

Charles Babbage, looking ahead to the sorts of problems his Analytical Engine would be able to solve, gave this example:

 What is the smallest positive integer whose square ends in the digits 269,696?

— Babbage, letter to Lord Bowden, 1837; see Hollingdale and Tootill, Electronic Computers, second edition, 1970, p. 125.

He thought the answer might be 99,736, whose square is 9,947,269,696; but he couldn't be certain.



Task[edit]
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].



Motivation
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.




;; Text from a semicolon to the end of a line is ignored
;; This lets the racket engine know it is running racket
#lang racket
 
;; “define” defines a function in the engine
;; we can use an English name for the function
;; a number ends in 269696 when its remainder when
;; divided by 1000000 is 269696 (we omit commas in
;; numbers... they are used for another reason).
(define (ends-in-269696? x)
  (= (remainder x 1000000) 269696))
 
;; we now define another function square-ends-in-269696?
;; actually this is the composition of ends-in-269696? and
;; the squaring function (which is called “sqr” in racket)
(define square-ends-in-269696? (compose ends-in-269696? sqr))
 
;; a for loop lets us iterate (it’s a long Latin word which
;; Victorians are good at using) over a number range.
;;
;; for/first go through the range and break when it gets to
;; the first true value
;;
;; (in-range a b) produces all of the integers from a (inclusive)
;; to b (exclusive). Because we know that 99736² ends in 269696,
;; we will stop there. The add1 is to make in-range include 99736
;;
;; we define a new variable, so that we can test the verity of
;; our result
(define first-number-that-when-squared-ends-in-269696
(for/first ((i ; “i” will become the ubiquetous looping variable of the future!
             (in-range 1 (add1 99736)))
            ; when returns when only the first one that matches
            #:when (square-ends-in-269696? i))
  i))
 
;; display prints values out; newline writes a new line (otherwise everything
;; gets stuck together)
(display first-number-that-when-squared-ends-in-269696)
(newline)
(display (sqr first-number-that-when-squared-ends-in-269696))
(newline)
(newline)
(display (ends-in-269696? (sqr first-number-that-when-squared-ends-in-269696)))
(newline)
(display (square-ends-in-269696? first-number-that-when-squared-ends-in-269696))
(newline)
;; that all seems satisfactory