Batch 3 - Class 162 - Three Jugs Problem

• What's your birthday? Can you find out which day of the week were you born on?

• There are two jugs each of capacity 8 liters. One of them has 8 liters of water, the other is empty. How much water can you pour out?
• What if these were measuring jars with integral liters marked - what would be the various fluid levels in both jugs?
• How do we represent them on a picture?

• Which points would be able to get to if the jars were unmarked, which ones if they were marked?
• What if you had jars with capacity of 5 and 7 liter capacity, but still 8 liter of water between them?
• Exercise: Suppose you have two jugs and 4 liter of water between them. Draw the diagram of possible distributions

• Problem: You have two empty jugs of 5 liter and 3 liter capacity, and a 8 liter jug full of water. All jugs are unmarked. Can you divide the water equally between two friends?
• If yes, can you determine the smallest number of steps?
• Whether or not, determine what measures can you divide
• Try with other combinations
• 2,6,8
• 4,6,10
• 5,6,11
• Can you find any patterns? Are there any methods that help you pour out different amounts?
• Charting: Lets come back and find a way of charting this three jug problem, starting with three jugs of 4 liter each and 4 liter of water. For a moment, lets assume that the jugs are marked
• Guide kids through each distribution and the points that represent them
• Note that each row really represents the two jug problem with the third jug held constant

• Since there are three jugs, why aren't there three dimensions? Where does this triangle show up on a 3D grid?
• Since A+B+C = 4, it reduces to two dimensions. On a 3D grid this would be a "diagonal cut"
• If we had infinite water, then the whole 3D grid would apply
• Charting Exercises: On a blank grid, mark
• All points that represent one jar being full and other two being empty
• Draw separate lines where jar C has 4 liters, 3 liters, 2 liters, 1 liter and 0 liter
• Label (3,1,0). Draw a line that represents what happens if you empty A into B. What would be the line if you emptied A into C
• What does the line parallel to one side mean?
• Label (1,2,1). Draw lines representing cases where you pour part or all of liquid from one jug to another. 
• Draw squares on points which can't be reached
• Draw circles on points which can be reached if ALL of the liquid has to be poured out
• Why are all these parallel to a side?
• Suppose capacity of the three jugs are 4,3,2 respectively - which points are now not possible?
• Back to our Three Jug Problem: Lets now try to chart our three jug problem
• Draw the chart, and mark out areas which are not possible because jug B and C are smaller capacity
• What shape are we left with?
• If we start with (8,0,0) where will any legal pouring keep us (boundary of the parallelogram)
• What would happen if that jars were marked? We could then stop in between and not necessarily go to the boundary of parallelogram.
• Which direction will any legal pouring move us in?
• Our goal is to get to 4 liters - mark out all the points that satisfy that. Which ones are on boundary of parallelogram?
• Can you now draw a sequence of moves which get you from (8,0,0) to (4,4,0) while always moving along one of the sides, and staying on boundary of the parallelogram?
• Write down the sequence of moves and compare with your solution

• What kind of problems are unsolvable here?
• What about any other combinations - which are the ones you can't pour out in this case?
• Lets go back to some others 
• 2,6,8 - what can you pour out?
• 4,6,10
• 5,6,11
• 6,7,8 with 10 liters of water - Can you get to 5 liters?
• Answer - No. If you draw the diagram, you will notice that all points with 5 liters form a closed triangle, which can't be reached from outside

• I have a 24L of water in a jar. Can you divide into 3 equal parts using only a 13L, a 5L and an 11 liter Jar and of course 24L jar.
• Answer