Determine the minimum value of a positive integer N such that the 2nd, 3rd, 4th and 5th digits in order immediately following the decimal point (reading left to right) in the base ten expansion of √N is 2013.

*** For an extra challenge, solve this puzzle without using a computer program.

A bit confused by "2nd, 3rd, 4th and 5th ... immediately following the decimal point", the following finds two numbers, one of whose "2013" immediately follows the decimal, and the other of whose "2013" is separated from the decimal by one arbitrary digit. In both cases, a sequence of arbitrary digits precede the decimal point and what is sought is the square of the "2013" version and the square of the "2014 version differing by 1 in their integer parts.

DEFDBL A-Z
CLS
FOR i = 1 TO 999
   low = i + .2013#: high = i + .2014#
   low2 = low * low: high2 = high * high
   IF INT(high2) <> INT(low2) THEN
     PRINT low; low * low, high; high * high
     EXIT FOR
   END IF
NEXT
PRINT

FOR i = 1 TO 999
   FOR j = 0 TO 9
     low = i + j / 10 + .02013#: high = i + j / 10 + .02014#
     low2 = low * low: high2 = high * high
     IF INT(high2) <> INT(low2) THEN
       PRINT low; low * low, high; high * high
       END
     END IF
   NEXT j
NEXT
PRINT i

finds

57.2013  3271.988721690001       57.2014  3272.00016196

67.32013  4531.999903216901      67.32014  4532.0012496196
 
The first row indicates that 57.2013 is just under 3272 while 57.2014 is just above that same number, so sqrt(3272) will have digits immediately following the decimal as 2013. In fact sqrt(3272) ~= 57.20139858430036.

However, if we want the 2nd through 5th places after the decimal point to be occupied by 2013 in proper order, the second line shows us that 67.32013 lies just below sqrt(4532), while 67.32014 lies just above, so N = 4532 is the sought answer, as sqrt(4532) ~= 67.32013071882734.

Posted by Charlie on 2013-01-09 12:48:13