Babbage problem
You are encouraged to solve this task according to the task description, using any language you may know.
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.
11l
<lang 11l>V n = 1 L n ^ 2 % 1000000 != 269696
n++
print(n)</lang>
360 Assembly
An assembler program always seems a bit tricky for non system engineer because it deals directly with the operating system and with the hardware instructions. Here we have a 32-bit computer with 16 32-bit registers. The caller (the operating system to keep it simple) is calling you giving your location address stored in register-15 and has stored in register-14 his return address. To save each program context, register-13 points to a 18 word save area. Do not spend time in understanding the context saving and restoring in the prologue and epilogue part of the program. What you have to know, “360” architecture uses 32-bit signed binary arithmetic, so here the maximum integer value is 2^31-1 (2147483647). Therefore the solution must be less than 2147483647. The multiplication and the division use a pair of registers; coding “MR 4,2” means multiply register-5 by register-2 and place result in the (register-4,register-5) pair; the same way “DR 4,2” means divide the (register-4,register-5) pair by register-2 and place the quotient in register-5 and the reminder in register-4. We use in the below program this intermediate 64-bit integers to find a solution with a value up to 2^31-1 even when we have to compute the square of this value. <lang 360asm>
- Find the lowest positive integer whose square ends in 269696
- The logic of the assembler program is simple :
- loop for i=524 step 2
- if (i*i modulo 1000000)=269696 then leave loop
- next i
- output 'Solution is: i=' i ' (i*i=' i*i ')'
BABBAGE CSECT beginning of the control section
USING BABBAGE,13 define the base register
B 72(15) skip savearea (72=18*4)
DC 17F'0' savearea (18 full words (17+1))
STM 14,12,12(13) prolog: save the caller registers
ST 13,4(15) prolog: link backwards
ST 15,8(13) prolog: link forwards
LR 13,15 prolog: establish addressability
LA 6,524 let register6 be i and load 524
LOOP LR 5,6 load register5 with i
MR 4,6 multiply register5 with i
LR 7,5 load register7 with the result i*i
D 4,=F'1000000' divide register5 with 1000000
C 4,=F'269696' compare the reminder with 269696
BE ENDLOOP if equal branch to ENDLOOP
LA 6,2(6) load register6 (i) with value i+2
B LOOP branch to LOOP
ENDLOOP XDECO 6,BUFFER+15 edit registrer6 (i)
XDECO 7,BUFFER+34 edit registrer7 (i squared)
XPRNT BUFFER,L'BUFFER print buffer
L 13,4(0,13) epilog: restore the caller savearea
LM 14,12,12(13) epilog: restore the caller registers
XR 15,15 epilog: set return code to 0
BR 14 epilog: branch to caller
BUFFER DC CL80'Solution is: i=............ (i*i=............)'
END BABBAGE end of the control section
</lang>
- Output:
Solution is: i= 25264 (i*i= 638269696)
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
Aime
<lang aime>integer i;
i = sqrt(269696); while (i * i % 1000000 != 269696) {
i += 1;
}
o_(i, "\n");</lang>
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:
25264
AppleScript
AppleScript's number types are at their limits here, but we can just get to the first Babbage number, after 638 integer root tests on suffixed numbers:
<lang AppleScript>-- BABBAGE -------------------------------------------------------------------
-- babbage :: Int -> [Int] on babbage(intTests)
script test
on toSquare(x)
(x * 1000000) + 269696
end toSquare
on |λ|(x)
hasIntRoot(toSquare(x))
end |λ|
end script
script toRoot
on |λ|(x)
((x * 1000000) + 269696) ^ (1 / 2)
end |λ|
end script
set xs to filter(test, enumFromTo(1, intTests))
zip(map(toRoot, xs), map(test's toSquare, xs))
end babbage
-- TEST ---------------------------------------------------------------------- on run
-- Try 1000 candidates
unlines(map(curry(intercalate)'s |λ|(" -> "), babbage(1000)))
--> "2.5264E+4 -> 6.38269696E+8"
end run
-- GENERIC FUNCTIONS ---------------------------------------------------------
-- curry :: (Script|Handler) -> Script on curry(f)
script
on |λ|(a)
script
on |λ|(b)
|λ|(a, b) of mReturn(f)
end |λ|
end script
end |λ|
end script
end curry
-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)
if m > n then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo
-- filter :: (a -> Bool) -> [a] -> [a] on filter(f, xs)
tell mReturn(f)
set lst to {}
set lng to length of xs
repeat with i from 1 to lng
set v to item i of xs
if |λ|(v, i, xs) then set end of lst to v
end repeat
return lst
end tell
end filter
-- hasIntRoot :: Int -> Bool on hasIntRoot(n)
set r to n ^ 0.5 r = (r as integer)
end hasIntRoot
-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- min :: Ord a => a -> a -> a on min(x, y)
if y < x then
y
else
x
end if
end min
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- unlines :: [String] -> String on unlines(xs)
intercalate(linefeed, xs)
end unlines
-- zip :: [a] -> [b] -> [(a, b)] on zip(xs, ys)
set lng to min(length of xs, length of ys)
set lst to {}
repeat with i from 1 to lng
set end of lst to {item i of xs, item i of ys}
end repeat
return lst
end zip</lang>
- Output:
2.5264E+4 -> 6.38269696E+8
ARM Assembly
<lang ARM Assembly>
/* ARM assembly Raspberry PI */ /* program babbage.s */
/************************************/ /* Constantes */ /************************************/ .equ STDOUT, 1 @ Linux output console .equ EXIT, 1 @ Linux syscall .equ WRITE, 4 @ Linux syscall
/*********************************/ /* Initialized data */ /*********************************/ .data sMessResult: .ascii "Result = " sMessValeur: .fill 11, 1, ' ' @ size => 11 szCarriageReturn: .asciz "\n"
/*********************************/ /* UnInitialized data */ /*********************************/ .bss /*********************************/ /* code section */ /*********************************/ .text .global main main: @ entry of program
ldr r4,iNbStart @ start number = 269696 mov r5,#0 @ counter multiply ldr r2,iNbMult @ value multiply = 1 000 000 mov r6,r4
1:
mov r0,r6 bl squareRoot @ compute square root umull r1,r3,r0,r0 cmp r3,#0 @ overflow ? bne 100f @ yes -> end cmp r1,r6 @ perfect square bne 2f @ no -> loop ldr r1,iAdrsMessValeur bl conversion10 @ call conversion decimal ldr r0,iAdrsMessResult bl affichageMess @ display message b 100f @ end
2:
add r5,#1 @ increment counter mul r3,r5,r2 @ multiply by 1 000 000 add r6,r3,r4 @ add start number b 1b
100: @ standard end of the program
mov r0, #0 @ return code mov r7, #EXIT @ request to exit program svc #0 @ perform the system call
iAdrsMessValeur: .int sMessValeur iAdrszCarriageReturn: .int szCarriageReturn iAdrsMessResult: .int sMessResult iNbStart: .int 269696 iNbMult: .int 1000000 /******************************************************************/ /* compute squareRoot */ /******************************************************************/ /* r0 contains n */ /* r0 return result or -1 */ squareRoot:
