You have two containers with capacities a = 5, b = 3 liters. The objective is to

Question

You have two containers with capacities a = 5, b = 3 liters. The objective is to obtain exactly c liters into one of the containers. You can only perform the following operations: • Fill a container to the top, • Empty a container completely, • Pour the liquid from one container to the other. (a) Describe how to solve the “Die Hard” problem when a = 5, b = 3 and c = 4. How do you solve the problem if c = 1? (b) Discuss how to solve the problem when a = 12, b = 9 and c = 4. (c) Describe the condition required of a and b so that you can always solve the problem for any 1 c max(a, b)?

Explanation / Answer

a. a = 5. b = 3. c = 4.

1. Fill 5 liter Jug. (5-liter jug holds 5 liter, and 3 liter jug is empty.)

2. Fill the 3 liter jug from 5 liter jug. (5-liter jug holds 2 liters now, and 3 liter jug is full.)

3. Empty 3 liter jug. (5-liter jug holds 2 liters now, and 3 liter jug is empty.)

4. Pour the water from 5-liter jug to 3-liter jug. (5-liter jug is empty, and 3-liter jug is holding 2-liter now.)

5. Fill the 5-liter jug. (5-liter jug is full with 5-liters and 3-liter jug is holding 2-liter now, with 1-liter space to hold.)

6. Finally, filling the 3-liter jug from 5-liter jug, will make the 5-liter jug to be left with 4-liter liquid.

If a = 5, b = 3, c = 1.

1. Fill the 3 liter jug. (5-liter jug is empty and 3-liter jug is full with 3-liter liquid.)

2. Pour the 3-liter liquid from 3-liter jug to 5-liter jug. (5-liter jug is holding 3-liter liquid, and 3-liter jug is empty.)

3. Fill the 3 liter jug again. (5-liter jug is holding 3-liter liquid, and 3-liter jug is full.)

4. Finally, filling the 5-liter jug from 3-liter jug will take 2 liters from it, and the 3-liter jug will be left with 1 liter.

b. a = 12. b = 9. c = 4.

This problem cannot be solved. i.e., using a 12-liter jug, and a 9-liter jug, you can't measure 4 liters, with whatever operations you do.

c. The condition required of a and b so that you can always solve the problem for any 1 <= c <= max(a,b) is, a and b should be co-primes.

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