A Solution To The 12 Ball Problem Statement of the problem: You have 12 balls, one of which is heavy or light. In 3 weighings, determine which ball is odd and whether it is light or heavy. Label the 12 balls 1, 2, etc. to 12. Perform the following 3 weighings: 1 2 3 4 5 6 7 8 1 4 8 9 2 3 11 12 3 7 9 12 1 2 5 10 If the left side goes Up, Up, Down, then 1 is light Up, Down, Down, then 2 is light Up, Down, Up, then 3 is light Up, Up, Even, then 4 is light Up, Even, Up, then 5 is heavy Even, Up, Up, then 9 is light Up, Even, Even, then 6 is heavy Even, Even, Up, then 10 is heavy Even, Up, Even, then 11 is heavy Down, Up, Even, then 8 is light Down, Even, Up, then 7 is light Even, Down, Up, then 12 is light Changing all words Up to Down and Down to Up (and leaving Even alone,) the 12 other possibilities are given by changing light to heavy and heavy to light. Note: There are many other ways of presenting the solution to this interesting problem. The one above reflects the particular way it came out the way I did it. I suppose I could have stated it so the right hand side always read light. This of course would mean I would simply have to change the 5th weighing for example to read Down, Even, Down, then 5 is light. etc. How I came up with the list above is a complicated thing to explain. Basically, I tried to remember how many balls you need to use on each side (for this problem was given to me many years ago and I have seen the solution; I played with 3 on each side for a while and then decided 4 on each side was needed.) Then I decided listing the unique Up/Down/Even possibilities would be useful. Then it was a matter of starting with the first weighing (1 2 3 4 5 6 7 8,) and arbitrarily deciding ball 1 was light (my first Up/Down/Even code happened to be Up, Up, Down, (or UUD in code form (see musings, below))) and this would happen provided ball 1 was light and I put ball 1 on the left side of the scale in the second weighing and on the right side in the 3rd. So that's just exactly what I did. The next Up/Down/Even code in my list similarly "generated" where ball 2 would need to be placed in the last two weighings. Instead of wasting many hours using my brain, guesswork etc. etc. I realized letting the three letter codes generate the placement of the odd ball was the way to go. In the whateverit'sworth department, doing all this reminded me of DNA, the four "bases" ATCG (Adenine, Thymine, Cytosine and Guanine) and how this 4 letter code dictates the helix of life and I eventually convinced myself that UUD, UDD, UDU, UUE, UEU, EUU, UEE, EEU, EUE, DUE, DEU and EDU were the building blocks of the 12 ball problem and well ... Problem solving is truly amazing ... A completely different approach is given in the graphic below. It was obtained from http://www.curiouser.co.uk/ and is shown here with permission. http://www.primepuzzle.com/leeslatest/12_ball_solution.html - Friday, June 09, 2006