Real Mathematics: Puzzle #2

There are certain movies I remember from my childhood which are among my favorites of all times. It would be though to make a top-10 list but if I were to make such a list, Die Hard 3 would make the list with ease. Crazy German terrorist Simon Gruber against our heroes John McClane and Zeus Carver whom were performed by Bruce Willis and Samuel Jackson respectively… Now, that is action!

die-hard-vengeance-laptop

There is a scene from this movie which will be the main topic of this article. In this particular scene, Simon hands out another life-or-death assignment for our heroes. Click here for the scene.

Simon says…

“There is a timed bomb in this briefcase. You have a 3-liter and a 5-liter jug that you can fill from the fountain. In order to stop the timer you must use those jugs and fill exactly 4 liters. It has to be precisely 4 liters because once you fill the jug; you should put it on the briefcase which has a scale.”

dayhard

Billiard Table

We could use try-and-see method to solve this riddle, but we might run out of time which would mean a sudden death for us. Obviously we don’t want that. To come up with a systematic method we have to set a few ground rules.

Rules

  • Let’s assume that vertices represents water amount in liters and lines represents how quantity can change.
  • Vertices will have number pairs. First number is for the 5 liter jug, second for the 3 liter jug.
  • Both jugs start from the vertex (0,0) as they are empty.

Example: Vertex (1,2) means that there are 1 liter in the 5-litered jug, 2 liters in the 3-litered jug.

According to our rules, we would have the following graph.

dayhard1

Solution with the method

We used equilateral triangles to build the billiard table for a reason. Using a billiard ball in such shaped table, ball would travel according to our rules. And that is making a regular reflection. In other words, ball moves on the table like a light ray reflecting from a mirror.

Ball’s paths are clear. When it starts travelling from the starting point which is (0,0) it could only follow either (0,3) or (5,0) path. Let’s assume we hit the ball into the direction of (0,3). Ball would pass from the vertices (0,1) and (0,2) and reach (0,3) until it makes a regular reflection which is in the direction of the vertices (3,3) or (3,0).

From now on I will refer to 5-liter jug as “jug A”, and 3-liter jug as “jug B”. To start the solution, I choose the direction of the vertex (5,0).  Then let’s follow the direction of the vertex (2,3).

dayhard2

From the vertex (2,3) let’s use the path to the vertex (2,0).

dayhard3

I will hit the ball towards vertex (0,2) and (5,2) respectively.

Final step is to make the reflection into the path of the vertex (4,3). This vertex means jug A has 4 liters of water which was the amount Simon wanted us to achieve.

Solution with words

  • Fill jug A.
  • Pour it into jug B.
  • Empty jug B.
  • Pour the 2 liters (which was inside jug A) into jug B.
  • Fill jug A completely and pour it on top of what jug B has (at the point jug B has only 1 liter of capacity).
  • Jug A has exactly 4 liters of water.

If only our heroes knew the power of vertices and lines, they would have solved this problem in a much shorter time. Although I would have forgotten my name if I was in front of a bomb. So, I congratulate you boys!

Serkan says…

  1. Is there a second way to find the 4 liters with our method?
  2. Find both ways to get 1 liter of water.
  3. If jugs were 6 and 15 liters in capacity, would it be possible to have 5 liters of water? Give your answer with a proof.

M. Serkan Kalaycıoğlu

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