Babbage problem

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

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.

For these purposes, Charles Babbage may be taken to be an intelligent person, familiar with mathematics and with the idea of a computer, who has never programmed—in fact, who has never so much as seen a single line of code.

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.

Ada

<lang Ada>-- The program is written in the programming language Ada. The name "Ada" -- has been chosen in honour of your friend, -- Augusta Ada King-Noel, Countess of Lovelace (née Byron). -- -- This is an program to search for the smallest integer X, such that -- (X*X) mod 1_000_000 = 269_696. -- -- In the Ada language, "*" represents the multiplication symbol, "mod" the -- modulo reduction, and the underscore "_" after every third digit in -- literals is supposed to simplify reading numbers for humans. -- Everything written after "--" in a line is a comment for the human, -- and will be ignored by the computer.

with Ada.Text_IO; -- We need this to tell the computer how it will later output its result.

procedure Babbage_Problem is

  -- We know that 99_736*99_736 is 9_947_269_696. This implies:
  -- 1. The smallest X with X*X mod 1_000_000 = 269_696 is at most 99_736.
  -- 2. The largest square X*X, which the program may have to deal with, 
  --    will be at most 9_947_269_69.

  type Number is range 1 .. 99_736*99_736;
  X: Number := 1; 
  -- X can store numbers between 1 and 99_736*99_736. Computations 
  -- involving X can handle intermediate results in that range.
  -- Initially the value stored at X is 1. 
  -- When running the program, the value will become 2, 3, 4, ect.

begin

  -- The program starts running. 
  
  -- The computer first squares X, then it truncates the square, such
  -- that the result is a six-digit number. 
  -- Finally, the computer checks if this number is 269_696. 
  while not (((X*X) mod 1_000_000) = 269_696) loop

     -- When the computer goes here, the number was not 269_696. 
     X := X+1; 
     -- So we replace X by X+1, and then go back and try again.

  end loop;
  
  -- When the computer eventually goes here, the number is 269_696.
  -- E.e., the value stored at X is the value we are searching for.
  -- We still have to print out this value. 
  
  Ada.Text_IO.Put_Line(Number'Image(X));
  -- Number'Image(X) converts the value stored at X into a string of 
  -- printable characters (more specifically, of digits). 
  -- Ada.Text_IO.Put_Line(...) prints this string, for humans to read.
  -- I did already run the program, and it did print out 25264.

end Babbage_Problem;</lang>

ALGOL 68

As with other samples, we use "simple" forms such as "a := a + 1" instead of "a +:= 1". <lang algol68>COMMENT text between pairs of words 'comment' in capitals are

       for the human reader's information and are ignored by the machine

COMMENT

COMMENT Define s to be the integer value 269 696 COMMENT INT s = 269 696;

COMMENT Name a location in the machine's storage area that will be

       used to hold integer values.
       The value stored in the location will change during the
       calculations.
       Note, "*" is used to represent the multiplication operator.
             ":=" causes the location named to the left of ":=" to
                  assume the value computed by the expression to the right.
             "sqrt" computes an approximation to the square root
                    of the supplied parameter
             "MOD" is an operator that computes the modulus of its
                   left operand with respect to its right operand
             "ENTIER" is a unary operator that yields the largest
                     integer that is at most its operand.

COMMENT INT v := ENTIER sqrt( s );

COMMENT the construct: WHILE...DO...OD repeatedly executes the

       instructions between DO and OD, the execution stops when
       the instructions between WHILE and DO yield the value FALSE.

COMMENT WHILE ( v * v ) MOD 1 000 000 /= s DO v := v + 1 OD;

COMMENT print displays the values of its parameters COMMENT print( ( v, " when squared is: ", v * v, newline ) )</lang>

Output:

     +25264 when squared is:  +638269696

APL

If at all possible, I would sit down at a terminal with Babbage and invite him to experiment with the various functions used in the program. <lang apl> ⍝ We know that 99,736 is a valid answer, so we only need to test the positive integers from 1 up to there:

     N←⍳99736
     ⍝ The SQUARE OF omega is omega times omega:
     SQUAREOF←{⍵×⍵}
     ⍝ To say that alpha ENDS IN the six-digit number omega means that alpha divided by 1,000,000 leaves remainder omega:
     ENDSIN←{(1000000|⍺)=⍵}
     ⍝ The SMALLEST number WHERE some condition is met is found by taking the first number from a list of attempts, after rearranging the list so that numbers satisfying the condition come before those that fail to satisfy it:
     SMALLESTWHERE←{1↑⍒⍵}
     ⍝ We can now ask the computer for the answer:
     SMALLESTWHERE (SQUAREOF N) ENDSIN 269696</lang>

Output:

AWK

A comment starts with a "#" and are ignored by the machine. They can be on a
line by themselves or at the end of an executable line.
A program consists of multiple lines or statements. This program tests
positive integers starting at 1 and terminates when one is found whose square
ends in 269696.
The next line shows how to run the program.
syntax: GAWK -f BABBAGE_PROBLEM.AWK

BEGIN { # start of program

this declares a variable named "n" and assigns it a value of zero

   n = 0

do what's inside the "{}" until n times n ends in 269696

   do {
     n = n + 1 # add 1 to n
   } while (n*n !~ /269696$/)

print the answer

   print("The smallest number whose square ends in 269696 is " n)
   print("Its square is " n*n)

terminate program

   exit(0)

} # end of program </lang>

Output:

The smallest number whose square ends in 269696 is 25264
Its square is 638269696

BBC BASIC

Clarity has been preferred over all other considerations. The line LET n = n + 1, for instance, would more naturally be written n% += 1, using an integer variable and a less verbose assignment syntax; but this optimization did not seem to justify the additional explanations Professor Babbage would probably need to understand it. <lang bbcbasic>REM Statements beginning 'REM' are explanatory remarks: the machine will ignore them.

REM We shall test positive integers from 1 upwards until we find one whose square ends in 269,696.

REM A number that ends in 269,696 is one that leaves a remainder of 269,696 when divided by a million.

REM So we are looking for a value of n that satisfies the condition 'n squared modulo 1,000,000 = 269,696', or 'n^2 MOD 1000000 = 269696' in the notation that the machine can accept.

LET n = 0

REPEAT

 LET n = n + 1

UNTIL n^2 MOD 1000000 = 269696

PRINT "The smallest number whose square ends in 269696 is" n

PRINT "Its square is" n^2</lang>

Output:

The smallest number whose square ends in 269696 is     25264
Its square is 638269696

Alternative method

Translation of: PowerShell

The algorithm given in the alternative PowerShell implementation may be substantially more efficient, depending on how long SQR takes, and I think could well be more comprehensible to Babbage. <lang bbcbasic>REM Lines that begin 'REM' are explanatory remarks addressed to the human reader.

REM The machine will ignore them.

LET n = 269696

REPEAT

 LET n = n + 1000000
 
 REM Find the next number that ends in 269,696.
 
 REM The function SQR finds the square root.
 
 LET root = SQR n
 
 REM The function INT truncates a real number to an integer.

UNTIL root = INT root

REM If the square root is equal to its integer truncation, then it is an integer: so we have found our answer.

PRINT "The smallest number whose square ends in 269696 is" root

PRINT "Its square is" n</lang>

Output:

Identical to the first BBC BASIC version.

C

<lang C> // This code is the implementation of Babbage Problem

include <stdio.h>
include <stdlib.h>
include <limits.h>

int main() { int current = 0, //the current number square; //the square of the current number

//the strategy of take the rest of division by 1e06 is //to take the a number how 6 last digits are 269696 while ((current*current % 1000000 != 269696) && (square<INT_MAX)) { current++; }

       //output

if (square>+INT_MAX) printf("Condition not satisfied before INT_MAX reached."); else printf ("The smallest number whose square ends in 269696 is %d\n", current);

       //the end

return 0 ; } </lang>

Output:

The smallest number whose square ends in 269696 is 25264

Common Lisp

<lang Lisp> (defun babbage-test (n)

"A generic function for any ending of a number"
 (when (> n 0)
   (do* ((i 0 (1+ i))
         (d (expt 10 (1+ (truncate (log n) (log 10))))) )
     ((= (mod (* i i) d) n) i) )))

</lang>

Output:

(babbage-test 269696)
25264

C++

<lang Cpp>#include <iostream>

int main( ) {

  int current = 0 ;
  while ( ( current * current ) % 1000000 != 269696 ) 
     current++ ;
  std::cout << "The square of " << current << " is " << (current * current) << " !\n" ;
  return 0 ;

}</lang>

Output:

The square of 25264 is 638269696 !