push {r1-r5,lr} @ save registers
cmp r0,#0
beq 100f @ if zero -> end
movlt r0,#-1 @ if negatif return - 1
blt 100f
cmp r0,#4 @ if < 4 return 1
movlt r0,#1
blt 100f
@ start
clz r2,r0 @ number of zeros on the left
rsb r2,#32 @ so many useful numbers right
bic r2,#1 @ to have an even number of digits
mov r3,#0b11 @ mask for extract 2 bits
lsl r3,r2
mov r1,#0 @ init résult with 0
mov r4,#0 @ raz remainder area
1: @ begin loop
and r5,r0,r3 @ extract 2 bits with mask add r4,r5,lsr r2 @ shift right and addition with remainder lsl r5,r1,#1 @ multiplication by 2 lsl r5,#1 @ shift left one bit orr r5,#1 @ bit right = 1 lsl r1,#1 @ shift left one bit subs r4,r5 @ sub remainder addmi r4,r4,r5 @ if negative restaur register addpl r1,#1 @ else add 1 subs r2,#2 @ decrement number bits movmi r0,r1 @ if end return result bmi 100f lsl r4,#2 @ no -> shift left remainder 2 bits lsr r3,#2 @ and shift right mask 2 bits b 1b @ and loop
100:
pop {r1-r5,lr} @ restaur registers
bx lr @return
/******************************************************************/ /* display text with size calculation */ /******************************************************************/ /* r0 contains the address of the message */ affichageMess:
push {r0,r1,r2,r7,lr} @ save registres
mov r2,#0 @ counter length
1: @ loop length calculation
ldrb r1,[r0,r2] @ read octet start position + index
cmp r1,#0 @ if 0 its over
addne r2,r2,#1 @ else add 1 in the length
bne 1b @ and loop
@ so here r2 contains the length of the message
mov r1,r0 @ address message in r1
mov r0,#STDOUT @ code to write to the standard output Linux
mov r7, #WRITE @ code call system "write"
svc #0 @ call systeme
pop {r0,r1,r2,r7,lr} @ restaur des 2 registres */
bx lr @ return
/******************************************************************/ /* Converting a register to a decimal unsigned */ /******************************************************************/ /* r0 contains value and r1 address area */ /* r0 return size of result (no zero final in area) */ /* area size => 11 bytes */ .equ LGZONECAL, 10 conversion10:
push {r1-r4,lr} @ save registers
mov r3,r1
mov r2,#LGZONECAL
1: @ start loop
bl divisionpar10U @ unsigned r0 <- dividende. quotient ->r0 reste -> r1
add r1,#48 @ digit
strb r1,[r3,r2] @ store digit on area
cmp r0,#0 @ stop if quotient = 0
subne r2,#1 @ else previous position
bne 1b @ and loop
@ and move digit from left of area
mov r4,#0
2:
ldrb r1,[r3,r2]
strb r1,[r3,r4]
add r2,#1
add r4,#1
cmp r2,#LGZONECAL
ble 2b
@ and move spaces in end on area
mov r0,r4 @ result length
mov r1,#' ' @ space
3:
strb r1,[r3,r4] @ store space in area add r4,#1 @ next position cmp r4,#LGZONECAL ble 3b @ loop if r4 <= area size
100:
pop {r1-r4,lr} @ restaur registres
bx lr @return
/***************************************************/ /* division par 10 unsigned */ /***************************************************/ /* r0 dividende */ /* r0 quotient */ /* r1 remainder */ divisionpar10U:
push {r2,r3,r4, lr}
mov r4,r0 @ save value
ldr r3,iMagicNumber @ r3 <- magic_number raspberry 1 2
umull r1, r2, r3, r0 @ r1<- Lower32Bits(r1*r0) r2<- Upper32Bits(r1*r0)
mov r0, r2, LSR #3 @ r2 <- r2 >> shift 3
add r2,r0,r0, lsl #2 @ r2 <- r0 * 5
sub r1,r4,r2, lsl #1 @ r1 <- r4 - (r2 * 2) = r4 - (r0 * 10)
pop {r2,r3,r4,lr}
bx lr @ leave function
iMagicNumber: .int 0xCCCCCCCD </lang>
- Output:
Result = 25264
AutoHotkey
<lang AutoKotkey>
- Give n an initial value
n = 519
- Loop this action while condition is not satisfied
while (Mod(n*n, 1000000) != 269696) { ; Increment n n++ }
- Display n as value
msgbox, %n% </lang>
- Output:
25264
AWK
<lang 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
Batch File
<lang dos>
- This line is only required to increase the readability of the output by hiding the lines of code being executed
@echo off
- Everything between the lines keeps repeating until the answer is found
- The code works by, starting at 1, checking to see if the last 6 digits of the current number squared is equal to 269696
- ----------------------------------------------------------------------------------
- loop
- Increment the current number being tested by 1
set /a number+=1
- Square the current number
set /a numbersquared=%number%*%number%
- Check if the last 6 digits of the current number squared is equal to 269696, and if so, stop looping and go to the end
if %numbersquared:~-6%==269696 goto end
goto loop
- ----------------------------------------------------------------------------------
- end
echo %number% * %number% = %numbersquared% pause>nul </lang>
- Output:
25264 * 25264 = 638269696
BASIC
Applesoft BASIC
This is an implementation based on the alternative solution for the BBC BASIC. We know that 269696 is not a perfect square, so we can safely start with N=1269696 and add 1000000 each time N is not a perfect square. The ST function returns the remainder of the difference between N and the square of the integer part of the root (R). There are a couple of quirks in AppleSoft BASIC, the largest integer is 32767, but the largest number not displayed on scientific notation is 999999999, which explains the strange IF statement in line 170.
<lang basic>
100 :
110 REM BABBAGE PROBLEM
120 :
130 DEF FN ST(A) = N - INT (A) * INT (A)
140 N = 269696
150 N = N + 1000000
160 R = SQR (N)
170 IF FN ST(R) < > 0 AND N < 999999999 THEN GOTO 150
180 IF N > 999999999 THEN GOTO 210
190 PRINT "SMALLESt NUMBER WHOSE
SQUARE ENDS IN"; CHR$ (13);
"269696 IS ";R;", AND THE
SQUARE IS"; CHR$ (13);N
200 END
210 PRINT "THERE IS NO SOLUTION
FOR VALUES SMALLER"; CHR$(13);
"THAN 999999999."
</lang>
- Output:
]RUN SMALLEST NUMBER WHOSE SQUARE ENDS IN 269696 IS 25264, AND THE SQUARE IS 638269696
Commodore BASIC
Based on the C language implementation <lang basic>10 rem This code is an implementation of Babbage Problem 20 num = 100 : rem We can safely start at 100 30 s = num*num 40 r = s - int(s/1000000)*1000000 : rem remainder when divided by 1,000,000 50 if r = 269696 then goto 100 : rem compare with 269,696 60 print "n="num"sq="s"rem="r 70 num = num+1 80 goto 30 90 rem Print out the result 100 print:print "The smallest number whose square ends in 269696 is:" 110 print num;"....";num;"squared = ";s 120 end</lang>
BASIC256
<lang BASIC256>
- This code is an implementation of Babbage Problem
number = 2 DO number += 2 UNTIL ((number^2) % 1000000) = 269696 PRINT "The smallest number whose square ends in 269696 is: "; number PRINT "It's square is "; number*number </lang>
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
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.
