[superprismatic]

Here are the first 1001 digits of pi:


 3.

 14159265358979323846264338327950288419716939937510

 58209749445923078164062862089986280348253421170679

 82148086513282306647093844609550582231725359408128

 48111745028410270193852110555964462294895493038196

 44288109756659334461284756482337867831652712019091

 45648566923460348610454326648213393607260249141273

 72458700660631558817488152092096282925409171536436

 78925903600113305305488204665213841469519415116094

 33057270365759591953092186117381932611793105118548

 07446237996274956735188575272489122793818301194912

 98336733624406566430860213949463952247371907021798

 60943702770539217176293176752384674818467669405132

 00056812714526356082778577134275778960917363717872

 14684409012249534301465495853710507922796892589235

 42019956112129021960864034418159813629774771309960

 51870721134999999837297804995105973173281609631859

 50244594553469083026425223082533446850352619311881

 71010003137838752886587533208381420617177669147303

 59825349042875546873115956286388235378759375195778

 18577805321712268066130019278766111959092164201989


We are to convert this into a string of letters.

We will use the assignment A=0, B=1, C=2, ..., Z=25

and these rules for making strings:


Let d(i) be the ith digit of pi (we ignore the

decimal point).  We start with an empty string and

an index, n, initialized to 1.

1. if d(n) is in the set {0,3,4,5,6,7,8,9}, we

   concatenate the letter corresponding to d(n)

   to the string, then we increment n by 1.

2. if d(n) is 1 then we have the choice:

   2a. concatinate B to our string, then

       increment n by 1.

   2b. concatenate the letter corresponding to

       10+d(n+1), then increment n by 2.

3. if d(n) is 2 and d(n+1) is greater than 5, we

   concatinate C to the string, then increment n

   by 1.

4. if d(n) is 2 and d(n+1) is less than 6, we

   have the choice:

   4a. concatinate C to the string, then increment

       n by 1.

   4b. concatinate the letter corresponding to

       20+d(n+1) to the string, then increment

       n by 2.


The challenge is to determine how many different

strings we can make in this way from the first 1001

digits of pi.

[/code]

[b]

Note well:

It may seem that this problem is a good one for

a programmer.  Not so!  The number of strings

are so numerous that you cannot enumerate them

with a program.  The real solution to this problem

rests with insight.  Owing to the size of the data

set (1001 digits), it may be helpful to massage the

data using a program, but this is by no means

necessary, and it may actually increase the time

to solution.

[/b]

[Guest]

If my logic is correct...

2^177 or 1.91561942608236E+53 different strings, then take into account the 1s and 2s that are next to each other, which will reduce your choice by one for each set that is together, you get (2^177)-37 or (1.91561942608236E+53)-37

[superprismatic]

If my logic is correct...

2^177 or 1.91561942608236E+53 different strings, then take into account the 1s and 2s that are next to each other, which will reduce your choice by one for each set that is together, you get (2^177)-37 or (1.91561942608236E+53)-37

First of all, I think you miscounted the 1s and 20s, 21s, 22s, 23s, 24s, and 25s. There are 176 of them, not 177. Even with that, I think your number is way too big.

[Guest]

I counted all combinations which allow more strings (i.e. 1 and 2 followed by 0,1,2,3,4,5). In pi we have:

1x 1121 = once 4 digits

3x 211

1x 122

2x 111 = six times 3 digits

3x 22

11x 21

3x 12

8x 11 = 25 times 2 digits

plus 104 single digits

total number of possible strings is:

2^104 * 3^25 * 5^6 * 8 = approx. 2.15 * 10^48

[Guest]

You beat me to it Hugo, I went back and found my mistake and counted the multiples again like you did,

My first answer was off by a factor of 5.

excel gave me...

I agree with 2^104 * 3^25 * 5^6 * 8 being the formula,

2^104 = 2.02824096036517 E+31

3^25 = 847,288,609,443

5^6 = 15625

8^1 = 8

Multiplied together you get 2.14813182865392E+48