C#

<lang csharp>namespace Babbage_Problem {

   class iterateNumbers
   {
       public iterateNumbers()
       {
           long baseNumberSquared = 0; //the base number multiplied by itself
           long baseNumber = 0;  //the number to be squared, this one will be iterated

           do  //this sets up the loop
           {
               baseNumber += 1; //add one to the base number
               baseNumberSquared = baseNumber * baseNumber; //multiply the base number by itself and store the value as baseNumberSquared
           }
           while (Right6Digits(baseNumberSquared) != 269696); //this will continue the loop until the right 6 digits of the base number squared are 269,696

           Console.WriteLine("The smallest integer whose square ends in 269,696 is " + baseNumber);
           Console.WriteLine("The square is " + baseNumberSquared);

       private long Right6Digits(long baseNumberSquared)
       {

           string numberAsString = baseNumberSquared.ToString(); //this is converts the number to a different type so it can be cut up

           if (numberAsString.Length < 6) { return baseNumberSquared; }; //if the number doesn't have 6 digits in it, just return it to try again.

           numberAsString = numberAsString.Substring(numberAsString.Length - 6);  //this extracts the last 6 digits from the number

           return long.Parse(numberAsString); //return the last 6 digits of the number

       }
   }

}}</lang>

Output:

The smallest integer whose square ends in 269,696 is 25264
The square is 638269696

Clojure

<lang clojure>; Defines function named babbage? that returns true if the

square of the provided number leaves a remainder of 269,696 when divided
by a million

(defn babbage? [n]

 (let [square (expt n 2)]
   (= 269696 (mod square 1000000))))

Given a range of positive integers up to 99,736, apply the above babbage?
function, returning only numbers that return true.

(filter babbage? (range 99736))</lang>

Output:

(25264)

COBOL

<lang cobol>IDENTIFICATION DIVISION. PROGRAM-ID. BABBAGE-PROGRAM.

A line beginning with an asterisk is an explanatory note.
The machine will disregard any such line.

DATA DIVISION. WORKING-STORAGE SECTION.

In this part of the program we reserve the storage space we shall
be using for our variables, using a 'PICTURE' clause to specify
how many digits the machine is to keep free.
The prefixed number 77 indicates that these variables do not form part
of any larger 'record' that we might want to deal with as a whole.

77 N PICTURE 99999.

We know that 99,736 is a valid answer.

77 N-SQUARED PICTURE 9999999999. 77 LAST-SIX PICTURE 999999. PROCEDURE DIVISION.

Here we specify the calculations that the machine is to carry out.

CONTROL-PARAGRAPH.

   PERFORM COMPUTATION-PARAGRAPH VARYING N FROM 1 BY 1
   UNTIL LAST-SIX IS EQUAL TO 269696.
   STOP RUN.

COMPUTATION-PARAGRAPH.

   MULTIPLY N BY N GIVING N-SQUARED.
   MOVE N-SQUARED TO LAST-SIX.

Since the variable LAST-SIX can hold a maximum of six digits,
only the final six digits of N-SQUARED will be moved into it:
the rest will not fit and will simply be discarded.

   IF LAST-SIX IS EQUAL TO 269696 THEN DISPLAY N.</lang>

Output:

D

<lang D>// It's basically the same as any other version. // What can be observed is that 269696 is even, so we have to consider only even numbers, // because only the square of even numbers is even.

import std.math; import std.stdio;

void main( ) {

   // get smallest number <= sqrt(269696)
   int k = cast(int)(sqrt(269696.0));

   // if root is odd -> make it even
   if (k % 2 == 1)
       k = k - 1;

   // cycle through numbers
   while ((k * k) % 1000000 != 269696) 
       k = k + 2;

   // display output
   writefln("%d * %d = %d", k, k, k*k);

}</lang>

Output:

25264 * 25264 = 638269696

Elixir

<lang elixir>defmodule Babbage do

 def problem(n) when rem(n*n,1000000)==269696, do: n
 def problem(n), do: problem(n+2)

end

IO.puts Babbage.problem(0)</lang> or <lang elixir>Stream.iterate(2, &(&1+2)) |> Enum.find(&rem(&1*&1, 1000000) == 269696) |> IO.puts</lang>