IS-BASIC
<lang IS-BASIC>100 PROGRAM "Babbage.bas" 110 LET N=2 120 DO 130 LET N=N+2 140 LOOP UNTIL MOD(N*N,1000000)=269696 150 PRINT "The smallest number whose square ends in 269696 is:";N 160 PRINT "It's square is";N^2</lang> Alternative method: <lang IS-BASIC>100 PROGRAM "Babbage.bas" 110 LET N=269696 120 DO 130 LET N=N+1000000 140 LET R=SQR(N) 150 LOOP UNTIL R=INT(R) 160 PRINT "The smallest number whose square ends in 269696 is:";R 170 PRINT "It's square is";N</lang>
Befunge
Befunge is not an easily readable language, but with a basic understanding of the syntax, I think an intelligent person should be able to follow the logic of the code below.
<lang befunge> 1+ ::* "d"::** % "V8":** -! #v_ > > > > >
v
increment n n*n modulo 1000000 equal to 269696? v if false, loop to right
v
v"Smallest number whose square ends in 269696 is "0 < else output n below >:#,_$ . 55+, @
ouput message then n newline exit
numeric constants explained:
"d" ascii value of 'd', i.e. 100
- duplicate twice: 100,100,100
- multiply twice: 100*100*100 = 1000000
"V8" ascii values of 'V' and '8', i.e. 86 and 56
- duplicate the '8' (56): 86,56,56
- multiply twice: 86*56*56 = 269696</lang>
- Output:
Smallest number whose square ends in 269696 is 25264
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 (((square=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
Alternate solution
I use Allegro CL 10.1
<lang lisp>
- Project
- Babbage problem
(setq n 1) (setq bab2 1) (loop while (/= bab2 269696)
do (setq n (+ n 1))
(setf bab1 (expt n 2))
(setf bab2 (mod bab1 1000000)))
(format t "~a" "The smallest number whose square ends in 269696 is: ") (write n) (terpri) (format t "~a" "Its square is: ") (write (* n n)) </lang> Output:
The smallest number whose square ends in 269696 is: 25264 Its square is: 638269696
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:
25264
Component Pascal
<lang oberon2> MODULE BabbageProblem; IMPORT StdLog;
PROCEDURE Do*; VAR i: LONGINT; BEGIN i := 2; WHILE (i * i MOD 1000000) # 269696 DO IF i MOD 10 = 4 THEN INC(i,2) ELSE INC(i,8) END END; StdLog.Int(i) END Do;
END BabbageProblem. </lang>
Execute: ^Q BabbageProble.Do
- Output:
25264
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
Dafny
<lang dafny> // Helper function for mask: does the actual computation. function method mask_(v:int,m:int):int
decreases v-m requires 0 <= v && 0 < m ensures v < mask_(v,m)
{
if v < m then m else mask_(v,m*10)
}
// Return the smallest power of 10 greater than v. function method mask(v:int):int
requires 0 <= v ensures v < mask(v)
{
mask_(v,10)
}
// Return true if the last digits of v == suffix. predicate method EndWith(v:int,suffix:int)
requires 0 <= suffix
{
v % mask(suffix) == suffix
}
method SmallestSqEndingWith(suffix:int) returns (s:int)
requires 0 < suffix ensures EndWith(s*s, suffix) // ensures forall i :: 0 <= i < s ==> !EndWith(i*i,suffix) decreases * // This method may not terminate.
{
s := 0;
// squares is the sequence of s*s. A ghost variable is only used by the
// verification process at compile time.
ghost var squares := [];
while !EndWith(s*s, suffix)
invariant s == |squares|
invariant forall i :: 0 <= i < s ==> squares[i] == i*i && !EndWith(squares[i], suffix)
decreases *
{
squares := squares + [s*s];
s := s + 1;
}
// Leaving the method:
// s*s ends with the suffix.
assert EndWith(s*s, suffix);
// The sequence squares contains i*i for i in [0..s]; none of the elements of
// squares ends with the suffix.
assert s == |squares|;
assert forall i :: 0 <= i < s ==> i*i == squares[i] && !EndWith(squares[i], suffix);
// That last assertion should imply the commented-out post-condition of the
// method, but I'm not sure how to express that.
//
// Conclusion: s is guaranteed to be the smallest number whose square ends
// with the suffix.
}
method Main() decreases * {
var suffix := 269696; var smallest := SmallestSqEndingWith(suffix); print smallest, "\n";
} </lang>
Dart
<lang dart>
main() {
var x = 0;
while((x*x)% 1000000 != 269696)
{ x++;}
print('$x'); } </lang>
Dyalect
<lang dyalect>for i in 2..Integer.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
EasyLang
<lang>while n * n mod 1000000 <> 269696
n += 1
. print n</lang>
Elena
ELENA 4.x : <lang elena>import extensions; import system'math;
public program() {
var n := 1;
until(n.sqr().mod:1000000 == 269696)
{
n += 1
};
console.printLine(n)
}</lang>
- Output:
25264
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:
25264
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>
Same as above, sans mutable state. <lang fsharp> Seq.initInfinite id |> Seq.skipWhile (fun n->(n*n % 1000000) <> 269696) |> Seq.head |> printfn "%d" </lang>
Factor
<lang factor>! Lines like this one are comments. They are meant for humans to ! read and have no effect on the instructions carried out by the ! computer (aside from Factor's parser ignoring them).
! Comments may appear after program instructions on the same ! line.
! Each word between USING: and ; is a vocabulary. By importing ! a vocabulary in this way, its words are made available for the ! program to use. This is a way to keep the space requirements ! down for deployed programs, and a nice side effect is that it ! gives readers a clue for where to look for documentation.
USING: kernel math math.ranges prettyprint sequences ;
! Before the program begins, it's incredibly helpful to have an ! understanding of Factor's dataflow model. Don't worry; it's ! not complicated, but it's confusing to read a Factor program ! without this knowledge.
! Factor is a stack-based language. What this means is that ! there is an implicit data stack in the background, waiting ! to recieve whatever manner of thing we wish to give it. Here ! is a simple arithmetic expression to demonstrate:
! language token | data stack ! ---------------+----------- ! 2 2 ! numbers place themselves on the stack. ! 1 2 1 ! 4 2 1 4 ! + 2 5 ! consume 1 and 4 and leave behind 5. ! * 10 ! consume 2 and 5 and leave behind 10.
! Thus the phrase
! 2 1 4 + *
! in Factor is a way to calculate 2 * (4 + 1). ! We could have also written this as
! 1 4 + 2 *
! with no change in meaning or outcome.
! Because of the way the data stack works, there is no need ! to specify order of operations in the language, because you do ! so inherently by the order you place things on the data stack.
! === BEGIN PROGRAM ============================================
518 99,736 2 <range> ! Here we place three numbers on the
! stack representing a range of numbers.
! The first, 518, represents the starting
! point of the sequence. 99,736
! represents the ending point of the
! sequence. 2 represents the "step" of
! the sequence, or a constant distance
! between members.
! <range> takes those three numbers and
! creates an object representing the
! described range of numbers. Computers
! of today are more than capable of
! storing that many numbers, but <range>
! doesn't store them all; it calculates
! the number that is needed at the
! current time.
! The rationale for the sequence is as
! follows. Odd squares are always odd, so
! we don't need to consider them. That's
! why the sequence starts with an even
! number and is incremented by 2. We
! choose 518 to start because it's the
! largest even square less than 269,696.
! We choose 99,736 to end because we
! know it's a solution.
[ sq 1,000,000 mod 269,696 = ]
! the [ ... ] form is called a quotation.
