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Batman Question


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Batman Question

  1. 1. Photo by flickr user J.Weissmahr
  2. 2. The napkin that we discovered had the mnemonic CD-BAG written on it. We believe that these are the letters for the piano key code that will grant us access to batman’s lair. 1. How many ways can those letters be played on the piano, assuming they must be played in that order? 2. How many ways can we play CDBA and G if we are limited to the 3 middle octaves and repetitions are allowed? 3. How many ways can we play the notes C, D, B, A and G if sharps and flats are allowed and we must play them in that order? a) If it takes us 5 seconds to enter one set of notes how long will it take us to try all of the combinations in the above question.
  3. 3. •This is part of the keyboard that makes up a piano. As you can see the keys are labeled using the first 7 letters of the alphabet. (Starting with A on the left hand side on a full piano) •The white keys are used to play normal notes, and the black keys are used for playing accidentals (sharps and flats). •The piano in Batman’s home has 52 white keys and 36 black keys, for a total of 88 keys. •This means there are 7 full octaves on his piano (one set of A to G is called an octave). There is also the start of an eighth octave. (the notes A, B, C) •Now that we know a little bit about piano’s let’s try and solve this question!
  4. 4. • Let’s think back to the keys on the piano… There are 52 white keys Starting with A on the left hand side, that means there will be 8 A’s, 8 B’s, 8 C’s, 7 D’s 7 E’s 7 F’s and 7 G’s. • So how many ways can we play the notes C D B A and G in that order? • Note: We can play the notes in different octaves, for example we can play a high C and then play a low D as long as we play the C immediately followed by the D.
  5. 5. 1. Let’s start off with 5 blank slots, because we have 5 letters to plug in. 2. We must play the notes C D B A and G in that order, so that is how we will fill in the slots. 3. Now we must look at how many ways we can play each note. We can play a C 8 ways, D 7 ways B 8 ways A 8 ways and G 7 ways. 4. Why do we multiply you ask? Well that’s simple! When you have M ways of doing one thing and N ways of doing another you have MN ways of doing both things! (:
  6. 6. • Lets write down what we know! • First of all: We are limited to 3 octaves! • Second: Order matters! However we are allowed repetitions, which means we do not have to play the notes C D B A and G in THAT order (: • We have to play 5 notes.
  7. 7. 1. Start with 5 empty slots because that is how many notes we must play. 2. In each octave there is one C, D, B, A and G, so in three octaves there will be 3 of each note. Therefore there are 15 ways to play five notes in 3 octaves. 3. We fill each slot with 15 because we are allowed repetitions.
  8. 8. Let’s write down what we know! 1. Flats and sharps are allowed, which means there are now 13 different notes we can play. C, #C, D, #D,♭D, B,♭B, A, #A,♭A, G, #G,♭G 2. The total number of ways to play each note is; C- 15, D- 21, B- 16, A- 23, G- 21 3. Order matters! This is a permutation!
  9. 9. 1. Start with 5 blank slots since that is how many notes we must play. 2. This is the order we’re playing the notes in. 3. Plug in the number of ways of playing each note.
  10. 10. If it takes us 5 seconds to enter one set of notes how long will it take us to try all of the combinations in the above question. • You might be wondering how come were only finding the time for the last question? • Well the answer to that is simple. Since we found the mnemonic CD- BAG on the napkin it stands to reason that all of those letters must be used in the actual code. It also stands to reason that they would be played in that order because the point is to help remember something! • You may also be wondering why we answered all those other questions if we knew they most likely would not be the codes we were looking for. The answer to that is also simple! Number B enjoys asking excess questions that do not particularly apply to the situation and Number A must then explain them. We displayed some of the answers to some of Number B’s questions… but there were many more… 0_0
  11. 11. How long will it take to play 2 434 320 different combinations? 2 434 320 x 5 = 12 171 600 Multiply the number of combinations by 5 because it takes 5 seconds to try each combination. 12 171 600/60= 202 860 Divide by 60 to find minutes. 202 860/60= 3381 Divide by 60 to find hours. 3381/ 24= 140.875 Divide by 24 to find days!
  12. 12. • There are 25 088 ways to play the note C, D, B, A and G, in that order. • There are 759 375 ways to play the notes if repetitions are allowed and you are restricted to 3 octaves. • There are 2 434 320 ways to play the notes if sharps and flats are allowed. • It would take us approximately 141 days to try the likely combinations. • That is of course without sleeping, bathroom breaks and breakfast, lunch and dinner. . .
  13. 13. Well I guess we won’t be able to break into his secret lair…. Looks like we’ll have to settle for plan B... PUTTING A STINK BOMB UNDER HIS CHAIR!!!