.
- If the right side is heavier, #2 is the odd ball.
- If they balance, #1 and #2 are both normal. The odd ball must be #3, #4, or #5. I will therefore test #3 (left) against #4 (right).

- If the left side is heavier, #3 is the odd ball.
- If the right side is heaver, #4 is the odd ball.
- If they balance, #3 and #4 are both normal. Since I already know tha #1 and #2 are normal, the odd ball must be #5.

#### Generalization

Using a solution appropriately is as important as finding it in the first place. It's in the nature of mathematics to apply solutions to similar problems and try to create more general, and therefore more powerful, rules. The multiplication of cases by successive events goes far beyond billiard balls and balance scales. But a grasp of these problems will allow an application of the rule based on understanding and not superstition.