! Think of it like a sequence that stores
! code. It's a way to place code on the
! data stack without executing it. This
! is so that it can be used by the find
! word. You could also think of it much
! like a function that hasn't been given
! a name.
find
! When we call the find word, there are
! two objects on the stack: a sequence
! and a quotation. find is a word that
! takes a sequence and a quotation and
! applies the quotation to one member of
! the sequence after another. It does
! so until the quotation returns a t
! value (denoting a boolean true) and
! then leaves that number, along with its
! index in the sequence, on the stack.
! Let's take a look at what happens
! for each iteration of find. Let's look
! at what happens with the first number
! in the sequence.
! language token | data stack ! ---------------+----------- ! 518 518 ! 518 is placed on the stack
! from the sequence by find.
! sq 268,324 ! square it ! 1,000,000 268,324 1,000,000 ! place a million on the stack ! mod 268,324 ! take modulus of 268,324
! and 1,000,000
! 269,696 268,324 269,696 ! place 269,696 on the stack ! = f ! test 268,324 and 269,696 for
! equality.
! So the square of the first number in
! the sequence, 518, does not end with
! 269,696. We'll try each number in the
! sequence until we get a t.
. ! Consume the top member of the data stack and print it out.
drop ! find leaves both the found element from the sequence
! and the index at which it was found on the data stack. ! We don't care about the index so we will call drop to ! remove it from the top of the data stack. All programs ! must end with an emtpy data stack.
! Putting the entire program together, it looks like this:
! 518 99,736 2 <range> [ sq 1,000,000 mod 269,696 = ] find . drop</lang>
- Output:
25264
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:
25264
Fortran
First FORTRAN
Mister Babbage,
I have been working for 2 years in New York on an IBM 704 computer.
I have just finished my work on a new language, a language for non-specialists.
I called it: FORTRAN (FORmula TRANslator).
And with it I solved your problem.
Sincerely,
John Backus - September 1956
<lang fortran> DO 3 N=1,99736
IF(MODF(N*N,1000000)-269696)3,4,3
3 CONTINUE
4 PRINT 5,N
5 FORMAT(I6)
STOP
</lang>
- Output:
25264
Mister Babbage, an addendum :
I was confident that my program will work because the FORTRAN for IBM 704 rely on a computer with a hardware of 36 bit integers.
You already proved than the number N must be less or equal to 99736. But if N were greater than 46340 (√ 231-1 ), half of the programs you see here use 32 bit integers and they would have failed with an overflow exception.
Sincerely - J. B.
Modern Fortran
<lang fortran> program babbage
implicit none integer :: n
n=1
do while (mod(n*n,1000000) .ne. 269696)
n = n + 1
end do
print*, n
end program babbage </lang>
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
next
print "The smallest number whose square ends in 269696 is"; i print "Its square is"; i ^ 2</lang>
- Output:
The smallest number whose square ends in 269696 is 25264 Its square is 638269696
Gambas
Click this link to run this code <lang gambas>Public Sub Main() Dim iNum As Long
For iNum = 1 To 100000
If Str(iNum * iNum) Ends "269696" Then Break
Next
Print "The lowest number squared that ends in '269696' is " & Str(iNum)
End</lang> Output:
The lowest number squared that ends in '269696' is 25264
Go
<lang go>package main
import "fmt"
func main() { const ( target = 269696 modulus = 1000000 ) for n := 1; ; n++ { // Repeat with n=1, n=2, n=3, ... square := n * n ending := square % modulus if ending == target { fmt.Println("The smallest number whose square ends with", target, "is", n, ) return } } }</lang>
- Output:
The smallest number whose square ends with 269696 is 25264
Groovy
<lang Groovy> int n=104; ///starting point while( (n**2)%1000000 != 269696 )
{ if (n%10==4) n=n+2;
if (n%10==6) n=n+8;
}
println n+"^2== "+n**2 ;
</lang>
- Output:
25264^2== 638269696
Haskell
head
<lang Haskell>--Calculate squares, testing for the last 6 digits findBabbageNumber :: Integer findBabbageNumber =
head (filter ((269696 ==) . flip mod 1000000 . (^ 2)) [1 ..])
main :: IO () main =
(putStrLn . unwords)
(zipWith
(++)
(show <$> ([id, (^ 2)] <*> [findBabbageNumber]))
[" ^ 2 equals", " !"])</lang>
- Output:
25264 ^ 2 equals 638269696 !
Safe.headMay
Or, if we incline to the nullius in verba approach, are not yet convinced that there really are any such numbers below 100,000, and look uncertainly at head – a partial function which simply fails on empty lists, we could import the Safe module, and use the headMay alternative, which, more cautiously and experimentally, returns a Maybe value:
<lang haskell>import Data.List (intercalate) import Data.Maybe (maybe) import Safe (headMay)
maybeBabbage :: Integer -> Maybe Integer maybeBabbage upperLimit =
headMay (filter ((269696 ==) . flip rem 1000000) ((^ 2) <$> [1 .. upperLimit]))
main :: IO () main = do
let upperLimit = 100000
putStrLn $
maybe
(intercalate (show upperLimit) ["No such number found below ", " ..."])
(intercalate " ^ 2 -> " .
fmap show . (<*>) [floor . sqrt . fromInteger, id] . pure)
(maybeBabbage upperLimit)</lang>
- Output:
25264 ^ 2 -> 638269696
Suffixes and integer roots
The inverse approach, which gets us to the first number in just 638 tests, is to append a 269696 suffix to each successive integer, filtering for results with integer square roots.
We can then harvest as many as we need from an infinite stream of babbages, Mr Babbage.
<lang haskell>import Data.List (intercalate)
babbagePairs :: Integer babbagePairs =
[0,1000000 ..] >>= -- Drawing from a succession of N * 10^6
\x ->
let y = (x + 269696) -- The next number ending in 269696,
r = (sqrt . fromIntegral) y -- its square root,
i = floor r -- and the integer part of that root.
in [ [i, y] -- Root and square harvested together,
| r == fromIntegral i ] -- only if that root is an integer.
main :: IO () main = do
let arrowed = intercalate " ^ 2 -> " . fmap show mapM_ putStrLn (arrowed <$> take 10 babbagePairs)</lang>
- Output:
25264 ^ 2 -> 638269696 99736 ^ 2 -> 9947269696 150264 ^ 2 -> 22579269696 224736 ^ 2 -> 50506269696 275264 ^ 2 -> 75770269696 349736 ^ 2 -> 122315269696 400264 ^ 2 -> 160211269696 474736 ^ 2 -> 225374269696 525264 ^ 2 -> 275902269696 599736 ^ 2 -> 359683269696
A quick glance at these results suggests that Mr Babbage would have done well to inspect more closely the way in which the final digits of the square constrain the final digits of the root.
We can get to the solution almost immediately, after only a handful of tests, well within the reach of pencil and paper, if we discern that the root itself, to produce the 269692 suffix in its square, must have one of only four different final digit sequences: (0264, 5264, 4736, or 9736).
With a machine, this approach can industrialise the babbage harvest, yielding thousands of pairs in less than a second: <lang haskell>import Data.List (intercalate)
babbagePairs :: Integer babbagePairs =
[0,10000 ..] >>=
\x ->
(<*>) [(:) <*> return . (^ 2)] ((+ x) <$> [0264, 5264, 9736, 4736]) >>=
\[a, b] ->
[ [a, b]
| ((269696 ==) . flip rem 1000000) b ]
main :: IO () main =
mapM_ putStrLn (intercalate " ^ 2 -> " . fmap show <$> take 4000 babbagePairs)</lang>
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.
