## Milkman's dilemma in giving 1 quart of milk each to a 4 pint and a 5 pint jug

A milkman has to deliver 1 quart of milk each to a 4 pint and a 5 pint jug from his two 10 gallon filled up milk cans. Time to solve 20 mins.

### The riddle of delivering 1 quart of milk each into two jugs

A milkman while doing his morning round met two women. One of them had a 4 pint jug, the other a 5 pint jug. They wanted the milkman to sell them 1 quart of milk each. The milkman carried two cans full of 10 gallons of milk each and no measuring tool. He had no way to measure the 1 quart of milk wanted by the women.

But, being a brainy fellow, he could finally measure the 1 quart of milk each into the two jugs of the two women.

Can you do it in least number of steps?

You may transfer any amount of milk between the four containers as many times as you want.

**Information:** 1 gallon equals 4 quarts and 1 quart equals 2 pints.

**Time to solve:** 20 minutes.

### Solution to the two jugs riddle of delivering 1 quart of milk each

How will you start? Can you measure 1 quart or 2 pints of milk easily in one of the two empty jugs?

Only sure way of measuring here is to

fill a jug or a can with known empty volume.

Let us measure what we can easily. We won’t think of the solution.

Fill 5 pint jug from first 10 gallon can, fill 4 pint jug from 5 pint jug leaving 1 pint of milk in 5 pint jug. Representing actions by arithmetic is easy to understand,

$5-4=1$ pint milk in 5 pint jug.

Empty 4 pint jug into first 10 gallon can. It has 1 pint of empty volume.

Transfer 1 pint from 5 pint jug to 4 pint jug. Fill 5 pint jug again from the first 10 gallon can. It has now 6 pints of empty volume.

Pour 3 pints from 5 pint jug into the 4 pint jug with 1 pint of milk in it. The 4 pint jug filled and 2 pints, or 1 quart of milk, remains in 5 pint jug.

$5-3=2$ pints.

We have 1 quart of milk in the 5 pint jug.

Till this point it has been easy, but how to proceed now? You avoid complexity in thinking this direction and decide to think about what must happen in the end.

This is a natural problem-solving technique of **Working backwards approach**. In this approach, you will imagine the last step, then last but one step and so on to the first.

To be more realistic, let's try to figure out what must be the last few steps to put 1 quart of milk into the 4 pint jug, with 1 quart of milk already measured in the 5 pint jug.

After the last step, there has to be 1 quart of milk in each of the 4 pint and 5 pint jugs, isn’t it? We are trying to understand how to measure 1 quart of milk in the 4 pint jug when the 5 pint jug has already 1 quart of milk in it. We have already solved the latter part.

**Assumption:** 1 quart of milk measured in the 5 pint jug.

**Aim:** To measure 1 quart of milk in the 4 pint jug.

You realize,

When 4 pint jug is empty and 5 pint jug contains 1 quart of milk, this amount of milk must miss from one of the two 10 gallon cans. Fill this partly full can from a container and you will get 2 pints or 1 quart of milk reduced from the container.

**This is the first key realization.**

**Last step**

Instead of filling this 1 quart of empty volume in a 10 gallon can from the other filled 10 gallon can, you will fill it from the filled 4 pint jug, leaving $4-2=2$ pints or 1 quart of milk in the 4 pint jug in the last step.

With 1 quart of milk already in the 5 pint jug, you will have your solution. **This is the second key realization.**

**Last but one step**

So in the step before the last, you must fill the 4 pint jug from the filled 10 gallon can. The other 10 gallon can already has 2 pints or 1 quart of empty volume in it.

Let us join the first few steps from start and the last few steps cleared up just now.

Take special care to get 1 quart of empty volume in one 10 gallon can, and second 10 gallon can filled when 4 pint jug is empty and 1 quart of milk is in 5 pint jug.

#### Series of steps to the solution to the riddle of delivering 1 quart of milk each to two jugs

Putting all the steps together,

**Step 1:**Fill 5 pint jug from the first 10 gallon can. The can has 5 pints of empty volume.**Step 2:**Fill 4 pint jug from 5 pint jug. 1 pint of milk remains in 5 pint jug.**Step 3:**;Empty filled 4 pint jug into the first 10 gallon can. It has 1 pint of empty volume.**Step 4:**Transfer 1 pint of milk from 5 pint jug to the 4 pint jug. Now 5 pint jug is empty and 1 pint empty volume is in the first can. The second 10 gallon can full.**Step 5:**Fill 5 pint jug from the fiirst can. This can has 6 pints of empty volume.**Step 6:**Fill 4 pint jug with 1 pint of milk in it from the filled 5 pint jug. 2 pints or 1 quart of milk remains in 5 pint jug, 4 pint jug filled. First can has 6 pints empty volume. $5-3=2$ pints in 5 pint jug.**Step 7:**Empty 4 pint jug into first 10 gallon can. The can has now 2 pints or 1 quart of empty volume. Other 10 gallon can is full. $6-4=2$ pints of empty volume in first 10 gallon can.**Step 8:**Fill 4 pint jug from the second filled 10 gallon can.**Step 9:**Fill the 2 pints or 1 quart of empty volume of first 10 gallon can from filled 4 pint jug. 2 pints or 1 quart of milk remains in 4 pint jug. $4-2=2$ pints in 4 pint jug.

**Result:** 1 quart of milk in 4 pint jug and 1 quart of milk in 5 pint jug. Both women and the milkman satisfied.

You have used **Problem breakdown technique** along with **Working backwards approach** and **End state analysis approach**, all three powerful problem-solving techniques.

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