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As Easy As ... (Posted on 2006-03-08) Difficulty: 3 of 5
What are the next three numbers in this sequence:

See The Solution Submitted by Vernon Lewis    
Rating: 4.0000 (2 votes)

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Solution Puzzle Solution With Explanation Comment 11 of 11 |
(In reply to Answer by K Sengupta)

At the outset, there are many objects which can correspond to the word that would go in the blank space in the problem title.

However, the given problem requires one to find the missing "numbers", and so we are looking for an appropriate word which is either a math word, or is phoenetically similar to a math word.

After a lot of trial and eror, we observe  "pie" is a valid word, since "pie" is indeed phoenetically similar to "pi", and "as easy as a pie" is a well known saying.

Now, "pie" is not exactly equivalent to the math word "pie", and a concatenation of the letter "e" is required to transform "pi" to "pie". It is well known that "e" is a well known mathematical constant, known as "Euler's Constant".

We now write the values of "pi" and "e" correct to 29 places
of decimals. Thus:

pi =~ 3.14159265358979323846264338328
e =~  2.71828182845904523536048747135

We also observe that "as easy as 1-2-3" is also a common and well known saying.

From the foregoing, it is now readily observed that ignoring the decimal points and denoting the pth digit from the left in each of the above decimal expansions of pi and e by A(p) and B(p), the pth term S(p) of the given sequence is given by S(p) = A(p)*B(p) + p

For example,
S(1) = A(1)*B(1) + 1 = 3*2 = 7
S(2) = A(2)*B(2) = 1*7 + 2 = 9
S(3) = A(3)*B(3) = 4*1 + 3 = 7, and so on

We observe that:

S(11) = 5*4 + 11 = 31
S(12) = 8*5 + 12 = 52
S(13) = 9*9 + 13 = 94

Thus, the required next three numbers are 31-52-94. 

Edited on April 3, 2008, 11:52 am
  Posted by K Sengupta on 2008-04-03 11:43:48

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