Alternatively, inspired by the APL example that makes the sentence sound natural
<lang J> NB. In the interactive environment. NB. First here, Mr Babbage, we'll make the computer's words more meaningful to an english speaker.
NB. The first is the "head" of a list, written with these inviting open arms that embrace one small dot :
first=: {.
NB. The small i. notation denotes "all integers up to 100000". You've already found a solution in that range.
n=: i. 100000
NB. This is how we write squaring.
squareof=: *:
NB. In our notation, a dyad is a word that takes an x value on the left and an y value on the right.
ends=: dyad : ' x = 1000000 | y '
NB. This dyad selects values from the list x, as marked by the list y
where=: dyad : ' y # x '
NB. Now that we defined our words, we can ask our question with them :
first n where 269696 ends squareof n
25264
NB. With a bit of habit, you won't need to define words in english anymore. NB. The following easily relates word for word to the sentence we've written :
{. (i.100000) #~ 269696 = 1000000 | *: i.100000
25264
NB. Like all mathematical notations, in J you see patterns that suggest simplification :
{. I. 269696 = 1000000 | *: i.100000
25264
</lang>
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>
25264
JavaScript
Iteration
<lang javascript>// Every line starting with a double slash will be ignored by the processing machine, // just like these two. // // 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>
Array constructor
JavaScript's Array constructor lets us filter an array automatically populated with a function of the element index. This proves faster than setting up and running a while loop test, and we can make it particularly efficient by testing the potential squares rather than the potential roots.
Starting with numbers which end in 269696, and filtering for those which have integer roots, so that we reach 25264 ^2 -> 638269696 after only 638 tests.
<lang JavaScript>(() => {
// BABBAGE ---------------------------------------------------------------
// babbageNumbers :: Int -> [(Int, Int)]
const babbageNumbers = intTests =>
map(x => [Math.sqrt(x), x], filter(
hasIntegerRoot,
takeNfromSeries(intTests, x => (x * 1000000) + 269696)
));
// hasIntegerRoot :: Int -> Bool
const hasIntegerRoot = n => {
const r = Math.sqrt(n);
return Math.floor(r) === r;
};
// takeNFromSeries :: Int -> (Int -> a) -> [a]
const takeNfromSeries = (n, f) =>
Array.from({
length: n
}, (_, i) => f(i));
// GENERIC FUNCTIONS -----------------------------------------------------
// curry :: ((a, b) -> c) -> a -> b -> c const curry = f => a => b => f(a, b);
// filter :: (a -> Bool) -> [a] -> [a] const filter = (f, xs) => xs.filter(f);
// intercalate :: String -> [a] -> String const intercalate = (s, xs) => xs.join(s);
// map :: (a -> b) -> [a] -> [b] const map = (f, xs) => xs.map(f);
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// TEST ------------------------------------------------------------------
return unlines(
map(
curry(intercalate)(' ^2 -> '),
babbageNumbers(1000000) // Testing 10^6 numbers ending in 269696
)
);
})();</lang>
- Output:
25264 ^2 -> 638269696 99736 ^2 -> 9947269696 150264 ^2 -> 22579269696 224736 ^2 -> 50506269696 275264 ^2 -> 75770269696 349736 ^2 -> 122315269696 400264 ^2 -> 160211269696 474736 ^2 -> 225374269696 525264 ^2 -> 275902269696 599736 ^2 -> 359683269696 650264 ^2 -> 422843269696 724736 ^2 -> 525242269696 775264 ^2 -> 601034269696 849736 ^2 -> 722051269696 900264 ^2 -> 810475269696 974736 ^2 -> 950110269696
jq
$ jq -n '1 | until( .*. | tostring | test("269696$"); .+1)'
25264
In words: start with n=1; if the decimal representation of n*n ends with 269696 then print n, otherwise increment n and restart.
Note for Mr Babbage.
Dear Sir. The answer to your most excellent problem is 25,264. For a demonstration, come join us in the 21st century and see for yourself: Ceci est une |."
Julia
<lang julia> function babbage(x::Integer)
i = big(0) d = floor(log10(x)) + 1 while i ^ 2 % 10 ^ d != x i += 1 end return i
end </lang>
- Output:
julia> babbage(269696) 25264
Kotlin
<lang scala>fun main(args: Array<String>) {
var number = 520L var square = 520 * 520L
while (true) {
val last6 = square.toString().takeLast(6)
if (last6 == "269696") {
println("The smallest number is $number whose square is $square")
return
}
number += 2
square = number * number
}
}</lang>
- Output:
The smallest number is 25264 whose square is 638269696
Liberty BASIC
Now Mr. Babbage -- May I call you Charlie? No. OK -- we'll first start with 'n' equal to zero, then multiply it by itself to square it. If the last six digits of the result are not 269696, we'll add one to 'n' then go back and square it again. On our modern computer it should only take a moment to find the answer... <lang lb> [start] if right$(str$(n*n),6)="269696" then
print "n = "; using("###,###", n);
print " n*n = "; using("###,###,###,###", n*n)
end if if n<100000 then n=n+1: goto [start] print "Program complete." </lang> Eureka! We found it! -- <lang sh> n = 25,264 n*n = 638,269,696 n = 99,736 n*n = 9,947,269,696 Program complete. </lang> Now my question for you, Sir, is how did you know that the square of ANY number would end in 269696?? Oh, and by the way, 99,736 is an answer too.
Limbo
<lang Limbo>implement Babbage;
include "sys.m"; sys: Sys; print: import sys; include "draw.m"; draw: Draw;
Babbage : module { init : fn(ctxt : ref Draw->Context, args : list of string); };
init (ctxt: ref Draw->Context, args: list of string) { sys = load Sys Sys->PATH; current := 0; while ((current * current) % 1000000 != 269696) current++; print("%d", current); } </lang>
Lua
<lang lua>-- get smallest number <= sqrt(269696) k = math.floor(math.sqrt(269696))
-- if root is odd -> make it even if k % 2 == 1 then
k = k - 1
end
-- cycle through numbers while not ((k * k) % 1000000 == 269696) do
k = k + 2
end
io.write(string.format("%d * %d = %d\n", k, k, k * k))</lang>
M2000 Interpreter
<lang M2000 Interpreter>Def Long k=1000000, T=269696, n n=Sqrt(269696) For n=n to k {
If n^2 mod k = T Then Exit
} Report format$("The smallest number whose square ends in {0} is {1}, Its square is {2}", T, n, n**2) </lang>
Mathematica
Solving up to his guess would show that there is indeed a smaller integer with that property. <lang Mathematica> Solve[Mod[x^2, 10^6] == 269696 && 0 <= x <= 99736, x, Integers]</lang>
- Output:
{{x->25264},{x->99736}}
Microsoft Small Basic
<lang smallbasic>' Babbage problem ' The quote (') means a comment ' The equals sign (=) means assign n = 500 ' 500 is stored in variable n*n ' 500 because 500*500=250000 less than 269696
' The nitty-gritty is in the 3 lines between "While" and "EndWhile". ' So, we start with 500, n is being incremented by 1 at each round ' while its square (n*n) (* means multiplication) does not have ' a remainder (function Math.Remainder) 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. ' (<>) means "not equal to" While Math.Remainder( n*n , 1000000 ) <> 269696
n = n + 1
EndWhile
' (TextWindow.WriteLine) displays the string to the monitor ' (+) concatenates strings or variables to be displayed TextWindow.WriteLine("The smallest positive integer whose square ends in 269696 is " + (n) + ".") TextWindow.WriteLine("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.
