Solve this alphametic integral puzzle, where each of the capital letters in bold represents a different decimal digit from 0 to 9, given that C and N are constants and, E is not zero.

B
∫ C*x^N dx = EDGE
A

Evaluating the integral, we have:

EDGE = C * [ B^(N+1) - A^(N+1)] / (N+1)

EDGE / C = [ B^(N+1) - A^(N+1)] / (N+1)

EDGE / C will have 3 or 4 digits.

A quick inspection shows that this occurs only for N =< 4.

I used a spreadsheet to obtain which follows.

The values on the rhs that end in 0, can be eliminated, and also N=4.

The possibilities to be analyzed are (calling r the value of the rhs):

(N, B, A, r) = (3, 9, 5, 1484), (3, 8, 0, 1024), (3, 7, 5, 444), (3, 6, 0, 324), (3, 5, 1, 156), (2, 9, 0, 243), (2, 9, 3, 234), (2, 9, 6, 171), (2, 8, 5, 129) and (2, 7, 1, 114).

Knowing that the r (the rhs) multiplied by C must yield EDGE, that is, the same digit at left and at right, we quickly found the unique possibility for (N, B, A): (3, 6, 0). The rest follows.

 

Posted by pcbouhid on 2009-03-26 11:45:03