Problem Statement

Charles Babbage, looking ahead to the sorts of problems his Analytical Engine would be able to solve, gave this example: He thought the answer might be 99,736, whose square is 9,947,269,696; but he couldn't be certain.

The task is to find out if Babbage had the right answer — and to do so, as far as your language allows it, in code that Babbage himself would have been able to read and understand. As Babbage evidently solved the task with pencil and paper, a similar efficient solution is preferred. For these purposes, Charles Babbage may be taken to be an intelligent person, familiar with mathematics and with the idea of a computer; he has written the first drafts of simple computer programmes in tabular form. [Babbage Archive Series L].

The aim of the task is to write a program that is sufficiently clear and well-documented for such a person to be able to read it and be confident that it does indeed solve the specified problem.

Let's start with the solution:

#!/usr/bin/perl
use strict ;
use warnings ;

my $current = 0 ;
while ( ($current ** 2 ) % 1000000  != 269696 ) {
   $current++ ;
}
print "The square of $current is " . ($current * $current) . " !\n" ;


# Dear Mr. Babbage,
# in the following code, $number is the variable to contain the number
# we're looking for. '%' is the modulo division operator and '!='
# means "not equal to".

# We start to search with 518, because no smaller number can be even
# squared to 269696. We also increment the number always by 2, as no
# odd number can be part of the solution. We do this, because
# computational power - even in 2022 - does not grow on trees.
# And even if it did, we have less of them than you did.

my $number = 518;
while ( ($number ** 2) % 1000000 != 269696 ) {
   $number += 2 ;
}
print "The square of $number is " . ($number ** 2) . " !\n";