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HandoutLecture 6Binary Search Weighing with a Balance A large container is known to hold 24 oz of nails. The hardware store has a balance, but no weights. Can you measure out 9 oz of nails for a customer? A chemist has a balance and fixed weights of 1, 2, 4, and 8 grams. Show that she can weigh any amount of material from 1 to 15 grams by placing the weights on one side and the material on the other. What is the best set of four fixed weights in the sense of maximizing the range of measurable weights in increments of 1 gram?

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HandoutLecture 6Binary Search Find the Lighter Counterfeit Coin We have three coins. Two are normal coins; one is a counterfeit coin that weighs less. Identify the counterfeit coin with one weighing on a balance. We have nine coins; eight normal coins and a counterfeit coin that weighs less. Identify the counterfeit with 2 weighings. Generalize: How many weighing with a balance are needed to find a light counterfeit coin among n coins? How should we change the procedures above if the counterfeit coin is known to be heavier than normal ones?

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HandoutLecture 6Binary Search 12 Coins with 1 Counterfeit: Lighter or Heavier We have 12 coins. Eleven are normal coins; one is a counterfeit coin that weighs less or more than a normal coin. Identify the counterfeit coin and its relative weight with a minimum number of weighings on a balance. Hint: First do it for 3 coins, one of which is a counterfeit, using only two weighing,

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HandoutLecture 6Binary Search Another Solution to the 12-Coin Puzzle We have 12 coins. Eleven are normal coins; one is a counterfeit coin 1 4 6 10 against 5 7 9 12 2 5 4 11 against 6 8 7 10 3 6 5 12 against 4 9 8 11 Each weighing has three possible outcomes: L -- Left heavier R -- Right heavier B -- Balance 1 2 4 3 5 6 8 7 9 10 12 11 Example: L L B -- Counterfeit coin is among 1, 2, 7, 10 7 is lighter L L Lnp L L B7– L L R L B L L B B L B R L R L L R B L R R B L L B L B B L R B B L B B Bnp B B R B R L B R B B R R R L L R L B R L R R B L R B B R B R R R L R R B7+ R R Rnp

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Discrete Structure Li Tak Sing( 李德成 ) Lectures 20-22 1.

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