The Mad Veterinarian A mad veterinarian has created

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. 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)

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 Dogs Mice 0 0 6 1 1 5 4 0 8 5 1 7 6 2 6 3 3 3 0 4 0

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 Dogs Mice Machine 0 0 6 III 1 1 5 II 4 0 8 III 5 1 7 III 6 2 6 -II 3 3 3 -II 0 4 0

Question 2: Can you convert 3 cats to 7 mice? Cats Dogs Mice Machine 3 0 0 I 2 2 5 -III 1 1 6 -III 0 0 7

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

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.

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) (3, -1, 3) (1, 1, -1) # cats = -x + 3 y + z #dogs = 2 x – y + z # mice = 5 x + 3 y - z (-x, 2 x, 5 x) (3 y, -y, 3 y) (z, z, -z)

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 + 3 y + z = 0 #dogs = 2 x – y + z = 4 # mice = 5 x + 3 y – z = -6 4 x + 6 y = -6 -21 x - 6 y = 6 17 x = 0 4 x + 6 y = -6 7 x + 2 y = -2 (multiply by -3) x=0 y = -1 z=3 # times Machine I runs # times Machine III runs

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 Dogs Mice Machine 0 0 6 III 1 1 5 II 4 0 8 III 5 1 7 III 6 2 6 -II 3 3 3 -II 0 4 0 x = 0 # times Machine I runs y = -1 # times Machine II runs z = 3 # times Machine III runs

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

Question 2: Can you convert 3 cats to 7 mice? Cats Dogs Mice Machine 3 0 0 I 2 2 5 -III 1 1 6 -III 0 0 7 x=1 y=0 z = -2 # times Machine I runs # times Machine III runs

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

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?

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?

Website applet: • http: //www. bumblebeagle. org/madvet/index. html Website for the Representation of Problem http: //groups. google. com/group/rec. puzzles/browse_thread/743 d 1 dd 03 ac 332 49/8 cc 7 fec 3 c 35 f 4848? hl=en&lnk=gst&q=mad+veterinarian#8 cc 7 fec 3 c 35 f 4848