Output:

Erlang

<lang Erlang> -module(solution1). -export([main/0]). babbage(N,E) when N*N rem 1000000 == 269696 -> io:fwrite("~p",[N]); babbage(N,E) -> case E of 4 -> babbage(N+2,6); 6 -> babbage(N+8,4) end. main()-> babbage(4,4). </lang>

F#

<lang f#> let mutable n=1 while(((n*n)%( 1000000 ))<> 269696) do

   n<-n+1

printf"%i"n </lang>

Forth

Can a Forth program be made readable to a novice, without getting into what a stack is? We shall see. <lang forth>( First we set out the steps the computer will use to solve the problem )

BABBAGE

   1 ( start from the number 1 )
   BEGIN ( commence a "loop": the computer will return to this point repeatedly )
       1+ ( add 1 to our number )
       DUP DUP ( duplicate the result twice, so we now have three copies )
       ( We need three because we are about to multiply two of them together to find the square, and the third will be used the next time we go around the loop -- unless we have found our answer, in which case we shall need to print it out )
       * ( * means "multiply", so we now have the square )
       1000000 MOD ( find the remainder after dividing it by a million )
       269696 = ( is it equal to 269,696? )
   UNTIL ( keep repeating the steps from BEGIN until the condition is satisfied )
   . ; ( when it is satisfied, print out the number that allowed us to satisfy it )

( Now we ask the machine to carry out these instructions )

BABBAGE</lang>

Output:

FreeBASIC

<lang freebasic>' version 25-10-2016 ' compile with: fbc -s console

' Charles Babbage would have known that only number ending ' on a 4 or 6 could produce a square ending on a 6 ' also any number below 520 would produce a square smaller than 269,696 ' we can stop when we have reached 99,736 ' we know it square and it ends on 269,696

Dim As ULong number = 524 ' first number to try Dim As ULong square, count

Do

   ' square the number
   square = number * number
   ' look at the last 6 digits, if they match print the number
   If Right(Str(square), 6) = "269696" Then Exit Do
   ' increase the number with 2, number end ons a 6
   number = number +2
   ' if the number = 99736 then we haved found a smaller number, so stop
   If number = 99736 Then Exit Do
   square = number * number
   ' look at the last 6 digits, if they match print the number
   If Right(Str(square),6 ) = "269696" Then Exit Do
   ' increase the number with 8, number ends on a 4
   number = number +8
   ' go to the first line under "Do"

Loop

If number = 99736 Then

   Print "No smaller number was found"

Else

   ' we found a smaller number, print the number and its square
   Print Using "The number = #####, and its square = ##########,"; number; square

End If

' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:

The number = 25,264 and its square = 638,269,696

FutureBasic

<lang qbasic>include "ConsoleWindow"

dim as long i

for i = 1 to 1000000

 if i ^ 2 mod 1000000 == 269696 then exit for

Output:

The smallest number whose square ends in 269696 is 25264
Its square is 638269696

Haskell

<lang Haskell>--calculate the squares of integer , testing for the last 6 digits findBabbageNumber :: Integer findBabbageNumber = head $ filter myCondition [1..]

  where 
     myCondition :: Integer -> Bool
     myCondition n = mod ( n ^ 2 ) 1000000 == 269696

--print out the result main :: IO ( ) main = do

  let babbageNumber = findBabbageNumber
  let squareOfNumber = babbageNumber ^ 2
  putStr $ show babbageNumber 
  putStr " ^ 2 equals "
  putStr $ show squareOfNumber
  putStrLn " !"</lang>

Output:

25264 ^ 2 equals 638269696 !

J

The key to understandability is a mix of hopefully adequate notation and some level of verifiability.

So let's break the implementation into some named pieces and present enough detail that a mathematician can verify that the result is both consistent and reasonable:

<lang J> square=: ^&2

  modulo1e6=: 1000000&|
  trythese=: i. 1000000                   NB. first million nonnegative integers
  which=: I.                              NB. position of true values
  which 269696=modulo1e6 square trythese  NB. right to left <-

25264 99736 150264 224736 275264 349736 400264 474736 525264 599736 650264 724736 775264 849736 900264 974736</lang>

The smallest of these values is 25264.

