given a word, print its index, words can be increased accordingly

Question

Imagine an alphabet of words.

Example:

 a ==> 1 
 b ==> 2 
 c ==> 3 

 z ==> 26 
 ab ==> 27 
 ac ==> 28 

 az ==> 51 
 bc ==> 52         


        

Answer 1

1. First make all the letters numbers:
   • 'aez' would become 1,5,26
2. Make these numbers variables called ...X3,X2,X1
   • 26 would be X1, 5 would be X2, 1 would be X3 (note, right to left)

Now for the magic formula:

Coded with examples and demonstration of speed even in worse case scenario:

def comb(n,k): #returns combinations
    p = 1 #product
    for i in range(k):
        p *= (n-i)/(i+1)
    return p

def solve(string):
    x = []
    for letter in string:
        x.append(ord(letter)-96)  #convert string to list of integers
    x = list(reversed(x))  #reverse the order of string
    #Next, the magic formula
    return x[0]+sum(comb(26,i)-comb(26-x[i-1]+1,i)*(1-i/(26-x[i-1]+1)) for i in range(2,len(x)+1))

solve('bhp')
764.0
>>> solve('afkp')
3996.0
>>> solve('abcdefghijklmnopqrstuvwxyz')
67108863.0
>>> solve('hpz')
2090.0
>>> solve('aez')
441.0
>>> if 1:
        s = ''
        for a in range(97,97+26):
                s += chr(a)
        t = time.time()
        for v in range(1000):
                temp = solve(s)
        print (time.time()-t)


0.1650087833404541

In order to understand my explanation to this formula, I need to go over a mathematical occurrence in pascal's triangle and the binomial theorem:

Here is pascal's triangle:

Going from top right to bottom left, first there is a sequence of 1s. Then a sequence of the counting numbers. The next sequence is the sum of the counting numbers. These are known as the triangular numbers. The next sequence is the sum of the triangular numbers, known as the tetrahedral numbers and this pattern goes on and on.

Now for the binomial theorem:

By combining the binomial theorem and pascals triangle, it can be seen that the nth triangular number is:

and nth tetrahedral number is:

the sum of the first n tetrahedral numbers is:

and on ...

Now for the explanation. For this explanation, I will only use 6 letters, a-f, and will replace these with the numbers 1-6. The procedure is the same with more letters

If the length is 1, then the possible sequences are:

1
2
3
4
5
6

In this the answer is simply the value

Now for a length of 2:

12  13  14  15  16
23  24  25  26
34  35  36
45  46
56

To solve this we split it into 3 parts:

1. Find the total number of elements in the rows above
   • In this case, there are 5 elements in the first row, 4 in the second, 3 in the 3rd and so forth. What we have to do is find a way to sum the first n elements of the sequence (5,4,3,2,1). In order to do this, we subtract triangular numbers. (1+2+3+4+5)-(1+2+3) = (4+5). Similarly (1+2+3+4+5)-(1+2) = 3+4+5. Therefore this value is equal to:
     • 
2. Now, we have accounted for the values above our target and are only concerned with the column it is in. To find this, we add x1-x2
3. Lastly, we need to add the amount of length 1 sequences there are. This is equal to 6. Therefore, our formula is:
   • 

Next we will repeat for sequences of length 3:

123  124  125  126
134  135  136
145  146
156

234  235  236
245  246
256

345  346
356

456

Once again we split this problem into steps:

1. Find how many elements are above each array. The arrays values are the backwards triangular numbers (10, 6, 3, 1). This time, instead of subtracting triangular numbers we subtract tetrahedral numbers:
   • 
2. Notice how each individual array has the shape of a length 2 sequence. By subtracting x3 from x1, and x2, we reduce the sequence to degree 2. For example, we will subtract 2 from the second array

This

234  235  236
245  246
256

becomes

12  13  14
23  24
34  

3. We can now use the length 2 formula, with 6-x3 instead of 6, because our sequences now have a different maximum value
   • 
4. Lastly, we add the total number of length 1 and length 2 sequences. It turns out there is a pattern for how many sequences of a particular length there are. The answer is combinations. There are sequences of length 1, of length 2, and so on.

Combining these our total formula for length 3 becomes:

We can follow this pattern of reduction for higher length sequences

Now we will right out our formulas to look for patterns:

Length 1: y1

Length 2:

Length 3:

Note: I also used length 4 to make sure the patterns held

With a bit of math, grouping of terms, and the change from 6 to 26 our formula becomes:

In order to simplify this further, more math must be done.
This identity holds true for all a and b. For a quick fun exercise, prove it (not really difficult):

This identity allows as to further group and negate terms to reach our much oversimplified formula: