The Contest Center - Scales & Balances

Puzzles with weights and measures

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Scales & Balances
The Contest Center
59 DeGarmo Hills Road
Wappingers Falls, NY 12590

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       There is a whole literature of puzzles involving scales, balances, rulers, liquid measures, and the like. This is a small sampling.
       If you are new to this type of puzzle, you will need to know that a pan balance compares the weight of two samples. If they have the same weight, the balance stays even, otherwise the balance tilts towards the heavier load. Several objects may be placed together on either pan.

       Most of these puzzles require only simple logic and elementary math. The harder puzzles are marked * and **. We will list the names of anyone who solves the marked puzzles, however you may submit the answers to any puzzle, and we will let you know if you are correct.

Pan Balance
You have a pan balance, and wish to be able to weigh any whole number of ounces. You can purchase standardized weights in any whole number of ounces. You can afford to buy only N such weights. What weights should you buy in order to be able to weigh up to the highest possible total weight?

Answer

Coins #1
       You have a number of gold coins of the same denomination. You believe that all of the coins are genuine, but it is possible that one coin is a fake. Genuine coins all weight 10 grams, but a fake coin will weigh either 9 or 11 grams. You have a pan balance, but you have only enough time to make 3 weighings.
       (A) How many coins can you have, and be able to tell whether any of them is fake, and if so, which one?
       (B) Suppose you know for certain one coin is fake. How many coins can you have, and be able to tell which one is the fake?
       (C) Suppose that you also have a large number of coins that you know are genuine. How many coins can you have, and be able to tell which one is the fake?

Answer

Coins #2
You have 20 large bags of coins and a digital scale that gives an exact reading of the weight. All of the bags but one contain genuine coins weighing 10 grams each. The remaining bag contains fake coins coins weighing 9 grams each. How can you identify the bag with the fakes using only 1 weighing?

Answer

* Coins #3
You have N large bags of coins and a digital scale that gives an exact reading of the weight, up to a maximum of 1000 grams. All of the bags but one contain genuine coins weighing 10 grams each. The remaining bag contains fake coins weighing 9 grams each. How many bags can you have and still be certain of identifying the bag with the fake coins using only 2 weighings?

Solved by: Nick McGrath, Andreas Abraham, S. Preethi Sudharsha, Denis Borris, Mark Rickert, P.M.A. Hakeem

** Coins #4
You have just bought 10 huge bags of coins from the notorious Numismo Foodlemyer. You suspect that as many as 3 of the 10 bags may contain fake coins. You know that genuine coins weigh exactly 10 grams, and that the fake coins, if any, all will weigh exactly 11 grams. You have a digital scale which will give you the precise weight of any load.
How can you identify the bags containing the false coins, if any, using only 1 weighing and the smallest possible total number of coins?

Solved by: Denis Borris, Mark Rickert, P.M.A. Hakeem

** The Faulty Scale
I have N bags containing 15 coins each. All but 1 of the bags contain genuine coins which weigh exactly 10 grams each. The other bag has fake coins that weigh either 9 or 11 grams. (The fake coins all have the same weight, and they are in all other respects identical to the real coins.) I have a digital scale which is accurate to within 1%. That is, if I weigh any amount up to 99 grams, the scale will give a correct reading, but if I weigh 100 grams to 199 grams the scale could read 1 gram low, the correct weight, or 1 gram high. Similarly, if the weight is 200 to 299 grams, the reading may be anywhere from 2 grams low to 2 grams high. And so forth.
What is the most bags I could have and still be able to identify the bag of fake coins in at most 5 weighings?

Solved by: Denis Borris, Jean Jacquelin, P.M.A. Hakeem

* U.S. Coins
       Suppose that you are the Supervisor of the U.S. Mint, and it is your responsibility to design a new set of currency. You have a free hand in choosing the denominations for the coins and bills. You need to provide denominations ranging from 1 cent to 100 dollars.
       You need to balance several important considerations. To save costs, you have to keep the number of different denominations, as well as the total number of coins and bills produced each year, small. You would like to make the process of making change easy, so that any amount of money from 1 cent to 100 dollars can be made with a small number of coins and bills. Finally, you need to choose denominations that make the calculations easy, even for young children. (You would not want to pick something like 1, 7, 19, and 54 cent coins.)
       What is the optimum set of denominations that best balances all 3 goals?

Answer

Cups
You have cups that hold exactly 4 gills and 9 gills. How can you use these to accurately measure 6 gills of fluid?

Answer

* The Sparse Ruler
It is not necessary for a straight ruler to have tick marks at every inch in order to be able to measure all of the integer distances. For example, a 3" ruler needs only one tick mark at 1" in order to measure objects of 1", 2" and 3". For 2" you measure from the 1" mark to the end of the ruler. Similarly, on a 6" ruler you need only 2 tick marks, at 1" and 4".
What is the longest ruler you can have with 4 tick marks that can measure all of the integer distances up to its full length? (Each distance must be measured in one operation.)

Solved by: Nick McGrath, Carlos Rivera, Janaki Mahalingam, Andreas Abraham, Mark Rickert, Jean-Charles Desjardins, P.M.A. Hakeem

** The Sparse Ruler #2
It is not necessary for a straight ruler to have tick marks at every inch in order to be able to measure all of the integer distances. For example, a 3" ruler needs only one tick mark at 1" in order to measure objects of 1", 2" and 3". For 2" you measure from the 1" mark to the end of the ruler. Similarly, on a 6" ruler you need only 2 tick marks, at 1" and 4".
If you have a ruler with 4 tick marks, what is the highest value of N such that you can measure any distance from 1 to N inches using at most 2 measurements?

Solved by: Mark Rickert, P.M.A. Hakeem

** Two Sparse Rulers
You have two rulers, each of which is a rigid rod with just 4 tick marks. You are allowed to make up to two measurements using either or both rulers.
What is the highest value of N such that you can measure any distance from 1 to N inches?

Solved by: P.M.A. Hakeem

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