MAXScript
<lang maxscript>-- MAXScript : Babbage problem : N.H. posInt = 1 while posInt < 1000000 do ( if (matchPattern((posInt * posInt) as string) pattern: "*269696") then exit posInt += 1 ) Print "The smallest number whose square ends in 269696 is " + ((posInt) as string) Print "Its square is " + (((pow posInt 2) as integer) as string) </lang>
- Output:
Output to MAXScript Listener:
"The smallest number whose square ends in 269696 is 25264" "Its square is 638269696"
Modula-2
<lang modula2>MODULE BabbageProblem; FROM FormatString IMPORT FormatString; FROM RealMath IMPORT sqrt; FROM Terminal IMPORT WriteString,ReadChar;
VAR
buf : ARRAY[0..63] OF CHAR; k : INTEGER;
BEGIN
(* Find the greatest integer less than the square root *) k := TRUNC(sqrt(269696.0));
(* Odd numbers cannot be solutions, so decrement *)
IF k MOD 2 = 1 THEN
DEC(k);
END;
(* Find a number that meets the criteria *)
WHILE (k*k) MOD 1000000 # 269696 DO
INC(k,2)
END;
FormatString("%i * %i = %i", buf, k, k, k*k);
WriteString(buf);
ReadChar
END BabbageProblem.</lang>
NetRexx
<lang NetRexx>/* NetRexx */ options replace format comments java crossref symbols nobinary utf8 numeric digits 5000 -- set up numeric precision
babbageNr = babbage() -- call a function to perform the analysis and capture the result babbageSq = babbageNr ** 2 -- calculate the square of the result -- display results using a library function System.out.printf("%,10d\u00b2 == %,12d%n", [Integer(babbageNr), Integer(babbageSq)]) return
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- A function method to answer Babbage's question: -- "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.)
method babbage() public static binary
n = int 104 -- (integer arithmatic)
-- begin a processing loop to determine the value
-- starting point: 104
loop while ((n * n) // 1000000) \= 269696
-- loop continues while the remainder of n squared divided by 1,000,000 is not equal to 269,696
if n // 10 == 4 then do
-- increment n by 2 if the remainder of n divided by 10 equals 4
n = n + 2
end
if n // 10 == 6 then do
-- increment n by 8 if the remainder of n divided by 10 equals 6
n = n + 8
end
end
return n -- end the function and return the result
</lang>
- Output:
25,264² == 638,269,696
Nim
<lang Nim> var n : int = 0 while n*n mod 1_000_000 != 269_696:
inc(n)
echo n </lang>
Objeck
<lang objeck>class Babbage {
function : Main(args : String[]) ~ Nil {
cur := 0;
do {
cur++;
}
while(cur * cur % 1000000 <> 269696);
cur_sqr := cur * cur;
"The square of {$cur} is {$cur_sqr}!"->PrintLine();
}
} </lang>
- Output:
The square of 25264 is 638269696!
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>
Ol
<lang scheme> (print (let loop ((i 2))
(if (eq? (mod (* i i) 1000000) 269696)
i
(loop (+ i 2)))))
</lang>
- Output:
25264
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:
25264
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
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:
<lang perl6>say $_ if ($_² % 1000000 == 269696) for 1..99736;</lang>
- 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:
25264
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>
PILOT
<lang pilot>Remark:Lines identified as "remarks" are intended for the human reader, and will be ignored by the machine. Remark:A "compute" instruction gives a value to a variable. Remark:We begin by making the variable n equal to 2. Compute:n = 2 Remark:Lines beginning with asterisks are labels. We can instruct the machine to "jump" to them, rather than carrying on to the next instruction as it normally would.
- CheckNextNumber
Remark:In "compute" instructions, "x * y" should be read as "x times y" and "x % y" as "x modulo y". Compute:square = n * n Compute:lastSix = square % 1000000 Remark:A "jump" instruction that includes an equation or an inequality in parentheses jumps to the designated label if and only if the equation or inequality is true. Jump( lastSix = 269696 ):*FoundIt Remark:If the last six digits are not equal to 269696, add 2 to n and jump back to "CheckNextNumber". Compute:n = n + 2 Jump:*CheckNextNumber
- FoundIt
Remark:Type, i.e. print, the result. The symbol "#" means that what follows is one of our variables and the machine should type its value. Type:The smallest number whose square ends in 269696 is #n. Its square is #square. Remark:The end. End:</lang>
PowerShell
<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:
25264
Alternative method
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:
25264
Prolog
Works with Swi-Prolog version 7+ <lang prolog>:- use_module(library(clpfd)).
babbage_(B, B, Sq) :- B * B #= Sq, number_chars(Sq, R), append(_, ['2','6','9','6','9','6'], R). babbage_(B, R, Sq) :- N #= B + 1, babbage_(N, R, Sq).
babbage :- once(babbage_(1, Num, Square)), format('lowest number is ~p which squared becomes ~p~n', [Num, Square]).</lang>
- Output:
1 ?- babbage. lowest number is 25264 which squared becomes 638269696 true.
PureBasic
<lang PureBasic>EnableExplicit Macro putresult(n)
If OpenConsole("Babbage_problem")
PrintN("The smallest number whose square ends in 269696 is " + Str(n))
Input()
EndIf
EndMacro
CompilerIf #PB_Processor_x64
#MAXINT = 1 << 63 - 1
CompilerElseIf #PB_Processor_x86
#MAXINT = 1 << 31 - 1
CompilerEndIf
- GOAL = 269696
- DIV = 1000000
Define n.i, q.i = Int(Sqr(#MAXINT))
For n = 2 To q Step 2
If (n*n) % #DIV = #GOAL : putresult(n) : Break : EndIf
Next</lang>
- Output:
The smallest number whose square ends in 269696 is 25264
Python
<lang 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
- short version
>>> [x for x in range(30000) if (x*x) % 1000000 == 269696] [0] 25264 </lang>
- Output:
25264
Or, generating a non-finite stream of numbers which are the sum of 269696 and some integer multiple of one million, and also have an integer square root:
<lang python>Babbage problem
from math import (floor, sqrt) from itertools import (islice)
- squaresWithSuffix :: Int -> Gen [Int]
def squaresWithSuffix(n):
Non finite stream of squares with a given suffix.
stem = 10 ** len(str(n))
i = 0
while True:
i = until(lambda x: isPerfectSquare(n + (stem * x)))(
succ
)(i)
yield n + (stem * i)
i = succ(i)
- isPerfectSquare :: Int -> Bool
def isPerfectSquare(n):
True if n is a perfect square. r = sqrt(n) return r == floor(r)
- TEST ----------------------------------------------------
- main :: IO ()
def main():
Smallest positive integers whose squares end in the digits 269,696
print(
fTable(main.__doc__ + ':\n')(
lambda n: str(int(sqrt(n))) + '^2'
)(repr)(identity)(
take(10)(squaresWithSuffix(269696))
)
)
- GENERIC -------------------------------------------------
- identity :: a -> a
def identity(x):
The identity function. return x
- succ :: Enum a => a -> a
def succ(x):
The successor of a value.
For numeric types, (1 +).
return 1 + x if isinstance(x, int) else (
chr(1 + ord(x))
)
- take :: Int -> [a] -> [a]
- take :: Int -> String -> String
def take(n):
The prefix of xs of length n,
or xs itself if n > length xs.
return lambda xs: (
xs[0:n]
if isinstance(xs, (list, tuple))
else list(islice(xs, n))
)
- until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
The result of repeatedly applying f until p holds.