Java

<lang java>public class Test {

   public static void main(String[] args) {

       // let n be zero
       int n = 0;

       // repeat the following action
       do {

           // increase n by 1
           n++;

       // while the modulo of n times n is not equal to 269696
       } while (n * n % 1000_000 != 269696);

       // show the result
       System.out.println(n);
   }

}</lang>

JavaScript

<lang javascript>// Every line starting with a double slash will be ignored by the processing machine. // // Since the square root of 269,696 is approximately 519, we create a variable named "n" // and give it this value.

 n = 519

// The while-condition is in parentheses // * is for multiplication // % is for modulo operation // != is for "not equal"

 while ( ((n * n) % 1000000) != 269696 )
   n = n + 1

// n is incremented until the while-condition is met, so n should finally be the // smallest positive integer whose square ends in the digits 269,696. To see n, we // need to send it to the monitoring device (named console).

 console.log(n)

</lang>

Liberty BASIC

Now Mr. Babbage -- May I call you Charlie? No. OK -- we'll first start with 'n' equal to zero, then multiply it by itself to square it. If the last six digits of the result are not 269696, we'll add one to 'n' then go back and square it again. On our modern computer it should only take a moment to find the answer... <lang lb> [start] if right$(str$(n*n),6)="269696" then

   print "n = "; using("###,###", n);
   print "   n*n = "; using("###,###,###,###", n*n)

end if if n<100000 then n=n+1: goto [start] print "Program complete." </lang> Eureka! We found it! -- <lang sh> n = 25,264 n*n = 638,269,696 n = 99,736 n*n = 9,947,269,696 Program complete. </lang> Now my question for you, Sir, is how did you know that the square of ANY number would end in 269696?? Oh, and by the way, 99,736 is an answer too.

OCaml

<lang OCaml> let rec f a= if (a*a) mod 1000000 != 269696 then f(a+1) else a in let a= f 1 in Printf.printf "smallest positive integer whose square ends in the digits 269696 is %d\n" a </lang>

PARI/GP

<lang parigp>m=269696; k=1000000; {for(n=1,99736,

 \\ Try each number in this range, from 1 to 99736
 if(denominator((n^2-m)/k)==1, \\ Check if n squared, minus m, is divisible by k
   return(n) \\ If so, return this number and STOP.
 )

)}</lang>

Pascal

<lang pascal>program BabbageProblem; (* Anything bracketed off like this is an explanatory comment. *) var n : longint; (* The VARiable n can hold a 'long', ie large, INTeger. *) begin

   n := 2; (* Start with n equal to 2. *)
   repeat
       n := n + 2 (* Increase n by 2. *)
   until (n * n) mod 1000000 = 269696;

(* 'n * n' means 'n times n'; 'mod' means 'modulo'. *)

   write(n)

end.</lang>

Output:

Perl

<lang Perl>#!/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" ;</lang>

Output:

The square of 25264 is 638269696 !

Perl 6

Works with: Rakudo version 2016.03

This could certainly be written more concisely. Extra verbiage is included to make the process more clear. <lang perl6># For all positives integers from 1 to Infinity for 1 .. Inf -> $integer {

   # calculate the square of the integer
   my $square = $integer²;
   
   # print the integer and square and exit if the square modulo 1000000 is equal to 269696
   print "{$integer}² equals $square" and exit if $square mod 1000000 == 269696;

}</lang>

Output:

25264² equals 638269696

Alternatively, the following just may be declarative enough to allow Babbage to understand what's going on:

Output:

25264
99736

Phix

We can omit anything odd, as any odd number squared is obviously always odd.
Mr Babbage might need the whole "i is a variable" thing explained, and that "?i" prints the value of i, nowt else springs to mind. <lang Phix>for i=2 to 99736 by 2 do

   if remainder(i*i,1000000)=269696 then ?i exit end if

end for</lang>

Output:

PicoLisp

<lang PicoLisp>: (for N 99736 # Iterate N from 1 to 99736

  (T (= 269696 (% (* N N) 1000000)) N) )    # Stop if remainder is 269696

-> 25264</lang>

PowerShell

Definitions:
Lines that begin with the "#" symbol are comments: they will be ignored by the machine.
-----------------------------------------------------------------------------------------
While
Run a command block based on the results of a conditional test.
Syntax
while (condition) {command_block}
Key
condition If this evaluates to TRUE the loop {command_block} runs.
when the loop has run once the condition is evaluated again.
command_block Commands to run each time the loop repeats.
As long as the condition remains true, PowerShell reruns the {command_block} section.
-----------------------------------------------------------------------------------------
* means 'multiplied by'
% means 'modulo', or remainder after division
-ne means 'is not equal to'
++ means 'increment variable by one'

Declare a variable, $integer, with a starting value of 0.

