The document discusses possible piano key combinations for a code to access Batman's lair based on the clue "CD-BAG" found on a napkin. It calculates: 1) 25,088 ways to play the notes in that exact order. 2) 759,375 ways if repetitions are allowed within 3 octaves. 3) 2,434,320 ways if sharps and flats are included. It would take approximately 141 days to try all the combinations, making the code impossible to crack without other clues. An alternative mischievous plan is proposed instead.

1. Photo by flickr user
J.Weissmahr

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. •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. • 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. 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. • 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. 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. 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. 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. 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. 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. • 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. 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!!!