The initial seed value is x.
def go(f, x):
v = x
while not p(v):
v = f(v)
return v
return lambda f: lambda x: go(f, x)
- FORMATTING ----------------------------------------------
- fTable :: String -> (a -> String) ->
- (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
def go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
Smallest positive integers whose squares end in the digits 269,696: 25264^2 -> 638269696 99736^2 -> 9947269696 150264^2 -> 22579269696 224736^2 -> 50506269696 275264^2 -> 75770269696 349736^2 -> 122315269696 400264^2 -> 160211269696 474736^2 -> 225374269696 525264^2 -> 275902269696 599736^2 -> 359683269696 [Finished in 0.285s]
As a footnote on what Babbage might have managed with pencil and paper – applying the squaresWithSuffix(n) function to shorter suffixes (6, 96, 696 ...) enables us to explore the way in which the final digits of the integer root constrain and determine those of the perfect square. It quickly becomes apparent that Mr Babbage need only have considered roots ending in the digit sequences 264 or 736, a constraint which, had he deduced it with pencil and paper, would have allowed him to reach 25264 after testing the squares of only 24 other numbers.
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:
25264 638269696 #t #t
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:
25264
Red
<lang Red>Red [] number: 510 ;; starting number
- repeat, until the last condition in the block is true
until [
number: number + 2 ;; only even numbers can have even squares ;; The word modulo computes the non-negative remainder of the ;; first argument divided by the second argument. ;; ** => Returns a number raised to a given power (exponent) 269696 = modulo (number ** 2) 1000000
] ?? number </lang>
- Output:
number: 25264
REXX
Each of the first three REXX programs were constructed to be as simple as possible so as to be easily understood by a mathematican (in Mr. Charles Babbage's time).
For instance, in Charles Babbage's era, multiplication of j and j would be jj (implied multiplication), so it would be necessary to explain that an asterisk (*) means multiplication. Another form of multiplication in his day would be: j x j or j ∙ j. Most mathematical or algebraic texts used simple letters for values (we would now call them variables when dealing with computer programs).
Fortunately, the REXX language uses decimal numbers, so binary values don't need to be explained.
So, with that in mind, the use of (REXX) arithermetic operators were explained within a comment, as well as trying to explain some statements in the REXX language. And, further comments probably should've been added to explain what a comment is in the REXX language (and for that matter, comments should've been included for each and every REXX statement explaining what the instruction (statement) does and what the nomenclature means. Fortunately, most REXX statements are easy understood (one of REXX's design goals) and (most likely) are intuitively understood (at least on a fundamental or basic level), although a do or for loop would be confusing without a detailed explanation of what a loop is.
A computer program (in his day) would've been undstood to be list of instructions to a (human) computer to be performed, quite literally.
For instance, Mr. Babbage would know of a typewriter (machine), and, to the human computers in his day and age, they would've known (or intuited) what a type or print instruction would do, and he would recognize that the (human) computer would use a typewriter. An idle monitor on the other hand (I suspect), he would be baffled and he wouldn't probably know of it's function. When not displaying anything, it looks like an ineffective mirror, possibly having a blinking cursor, whatever that is.
(In those days of yore, a cursor was the movable transparent slide on a slide rule.)
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
-
whowhat 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 six digits of square
<lang rexx>/*REXX program finds the lowest (positive) integer whose square ends in 269,696. */
do j=2 by 2 until right(j * j, 6) == 269696 /*start J at two, increment by two. */
end /*◄── signifies the end of the DO loop.*/
/* [↑] * means multiplication. */
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 one million
<lang rexx>/*REXX program finds the lowest (positive) integer whose square ends in 269,696. */
do j=2 by 2 /*start J at two, increment by two. */
if ((j * j) // 1000000) == 269696 then leave /*is square mod one million our target?*/
end /*◄── signifies the end of the DO loop.*/
/* [↑] // is division remainder.*/
say "The smallest integer whose square ends in 269,696 is: " j</lang>
- output is identical to the 1st 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 will 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 right-most 6 decimal digits. */
/* == means exactly equal to. */
k = j+2 /*set K to J+2 value. */
if right(k * k, 6) == 269696 then leave /*examine right-most 6 decimal digits. */
end /*◄── signifies the end of the DO loop.*/
/* [↑] * means multiplication. */
say "The smallest integer whose square ends in 269,696 is: " k</lang>
- output is identical to the 1st 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 print 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 = 0;
while (current * current) % 1_000_000 != 269_696 {
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>
Scheme
<lang Scheme> (define (digits n)
(string->list (number->string n)))
(define (ends-with list tail)
;; does list end with tail?
(starts-with (reverse list)
(reverse tail)))
(define (starts-with list head)
(cond ((null? head)
#t)
((null? list)
#f)
((equal? (car list) (car head))
(starts-with (cdr list) (cdr head)))
(else
#f)))
(let loop ((i 1))
(if (ends-with (digits (* i i)) (digits 269696))
i
(loop (+ i 1))))
- 25264
</lang>
Scilab
<lang>n=2; flag=%F
while ~flag
n = n+2;
if pmodulo(n*n,1000000)==269696 then
flag=%T;
end
end disp(n);</lang>
- Output:
25264.
Seed7
<lang seed7>$ include "seed7_05.s7i";
const proc: main is func
local
var integer: current is 0;
begin
while current ** 2 rem 1000000 <> 269696 do
incr(current);
end while;
writeln("The square of " <& current <& " is " <& current ** 2);
end func;</lang>
- Output:
The square of 25264 is 638269696
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
Shen
<lang shen>(define babbage
N -> N where (= 269696 (shen.mod (* N N) 1000000))) N -> (babbage (+ N 1))
(babbage 1)</lang>
- Output:
25264
Sidef
<lang ruby>var n = 0 while (n*n % 1000000 != 269696) {
n += 2
} say n</lang>
- Output:
25264
Simula
<lang simula>BEGIN
INTEGER PROBE, SQUARE; BOOLEAN DONE;
WHILE NOT DONE DO BEGIN
PROBE := PROBE + 1;
SQUARE := PROBE * PROBE;
IF MOD(SQUARE, 1000000) = 269696 THEN BEGIN
OUTTEXT("THE SMALLEST NUMBER: ");
OUTINT(PROBE,0);
OUTIMAGE;
OUTTEXT("THE SQUARE : ");
OUTINT(SQUARE,0);
OUTIMAGE;
DONE := TRUE;
END;
END;
END</lang>
- Output:
THE SMALLEST NUMBER: 25264 THE SQUARE : 638269696
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:
25264
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
Tcl
Hope Mr Babbage can understand this one-liner... <lang Tcl>for {set i 1} {![string match *269696 [expr $i*$i]]} {incr i} {} puts "$i squared is [expr $i*$i]"</lang>
25264 squared is 638269696
TI-83 BASIC
In 1996 was manufactured the TI-83, a handheld graphing calculators with a basic language called TI-83 Basic.
The language is small and neat. For example to store 500 into a variable, it is done without twisting the meaning in mathematics of the equal sign (=).