$integer = 0

while (($integer * $integer) % 1000000 -ne 269696) {

   $integer++

}

Show the result.

$integer </lang>

Output:

Alternative method

Works with: PowerShell version 2

By looping through potential squares instead of potential square roots, we reduce the number of loops by a factor of 40. <lang PowerShell># Start with the smallest potential square number $TestSquare = 269696

Test if our potential square is a square
by testing if the square root of it is an integer
Test if the square root is an integer by testing if the remainder
of the square root divided by 1 is greater than zero
% is the remainder operator
-gt is the "greater than" operator

While the remainder of the square root divided by one is greater than zero

While ( [Math]::Sqrt( $TestSquare ) % 1 -gt 0 )

   {
   #  Add 100,000 to get the next potential square number
   $TestSquare = $TestSquare + 1000000
   }

This will loop until we get a value for $TestSquare that is a square number

Caclulate the root

$Root = [Math]::Sqrt( $TestSquare )

Display the result and its square

$Root $TestSquare</lang>

Output:

25264
638269696

Processing

<lang java>// Lines that begin with two slashes, thus, are comments: they // will be ignored by the machine.

// First we must declare a variable, n, suitable to store an integer:

int n;

// Each statement we address to the machine must end with a semicolon.

// To begin with, the value of n will be zero:

n = 0;

// Now we must repeatedly increase it by one, checking each time to see // whether its square ends in 269,696.

// We shall do this by seeing whether the remainder, when n squared // is divided by one million, is equal to 269,696.

do {

   n = n + 1;

} while (n * n % 1000000 != 269696);

// To read this formula, it is necessary to know the following // elements of the notation: // * means 'multiplied by' // % means 'modulo', or remainder after division // != means 'is not equal to'

// Now that we have our result, we need to display it.

// println is short for 'print line'

println(n);</lang>

Output:

Python

Lines that start by # are a comments:
they will be ignored by the machine

n=0 # n is a variable and its value is 0

we will increase its value by one until
its square ends in 269,696

while n**2 % 1000000 != 269696:

   # n**2 -> n squared
   # %    -> 'modulo' or remainer after division
   # !=   -> not equal to
   
   n += 1 # += -> increase by a certain number

print(n) # prints n </lang>

Output:

Racket

<lang racket>;; 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</lang>

Output:

R

<lang R> babbage_function=function(){

 n=0
 while (n**2%%1000000!=269696) {
   n=n+1
 }
 return(n)

} babbage_function()[length(babbage_function())] </lang>

Output:

REXX

If this were a computer program to be shown to a computer programming novice (albeit a very
intelligent polymath novice), the computer program would also have a lot more comments,
notes, and accompanying verbiage which would/could/should explain:

what a (computer program) comment looks like
what a computer is
what a computer program is
how a computer stores numbers and such
what are variables and how to store stuff in them
how the do loop works (initial value, incrementation, etc)
how an assignment = operator works
how a comparison == operator works
how an if statement works
what a (computer program) statement is
what the * operator is and how it does multiplication
what the + operator is and how it does addition
what the // operator is and how it does division remainder
what the right BIF does
~~who~~ what a BIF is and how it returns a value
how/when the then cause gets executed (after an if)
explain how/why an end statement is needed for a do loop
explain how a leave statement works
··· the say is probably the only statement that is self-explanatory

examine the right-most 6 digits of square

<lang rexx>/*REXX program finds the lowest (positive) integer whose square ends in 269,696. */

                                                /*operator   *    is multiplication.   */
  do j=2  by 2                                  /*start  J  at two,  increment by two. */
  if right(j*j, 6)==269696  then leave          /*is six right-most digits our target? */
  end                                           /*this signifies the end of the DO loop*/

say "The smallest integer whose square ends in 269,696 is: " j</lang> output

The smallest integer whose square ends in  269,696  is:  25264

examine remainder after dividing by 1 million

<lang rexx>/*REXX program finds the lowest (positive) integer whose square ends in 269,696. */

