A chunk of pages is missing from a book. The sum of the missing page numbers is 9808. What are those page numbers?

If a is the last missing page and b is the first, then the sum of the missing page numbers will be

(a+b)(a-b+1)/2

Multiplied out:

(a^2 - b^2 + a + b) / 2 = 9808

a^2 + a + b(1-b) - 19616 = 0

Using the quadratic formula to solve for a, when b is given, and assuming that a should be positive:

a = (-1 + sqrt(1 - 4b(1-b)) / 2

Then trying all the odd b values between 1 and 999, seeking a positive integer for a, hopefully even:

DEFDBL A-Z
FOR b = 1 TO 999 STEP 2
  c = b * (1 - b) - 19616
  disc = 1 - 4 * c
  IF disc >= 0 THEN
    sr = INT(SQR(disc) + .5)
    IF sr * sr = disc THEN
      a1 = (-1 + sr) / 2
      IF a1 = INT(a1) THEN
        PRINT b, a1
      END IF
    END IF
  END IF
NEXT b

we find b = 291 and a = 322, as the first and last of the missing pages.

Posted by Charlie on 2009-11-13 16:48:23