Batch 2 - Class 65 - Three Jugs Problem

• (Geoffrey - 143) Three dutchmen and their wives go shopping hogs, and each buys as many hogs as he or she pays for each hog. Each husband spends Rs 63 more than respective wife. Husbands are Hendrick, Elas and Cornelius. Wives are Gurtrun, Katrun and Anna. Hendrick buys 23 more hogs than Katrun, while Elas buys 11 more than Gurtrun. What are the names of each man's wife?

  Answer: Pairs are 32/31, 12/9, and 8/1 - we can now see integers that differ by 23 and 11. So its Hendrick-Anna, Elas-Katrun and Cornelius-Gurtrun

• 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

  2,6,84,6,105,6,11
• 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 infinite water. For a moment, lets assume that the jugs are marked

  Guide kids through each distribution and the points that represent themNote that each row really represents the two jug problem with the third jug held constant

• Charting Exercises: On a blank grid, mark

  All points that represent one jar being full and other two being emptyDraw separate lines where jar C has 4 liters, 3 liters, 2 liters, 1 liter and 0 literLabel (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 CWhat 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 reachedDraw circles on points which can be reached in ALL of the liquid has to be poured outWhy 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 capacityWhat shape are we left with?If we start with (8,0,0) where will any legal pouring keep us (boundary of the 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?4,6,105,6,116,7,8 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

• Solve YEAR * 12 = MONTH by replacing each letter by a unique digit 1..9. H > T

  Answer:  6129 * 12 = 73548
• 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

  24 13 11 5

  ----------

  24 0 0 0 

  11 13 0 0 

  6 13 0 5

  6 2 11 5

  8 0 11 5

  8 5 11 0

  8 13 3 0

  8 8 3 5

  8 8 8 0