You have 12 coins, six weigh 24 grams and six weigh 25 grams. You also have the broken scale from Five Weights and a Broken Scale.

Sort the 12 coins into the group of 24g coins and the group of 25g coins using that broken scale no more than 9 times.

First split the coins into rows of 3 each. Compare the weights of two of the four rows. Do it for the other 2 as well. We have:

Case 1

24,24,24 = 24,24,24

25,25,25 = 25,25,25

Case 2

24,24,25 x= 24,24,24

25,25,24 x= 25,25,25

Case 3

24,25,25 > 24,24,24

24,24,25 < 25,25,25

Case 4

25,25,25 > 24,24,24

24,24,24 < 25,25,25

Case 1 and 2 hold for a scenario in which both sides are balanced, Case 3 and 4 are unbalanced. Note also that it takes a difference of 2g to cause tipping. This principle applies throughout. The above procedure takes 2 steps.

If balanced, switch the rows (i.e do a cross exam)

Case A

24,24,24 < 25,25,25

25,25,25 > 24,24,24

Case B

24,24,25 < 25,25,25

25,25,24 > 24,24,24

This takes 1 step. Now, do a switch between the individual coins of two of the matched (the ones being compared) rows. Switch the first coin of each row with the first coin of the other row. Repeat for the second and third coins. We should have:

24,24,25 < 25,25,24

25,24,25 > 24,25,24

24,25,25 > 25,24,24

24,24,24 < 25,25,25

This takes 3 steps. You should be able to deduce the weight by the position of the coins. If by switching the ALL positions of the coins their balance does not change, one can conclude Case A.

Now back to Step 3. The scenario if the scale is unbalanced. Simply apply the above Steps 4-6 for Case 3 and 4 respectively. This should take 7 steps in addition to Step 3. (1+3+3)

Altogether 9 steps. =)

Btw.. the cases that i have stated out might not be exhuastive, especially for Case 1-4. However, the solution method provided should give one the proper answers in the exact number of steps if followed properly. 

 

 

Posted by Eugene on 2005-06-30 06:34:55