<lang ti83b>536→N</lang>
Do not be attracted by brute force, let's do some basic maths:
As
N²=1000000·A+269696
269696=27×72×43
1000000=26×56
269696 mod 64 = 0 & 1000000 mod 64 = 0 ⇒ N² mod 64 = 0 ⇒ N mod 8 = 0
√ 269696 =519.32 => N≥520
520=8×5×13
528=16×3×11
536=8×67
N must ends by 4 or 6 ⇒ N≥536
So, Lord Babbage here is your program: <lang ti83b>536→N While remainder(N*N,1000000)≠269696
i+8→N
End Disp N</lang> And within a minute you have the answer:
- Output:
25264
Done
And you can check the square:
- Input:
25264²
- Output:
638269696
UNIX Shell
<lang bash># Program to determine the smallest positive integer whose square
- has a decimal representation ending in the digits 269,696.
- Start with the smallest positive integer of them all
let trial_value=1
- Compute the remainder when the square of the current trial value is divided
- by 1,000,000.␣
while (( trial_value * trial_value % 1000000 != 269696 )); do
# As long as this value is not yet 269,696, increment # our trial integer and try again. let trial_value=trial_value+1
done
- To get here we must have found an integer whose square meets the
- condition; display that final result
echo $trial_value</lang>
- Output:
25264
Efficient version, pen-and-pencil method
The simple method above requires more than 20000 multiplications and would run days on Babbage's Analytical Engine (AE), if he would have managed to build such a machine.
As he had found a solution with pen and paper, and would have programmed the AE correspondingly, the following solution could have been used on the AE.
It should run on any shell, including the original Bourne Shell, if the arithmetic expressions are replaced by "expr".
<lang bash>#!/bin/dash
- Babbage problem:
- What is the smallest (positive) integer whose square ends in the digits 269,696?
- He found the second to smallest number (99736 instead of 25264) using pencil and paper,
- and would not have wasted hours of computing time on his (planned) Analytical Engine (AE).
- As most human computers know, a square must end in 0, 1, 4, 5, 6 or 9.
- because the squares of 0 to 9 end in 0, 1, 4, 9, 6, 5, 6, 9, 4, 1.
- Thus, the result must have the last digits 14 or 16 in the above case.
- So the algorithm starts with the set {0} and an increment of 1,
- squaring all numbers of 0+i, and keeping only those that have
- the correct end digit.
- Then, the new i is 10*i, and a new set of two digit endings
- created from the old set of one digit endings, and so on.
- As the AE did not have arrays or the like, the sets must be punched
- on cards and read in for the next round.
- The classical (original) Bourne Shell did not have arrays,
- so you may use this script on very old machines, if 'expr' is used
- instead of arithmetic expansion.
- And so his script works with 'dash', the standard command interpreter
- for non-interactive use.
- To prove the speed, try 1234554321 instead of 269696,
- the practicall immedidate answer should be 1250061111,
- while the simple method will take hours.
- Note that this method will stop if there is no solution,
- while the simple method continues endlessly.
- filename for workfile(s)
wrk=$(basename $0 .sh).data
- set $e to desired ending. Leading zeroes are ignored.
e=${1:-269696}
- set the modulus $m to the power of 10 above $e
m=1 while test $m -le $e do m=$((m*10)) done
- $a is number to add in each round (power of 10)
a=1
- first workfile contains just the number 0
echo 0 >$wrk
- test all workfile numbers with another digit in front
while test $a -lt $m # until the increment excees the modulus do mm=$((a*10)) # modulus in this round
ee=$((e % mm)) # ending in this round
cat $wrk | # numbers from current workfile
while read x
do y=$x # first number to test is the number read
while test $y -le $((x+mm-1)) do z=$(($y * $y)) # calculate the square z=$(($z % $mm)) # ending in this round if test $z -eq $ee then echo $y # candidate for next round fi y=$(($y + $a)) # advance leftmost digit done
done >$wrk.new # create new workfile # next round a=$((a*10)) # another leftmost digit mv $wrk.new $wrk # cycle workfiles
done
- find each number in the last workfile if x*x mod m = e
- ending in $e and modulus in $m
cat $wrk | # numbers from last workfile
while read x
do y=$(($x * $x)) # check
y=$(($y % $m))
if test $y -eq $e
then echo $x # solution found
fi
done |
sort -n | # numbers in ascending order
head -n 1 # show only smallest
</lang>
- Output:
$ sh babbage.sh 25264 $ sh babbage.sh 1234554321 1250061111
UTFool
<lang UTFool> ··· http://rosettacode.org/wiki/Babbage_problem ··· ■ BabbageProblem
§ static
▶ main
• args⦂ String[]
for each number from √269696 up to √Integer.MAX_VALUE
if ("⸨number × number⸩").endsWith "269696"
System.exit number
</lang>
VAX Assembly
<lang VAX Assembly> 36 35 34 33 32 31 00000008'010E0000' 0000 1 result: .ascid "123456" ;output buffer
0000 000E 2 retlen: .word 0 ;$fao_s bytes written
4C 55 21 00000018'010E0000' 0010 3 format: .ascid "!UL" ;unsigned decimal
001B 4
0000 001B 5 .entry bab,0
55 D4 001D 6 clrl r5 ;result
001F 7 10$:
55 D6 001F 8 incl r5
56 00 55 55 7A 0021 9 emul r5,r5,#0,r6 ;mulr.rl, muld.rl, add.rl, prod.wq
51 50 56 000F4240 8F 7B 0026 10 ediv #1000000,r6,r0,r1 ;divr.rl, divd.rq, quo.wl, rem.wl
51 00041D80 8F D1 002F 11 cmpl #269696,r1
E7 12 0036 12 bneq 10$ ;not equal - try next
0038 13
0038 14 $fao_s - ;convert integer to text
0038 15 ctrstr = format, -
0038 16 outlen = retlen, -
0038 17 outbuf = result, -
0038 18 p1 = r5
B1 AF C1 AF B0 004A 19 movw retlen, result ;adjust length
AE AF 7F 004F 20 pushaq result
00000000'GF 01 FB 0052 21 calls #1, g^lib$put_output
04 0059 22 ret
005A 23 .end bab
$ run bab 25264 </lang>
VBA
<lang vb> Sub Baggage_Problem() Dim i As Long
'We can start at the square root of 269696
i = 520
'269696 is a multiple of 4, 520 too
'so we can increment i by 4
Do While ((i * i) Mod 1000000) <> 269696
i = i + 4 'Increment by 4
Loop
Debug.Print "The smallest positive integer whose square ends in the digits 269 696 is : " & i & vbCrLf & _
"Its square is : " & i * i
End Sub </lang>
The smallest positive integer whose square ends in the digits 269 696 is : 25264 Its square is : 638269696
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.
Visual Basic .NET
<lang vbnet>Module Module1
Function Right6Digits(num As Long) As Long
Dim asString = num.ToString()
If asString.Length < 6 Then
Return num
End If
Dim last6 = asString.Substring(asString.Length - 6)
Return Long.Parse(last6)
End Function
Sub Main()
Dim bnSq = 0 'the base number squared
Dim bn = 0 'the number to be squared
Do
bn = bn + 1
bnSq = bn * bn
Loop While Right6Digits(bnSq) <> 269696
Console.WriteLine("The smallest integer whose square ends in 269,696 is {0}", bn)
Console.WriteLine("The square is {0}", bnSq)
End Sub
End Module</lang>
- Output:
The smallest integer whose square ends in 269,696 is 25264 The square is 638269696
x86 Assembly
AT&T syntax
<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.
next:
- 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:
25264
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
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:
25264
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:
25264
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