                                                /*operator  //   is division remainder.*/
  do j=2  by 2                                  /*start  J  at two,  increment by two. */
  if j*j // 1000000 == 269696  then leave       /*is square mod 1-million our target ? */
  end                                           /*this signifies the end of the DO loop*/

say "The smallest integer whose square ends in 269,696 is: " j</lang> output is identical as the 1^st REXX version.

examine only numbers ending in 4 or 6

<lang rexx>/*REXX program finds the lowest (positive) integer whose square ends in 269,696. */

                                                /*we'll only examine integers that are */
                                                /*  ending in  four  or  six.          */

  do j=4  by 10                                 /*start  J  at four,  increment by ten.*/
  k=j                                           /*set    K  to  J's  value.            */
  if right(k*k, 6)==269696  then leave          /*examine the right-most six digits.   */

  k=j+2                                         /*set    K  to  J+2  value.            */
  if right(k*k, 6)==269696  then leave          /*examine the right-most six digits.   */
  end                                           /*this signifies the end of the DO loop*/

say "The smallest integer whose square ends in 269,696 is: " k</lang> output is identical as the 1^st REXX version.

start with smallest possible number

<lang rexx>/*REXX ----------------------------------------------------------------

The solution must actually be larger than sqrt(269696)=519.585
--------------------------------------------------------------------*/

z=0 Do i=524 By 10 Until z>0

 If right(i*i,6)==269696  then z=i
 Else Do
  j=i+2
  if right(j*j,6)==269696  then z=j
  End
End

Say "The smallest integer whose square ends in 269696 is:" z Say ' 'z'**2 =' z**2</lang>

Output:

The smallest integer whose square ends in 269696 is: 25264
                            25264**2 = 638269696

Ring

<lang ring> n = 0 while pow(n,2) % 1000000 != 269696

       n = n + 1

end

see "The smallest number whose square ends in 269696 is : " + n + nl see "Its square is : " + pow(n,2) </lang> Output:

The smallest number whose square ends in 269696 is : 25264
Its square is : 638269696

Ruby

<lang ruby>n = 0 n = n + 2 until (n*n).modulo(1000000) == 269696 puts n </lang>

Run BASIC

<lang runbasic>for n = 1 to 1000000 if n^2 MOD 1000000 = 269696 then exit for next

PRINT "The smallest number whose square ends in 269696 is "; n PRINT "Its square is "; n^2</lang>

The smallest number whose square ends in 269696 is 25264
Its square is 638269696

Rust

<lang rust> fn main() {

  let mut current = 0i32;
  while (current * current)%1000000!=269696
  {current+=1;
  }
  println!("The smallest number whose square ends in 269696 is {}",current);

} </lang>

Output:

The smallest number whose square ends in 269696 is 25264

Scala

<lang scala>//Babbage Problem

object babbage{ def main( args:Array[String] ){

var x:Int = 524 //Sqrt of 269696 = 519.something

while( (x*x) % 1000000 != 269696 ){

if( x%10 == 4 ) x = x+2 else x = x+8 }

println("The smallest positive integer whose square ends in 269696 = " + x ) } } </lang>

Sidef

<lang ruby>var n = 0 while (n*n % 1000000 != 269696) {

   n += 2

} say n</lang>

Output:

SequenceL

<lang sequencel>main() := babbage(0);

babbage(current) :=

       current when current * current mod 1000000 = 269696
   else
       babbage(current + 1);</lang>

Output:

cmd:> babbage.exe
25264

Smalltalk

<lang smalltalk>"We use one variable, called n. Let it initially be equal to 1. Then keep increasing it by 1 for only as long as the remainder after dividing by a million is not equal to 269,696; finally, show the value of n." | n | n := 1. [ n squared \\ 1000000 = 269696 ] whileFalse: [ n := n + 1 ]. n</lang>

Output:

Swift

<lang Swift>import Swift

for i in 2...Int.max { if i * i % 1000000 == 269696 { print(i, "is the smallest number that ends with 269696") break } }</lang>

Output:

25264 is the smallest number that ends with 269696

VBScript

<lang vb>'Sir, this is a script that could solve your problem.

'Lines that begin with the apostrophe are comments. The machine ignores them.

