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Balls and Boxes

7

6

$begingroup$

Three boxes contain balls. Each box is large enough to contain all balls. We call $bftarget box$, box that receive balls from one of the others boxes. We allow only one process: moving balls from a box to a $bftarget box$ as long as the numbers of balls in the $bftarget box$ doubles.

Show that it is possible, whatever the initial configuration was, to have an empty box after processing.

abstract-algebra number-theory arithmetic

share|cite|improve this question

edited Mar 1 at 16:41

HAMIDINE SOUMARE

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the number of balls in the third box doubles at a stage then can we put remaining balls of the second container to the first one so that the second container is empty?
  $endgroup$
  – Sahil Silare
  Feb 14 at 18:18
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  This looks like an interesting question. Please give us some context, such as where it comes from, what you've tried so far, etc. Thanks.
  $endgroup$
  – John Omielan
  Feb 14 at 18:28
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @almagest The question does appear to be worded weirdly. I think it means that there are three boxes and you are allowed to move balls from one box to another box as long as the number of balls in the target box doubles. Thus I suspect that valid moves are moving two balls from either box 1 or 3 to box 2 and moving two balls from either box 2 or 3 to box 1.
  $endgroup$
  – jgon
  Feb 14 at 22:18
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  I generated some interesting thoughts: the ball transitions between boxes can be expressed as six $3times 3$ matrices operating on $mathbb Z^3$. Since the number of balls is constant, these transformations are pushing around points of the plane $x+y+z=n$ where $n$ is the number of balls. The goal is to push a given point to the edge of the triangle this plane cuts in the first quadrant. Of course, this is all overkill if there is an arithmetic answer, but I couldn't help but try to look at what's happening with the geometry...
  $endgroup$
  – rschwieb
  Feb 15 at 14:24
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @HAMIDINESOUMARE What would be the point of writing an answer with no solution given? Besides, you have not mentioned anything about your progress, yet.
  $endgroup$
  – rschwieb
  Feb 15 at 17:37
  
  

  
  
  
  

 | 
show 7 more comments

7

6

$begingroup$

Three boxes contain balls. Each box is large enough to contain all balls. We call $bftarget box$, box that receive balls from one of the others boxes. We allow only one process: moving balls from a box to a $bftarget box$ as long as the numbers of balls in the $bftarget box$ doubles.

Show that it is possible, whatever the initial configuration was, to have an empty box after processing.

abstract-algebra number-theory arithmetic

share|cite|improve this question

edited Mar 1 at 16:41

HAMIDINE SOUMARE

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the number of balls in the third box doubles at a stage then can we put remaining balls of the second container to the first one so that the second container is empty?
  $endgroup$
  – Sahil Silare
  Feb 14 at 18:18
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  This looks like an interesting question. Please give us some context, such as where it comes from, what you've tried so far, etc. Thanks.
  $endgroup$
  – John Omielan
  Feb 14 at 18:28
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @almagest The question does appear to be worded weirdly. I think it means that there are three boxes and you are allowed to move balls from one box to another box as long as the number of balls in the target box doubles. Thus I suspect that valid moves are moving two balls from either box 1 or 3 to box 2 and moving two balls from either box 2 or 3 to box 1.
  $endgroup$
  – jgon
  Feb 14 at 22:18
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  I generated some interesting thoughts: the ball transitions between boxes can be expressed as six $3times 3$ matrices operating on $mathbb Z^3$. Since the number of balls is constant, these transformations are pushing around points of the plane $x+y+z=n$ where $n$ is the number of balls. The goal is to push a given point to the edge of the triangle this plane cuts in the first quadrant. Of course, this is all overkill if there is an arithmetic answer, but I couldn't help but try to look at what's happening with the geometry...
  $endgroup$
  – rschwieb
  Feb 15 at 14:24
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  @HAMIDINESOUMARE What would be the point of writing an answer with no solution given? Besides, you have not mentioned anything about your progress, yet.
  $endgroup$
  – rschwieb
  Feb 15 at 17:37
  
  

  
  
  
  

 | 
show 7 more comments

$begingroup$

Three boxes contain balls. Each box is large enough to contain all balls. We call $bftarget box$, box that receive balls from one of the others boxes. We allow only one process: moving balls from a box to a $bftarget box$ as long as the numbers of balls in the $bftarget box$ doubles.

Show that it is possible, whatever the initial configuration was, to have an empty box after processing.

abstract-algebra number-theory arithmetic

share|cite|improve this question

edited Mar 1 at 16:41

HAMIDINE SOUMARE

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

$endgroup$

share|cite|improve this question

edited Mar 1 at 16:41

HAMIDINE SOUMARE

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

share|cite|improve this question

edited Mar 1 at 16:41

HAMIDINE SOUMARE

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212

asked Feb 14 at 17:48

HAMIDINE SOUMAREHAMIDINE SOUMARE

2,076212