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The Mad Veterinarian

The Mad Veterinarian. A mad veterinarian has created three animal transmogrifying machines : Machine I: Place a cat in the input bin. Open the output bin to find 2 dogs and 5 mice. Machine II: Place a dog in the input bin. Open the output bin to find 3 cats and 3 mice.

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The Mad Veterinarian

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  1. The Mad Veterinarian

  2. A mad veterinarian has created three animal transmogrifying machines: Machine I: Place a cat in the input bin. Open the output bin to find 2 dogs and 5 mice. Machine II: Place a dog in the input bin. Open the output bin to find 3 cats and 3 mice. Machine III: Place a mouse in the input bin. Open the output bin to find a cat and a dog. Note: each machine can also operate in reverse! (i.e., if you have 3 cats and 3 mice, then you can convert them into 1 dog)

  3. Question 1: The mad veterinarian has handed you 6 mice and expects you to convert them into only 4 dogs. Can you do it?

  4. Question 1: The mad veterinarian has handed you 6 mice and expects you to convert them into only 4 dogs. Can you do it?

  5. Question 2: Can you convert 3 cats to 7 mice?

  6. Question 3: Can you convert 1 cat to 7 mice?

  7. Representation of Problem • Associate with each machine a triplet which gives the change in cats, dogs, mice respectively Machine I: (-1, 2, 5) Machine II: (3, -1, 3) Machine III: (1, 1, -1) • Negative animal represents animals that were not replaced as a result of the process.

  8. Representation of Problem • Let x = number of times we run Machine I Let y = number of times we run Machine II Let z = number of time we run Machine II • Triplets: (-1, 2, 5) (-x, 2x, 5x) (3, -1, 3) (3y, -y, 3y) (1, 1, -1) (z, z, -z) # cats = -x + 3y + z #dogs = 2x – y + z # mice = 5x + 3y - z

  9. Question 1: The mad veterinarian has handed you 6 mice and expects you to convert them into only 4 dogs. Can you do it? # cats = -x + 3y + z = 0 #dogs = 2x – y + z = 4 # mice = 5x + 3y – z = -6 4x + 6y = -6 7x + 2y = -2 (multiply by -3) 4x + 6y = -6 17x = 0 x = 0 y = -1 z = 3 # times Machine I runs -21x - 6y = 6 # times Machine II runs # times Machine III runs

  10. Question 1: The mad veterinarian has handed you 6 mice and expects you to convert them into only 4 dogs. Can you do it? x = 0 # times Machine I runs y = -1 # times Machine II runs z = 3 # times Machine III runs

  11. Question 2: Can you convert 3 cats to 7 mice? # cats = -x + 3y + z #dogs = 2x – y + z # mice = 5x + 3y - z = -3 = 0 = 7 4x + 6y = 4 7x + 2y = 7 (multiply by -3) 4x + 6y = 4 -21x – 6y = -21 -17x = -17 x = 1 y = 0 z = -2 # times Machine I runs # times Machine II runs # times Machine III runs

  12. Question 2: Can you convert 3 cats to 7 mice? x = 1 y = 0 z = -2 # times Machine I runs # times Machine II runs # times Machine III runs

  13. Question 3:Can you convert 1 cat to 7 mice? # cats = -x + 3y + z #dogs = 2x – y + z # mice = 5x + 3y - z = -1 = 0 = 7 4x + 6y = 6 7x + 2y = 7 (multiply by -3) 4x + 6y = 6 -21x – 6y = -21 -17x = -15 No integer solution, therefore we cannot convert 1 cat to 7 mice.

  14. Other questions to ponder: • Can you convert 1 cat into a pack of dogs, with no mice or cats left over? • How many mice can be created from a single cat? • What is the smallest number of cats that can be turned into just dogs?

  15. Extension: • What about 4 machines? These machines also operate in reverse. Machine I: converts 1 cat to one dog and two mice. Machine II: converts 1 dog into a mouse and 3 rabbits Machine III: converts 1 mouse into 2 cats and 3 rabbits Machine IV: converts 1 rabbit into 1 cat and 5 dogs The mad veterinarian has handed you three cats, and expects you to convert them into four mice. Can you do it?

  16. Website applet: • http://www.bumblebeagle.org/madvet/index.html Website for the Representation of Problem http://groups.google.com/group/rec.puzzles/browse_thread/thread/743d1dd03ac33249/8cc7fec3c35f4848?hl=en&lnk=gst&q=mad+veterinarian#8cc7fec3c35f4848

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