SOLUTION FOR BAGS AND BEADS:
Label the bags from 1 to 5. Take 1 bead out of Bag 1, and label it 1. Take 2 beads out of Bag 2, and label them both with a 2. Take 3 beads out of Bag 3, and label each with a 3. Continue this pattern with Bags 4 and 5. Put these 15 beads on the tray of the scale.
If all 15 weighed 10 grams, the scale would register 150 grams. But since one or more of the beads weighs only 9 grams, the scale will register less than 150. Subtract the number on the scale from 150. Your answer will tell you the number of the bag with the lighter beads. (If the scale registers 148, it’s bag #2. If the scale registers 145, it’s bag #5.)
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SOLUTION FOR GOLD COINS:
step 1: we name all the bag of gold coins as #1, #2, #3......#8, #9, and #10
step 2: we put 1 coin from bag #1, 2 coins from bag #2, 3 coins from bag #3.........8 coins from bag #8, 9 coins from bag #9, and 10 coins from bag 10 onto the scale. Find out the total weight.
step 3: the total weight should have been 10 grams X
(1+2+3+4+5+6+7+8+9+10=55) = 550 grams if all coins are the same (10grams each).
step 4: Subtract the total of step 2 from total of step 3.
Conclusion: If step 4 results 1 gram, then bag #1 is the low quality coins, if step 4 results 2 grams, then bag #2 is the one, if step 4 results 3 grams, then bag #3 is the one.......etc.
Thanks to Jason Vuong for this variation!
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SOLUTION FOR NINE MARBLES:
Place 3 marbles on each tray.
- If the first weighing doesn't balance, remove all the marbles from the lighter side, and place one marble on each tray from the heavier tray. The heavier side is the 1.1 ounce marble, but if they balance, then the marble from the heavier tray from the first weighing that was not weighed in the second weighing is the heavier one(1.1).
- If the marbles balance on the first weighing, remove the marbles from the trays, and place 2 of the remaining unweighed marbles on the trays, one on each tray. If one is heavier, it is the heavier marble(1.1), but if they balance, the remaining unweighed marble is the heavier one.
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SOLUTION FOR NINE GOLD COINS
Step One: Take any 8 of the 9 coins, and load the scale up with four coins on either side. If the two sides are equal, then the remaining coin is the fake.
If however, the two sides are not equal, then that remaining coin is a real coin and the fake is in one side or the other of the scale. The trick is that now you must unload at the same time a single coin from each side of the scales. If the scales now balance, the bad coin is one of the two which you just withdrew. If the scales remain unbalanced, the fake is still on the scales. As you remove good coins you can just add them to the good coin pile which began with the first isolated coin. Once you have found the two coins which when removed balance the scales, or if they are the final two and the scales are still unbalanced, you take one of those two and weigh it against a known good coin. If they balance on this second loading of the scales, or if the don't, you have now with only two loadings of the scales correctly identified the fake coin.
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