'The next line declares a variable n and sets it to 0. Note that the 'equals sign "assigns", not just "relates". So in here, this is more 'of a command, rather than just a mere proposition. n = 0

'Starting from the initial value, which is 0, n is being incremented 'by 1 while its square, n * n (* means multiplication) does not have 'a modulo of 269696 when divided by one million. This means that the 'loop will stop when the smallest positive integer whose square ends 'in 269696 is found and stored in n. Before I forget, "<>" basically 'means "not equal to". Do While ((n * n) Mod 1000000) <> 269696

   n = n + 1 'Increment by 1.

Loop

'The function "WScript.Echo" displays the string to the monitor. The 'ampersand concatenates strings or variables to be displayed. WScript.Echo("The smallest positive integer whose square ends in 269696 is " & n & ".") WScript.Echo("Its square is " & n*n & ".")

'End of Program.</lang>

Output:

The smallest positive integer whose square ends in 269696 is 25264.
Its square is 638269696.

x86 Assembly

AT&T syntax

Works with: gas

<lang asmatt># What is the lowest number whose square ends in 269,696?

At the very end, when we have a result and we need to print it, we shall use for the purpose a program called PRINTF, which forms part of a library of similar utility programs that are provided for us. The codes given here will be needed at that point to tell PRINTF that we are asking it to print a decimal integer (as opposed to, for instance, text):

.data decin: .string "%d\n\0"

This marks the beginning of our program proper:

.text .global main

main:

We shall test numbers from 1 upwards to see whether their squares leave a remainder of 269,696 when divided by a million.

We shall be making use of four machine 'registers', called EAX, EBX, ECX, and EDX. Each can hold one integer.

Move the number 1,000,000 into EBX:

       mov    $1000000, %ebx

The numbers we are testing will be stored in ECX. We start by moving a 1 there:

       mov    $1,       %ecx

Now we need to test whether the number satisfies our requirements. We shall want the computer to come back and repeat this sequence of instructions for each successive integer until we have found the answer, so we put a label ('next') to which we can refer.

We move (in fact copy) the number stored in ECX into EAX, where we shall be able to perform some calculations upon it without disturbing the original:

       mov    %ecx,     %eax

Multiply the number in EAX by itself:

       mul    %eax

Divide the number in EAX (now the square of the number in ECX) by the number in EBX (one million). The quotient -- for which we have no use -- will be placed in EAX, and the remainder in EDX:

       idiv   %ebx

Compare the number in EDX with 269,696. If they are equal, jump ahead to the label 'done':

       cmp    $269696,  %edx
       je     done

Otherwise, increment the number in ECX and jump back to the label 'next':

       inc    %ecx
       jmp    next

If we get to the label 'done', it means the answer is in ECX.

done:

Put a reference to the codes for PRINTF into EAX:

       lea    decin,    %eax

Now copy the number in ECX, which is our answer, into an area of temporary storage where PRINTF will expect to find it:

       push   %ecx

Do the same with EAX -- giving the code for 'decimal integer' -- and then call PRINTF to print the answer:

       push   %eax
       call   printf

The pieces of information we provided to PRINTF are still taking up some temporary storage. They are no longer needed, so make that space available again:

       add    $8,       %esp

Place the number 0 in EAX -- a conventional way of indicating that the program has finished correctly -- and return control to whichever program called this one:

       mov    $0,       %eax
       ret

The end.</lang>

Output:

XLISP

<lang scheme>; The computer will evaluate expressions written in -- possibly nested -- parentheses, where the first symbol gives the operation and any subsequent symbols or numbers give the operands.

For instance, (+ (+ 2 2) (- 7 5)) evaluates to 6.

We define our problem as a function

(define (try n)

We are looking for a value of n that leaves 269,696 as the remainder when its square is divided by a million.

The symbol * stands for multiplication.

   (if (= (remainder (* n n) 1000000) 269696)

If this condition is met, the function should give us the value of n

If not, it should try n+1

       (try (+ n 1))))

We supply our function with 1 as an initial value to test, and ask the computer to print the final result.

(print (try 1))</lang>

Output:

zkl

<lang zkl>// The magic number is 269696, so, starting about its square root, // find the first integer that, when squared, its last six digits are the magic number. // The last digits are found with modulo, represented here by the % symbol const N=269696; [500..].filter1(fcn(n){ n*n%0d1_000_000 == N })</lang>

Output: