Given a number N and a digit D, we have to form an expression or equation that contains only D and that expression evaluates to N. Allowed operators in an expression are +, -, *, and / . Find the minimum length expression that satisfies the condition above and D can only appear in the expression at most 10(limit) times. Hence limit the values of N (Although the value of limit depends upon how far you want to go. But a large value of limit can take a longer time for the below approach). Remember, there can be more than one minimum expression of D that evaluates to N but the length of that expression will be minimum. Examples:
Input : N = 7, D = 3 Output : 3/3+ 3 + 3 Explanation : 3/3 = 1, and 1+3+3 = 7 This is the minimum expression. Input : N = 7, D = 4 Output : (4+4+4)/4 + 4 Explanation : (4+4+4) = 12, and 12/4 = 3 and 3+4 = 7 Also this is the minimum expression. Although you may find another expression but that expression can have only five 4's Input : N = 200, D = 9 Output : Expression not found! Explanation : Not possible within 10 digits.
The approach we use is Backtracking. We start with the given Digit D and start multiplying, adding, subtracting, and dividing if possible. This process is done until we find the total as N or we reach the end and we backtrack to start another path. To find the minimum expression, we find the minimum level of the recursive tree. And then apply our backtracking algorithm. Let’s say N = 7, D = 3
The above approach is exponential. At every level, we recurse 4 more ways (at-most). So, we can say the time complexity of the method is
// CPP Program to generate minimum expression containing // only given digit D that evaluates to number N. #include <climits> #include <iostream> #include <map> #include <sstream> #include <stack> // limit of Digits in the expression #define LIMIT 10 using namespace std;
// map that store if a number is seen or not map< int , int > seen;
// stack for storing operators stack< char > operators;
int minimum = LIMIT;
// function to find minimum levels in the recursive tree void minLevel( int total, int N, int D, int level) {
// if total is equal to given N
if (total == N) {
// store if level is minimum
minimum = min(minimum, level);
return ;
}
// if the last level is reached
if (level == minimum)
return ;
// if total can be divided by D.
// recurse by dividing the total by D
if (total % D == 0)
minLevel(total / D, N, D, level + 1);
// recurse for total + D
minLevel(total + D, N, D, level + 1);
// if total - D is greater than 0
if (total - D > 0)
// recurse for total - D
minLevel(total - D, N, D, level + 1);
// recurse for total multiply D
minLevel(total * D, N, D, level + 1);
} // function to generate the minimum expression bool generate( int total, int N, int D, int level) {
// if total is equal to N
if (total == N)
return true ;
// if the last level is reached
if (level == minimum)
return false ;
// if total is seen at level greater than current level
// or if we haven't seen total before. Mark the total
// as seen at current level
if (seen.find(total) == seen.end() ||
seen.find(total)->second >= level) {
seen[total] = level;
int divide = INT_MAX;
// if total is divisible by D
if (total % D == 0) {
divide = total / D;
// if divide isn't seen before
// mark it as seen
if (seen.find(divide) == seen.end())
seen[divide] = level + 1;
}
int addition = total + D;
// if addition isn't seen before
// mark it as seen
if (seen.find(addition) == seen.end())
seen[addition] = level + 1;
int subtraction = INT_MAX;
// if D can be subtracted from total
if (total - D > 0) {
subtraction = total - D;
// if subtraction isn't seen before
// mark it as seen
if (seen.find(subtraction) == seen.end())
seen[subtraction] = level + 1;
}
int multiply = total * D;
// if multiply isn't seen before
// mark it as seen
if (seen.find(multiply) == seen.end())
seen[multiply] = level + 1;
// recurse by dividing the total if possible
if (divide != INT_MAX)
if (generate(divide, N, D, level + 1)) {
// store the operator.
operators.push( '/' );
return true ;
}
// recurse by adding D to total
if (generate(addition, N, D, level + 1)) {
// store the operator.
operators.push( '+' );
return true ;
}
// recurse by subtracting D from total
if (subtraction != INT_MAX)
if (generate(subtraction, N, D, level + 1)) {
// store the operator.
operators.push( '-' );
return true ;
}
// recurse by multiplying D by total
if (generate(multiply, N, D, level + 1)) {
// store the operator.
operators.push( '*' );
return true ;
}
}
// expression is not found yet
return false ;
} // function to print the expression void printExpression( int N, int D) {
// find minimum level
minLevel(D, N, D, 1);
// generate expression if possible
if (generate(D, N, D, 1)) {
// stringstream for converting to D to string
ostringstream num;
num << D;
string expression;
// if stack is not empty
if (!operators.empty()) {
// concatenate D and operator at top of stack
expression = num.str() + operators.top();
operators.pop();
}
// until stack is empty
// concatenate the operator with parenthesis for precedence
while (!operators.empty()) {
if (operators.top() == '/' || operators.top() == '*' )
expression = "(" + expression + num.str() + ")" + operators.top();
else
expression = expression + num.str() + operators.top();
operators.pop();
}
expression = expression + num.str();
// print the expression
cout << "Expression: " << expression << endl;
}
// not possible within 10 digits.
else
cout << "Expression not found!" << endl;
} // Driver's Code int main() {
int N = 7, D = 4;
// print the Expression if possible
printExpression(N, D);
// print expression for N =100, D =7
minimum = LIMIT;
printExpression(100, 7);
// print expression for N =200, D =9
minimum = LIMIT;
printExpression(200, 9);
return 0;
} |
// Java Program to generate minimum expression containing // only given digit D that evaluates to number N. import java.util.HashMap;
import java.util.Stack;
public class SingleDigit {
// limit of Digits in the expression
static int LIMIT = 10 ;
// map that store if a number is seen or not
static HashMap<Integer, Integer> seen
= new HashMap<Integer, Integer>();
// stack for storing operators
static Stack<Character> operators
= new Stack<Character>();
static int minimum = LIMIT;
// function to find minimum levels in the recursive tree
static void minLevel( int total, int N, int D, int level)
{
// if total is equal to given N
if (total == N) {
// store if level is minimum
minimum = Math.min(minimum, level);
return ;
}
// if the last level is reached
if (level == minimum)
return ;
// if total can be divided by D.
// recurse by dividing the total by D
if (total % D == 0 )
minLevel(total / D, N, D, level + 1 );
// recurse for total + D
minLevel(total + D, N, D, level + 1 );
// if total - D is greater than 0
if (total - D > 0 )
// recurse for total - D
minLevel(total - D, N, D, level + 1 );
// recurse for total multiply D
minLevel(total * D, N, D, level + 1 );
}
// function to generate the minimum expression
static boolean generate( int total, int N, int D,
int level)
{
// if total is equal to N
if (total == N)
return true ;
// if the last level is reached
if (level == minimum)
return false ;
// if total is seen at level greater than current
// level or if we haven't seen total before. Mark
// the total as seen at current level
if ((seen.containsKey(total)
&& seen.get(total) >= level)
|| !seen.containsKey(total)) {
seen.put(total, level);
int divide = Integer.MAX_VALUE;
// if total is divisible by D
if (total % D == 0 ) {
divide = total / D;
// if divide isn't seen before
// mark it as seen
if (seen.containsKey(divide))
seen.put(divide, level + 1 );
}
int addition = total + D;
// if addition isn't seen before
// mark it as seen
if (seen.containsKey(addition))
seen.put(addition, level + 1 );
int subtraction = Integer.MAX_VALUE;
// if D can be subtracted from total
if (total - D > 0 ) {
subtraction = total - D;
// if subtraction isn't seen before
// mark it as seen
if (seen.containsKey(subtraction))
seen.put(subtraction, level + 1 );
}
int multiply = total * D;
// if multiply isn't seen before
// mark it as seen
if (seen.containsKey(multiply))
seen.put(multiply, level + 1 );
;
// recurse by dividing the total if possible
if (divide != Integer.MAX_VALUE)
if (generate(divide, N, D, level + 1 )) {
// store the operator.
operators.push( '/' );
return true ;
}
// recurse by adding D to total
if (generate(addition, N, D, level + 1 )) {
// store the operator.
operators.push( '+' );
return true ;
}
// recurse by subtracting D from total
if (subtraction != Integer.MAX_VALUE)
if (generate(subtraction, N, D,
level + 1 )) {
// store the operator.
operators.push( '-' );
return true ;
}
// recurse by multiplying D by total
if (generate(multiply, N, D, level + 1 )) {
// store the operator.
operators.push( '*' );
return true ;
}
}
// expression is not found yet
return false ;
}
// function to print the expression
static void printExpression( int N, int D)
{
// find minimum level
minLevel(D, N, D, 1 );
// generate expression if possible
if (generate(D, N, D, 1 )) {
String expression = "" ;
// if stack is not empty
if (!operators.empty()) {
// concatenate D and operator at top of
// stack
expression = Integer.toString(D)
+ operators.peek();
operators.pop();
}
// until stack is empty
// concatenate the operator with parenthesis for
// precedence
while (!operators.empty()) {
if (operators.peek() == '/'
|| operators.peek() == '*' )
expression = "(" + expression
+ Integer.toString(D) + ")"
+ operators.peek();
else
expression = expression
+ Integer.toString(D)
+ operators.peek();
operators.pop();
}
expression = expression + Integer.toString(D);
// print the expression
System.out.println( "Expression: " + expression);
}
// not possible within 10 digits.
else
System.out.println( "Expression not found!" );
}
// Driver's Code
public static void main(String[] args)
{
int N = 7 , D = 4 ;
// print the Expression if possible
printExpression(N, D);
// print expression for N =100, D =7
minimum = LIMIT;
printExpression( 100 , 7 );
// print expression for N =200, D =9
minimum = LIMIT;
printExpression( 200 , 9 );
}
} // This code is contributed by Lovely Jain |
# Python3 program to generate minimum expression containing # only the given digit D that evaluates to number N. LIMIT = 10
# to restrict the maximum number of times # the number D can be used to obtain N minimum = LIMIT
# to store the value of intermediate D # and the D of operands used to get that intermediate D, ie # seen[intermediateNumber] = numberOfOperandsUsed seen = {}
# stack to store the operators used to print the expression operators = []
# function to obtain minimum D of operands in recursive tree def minLevel(total, N, D, level):
global minimum
# if total is equal to given N
if total = = N:
# store if D of operands is minimum
minimum = min (minimum, level)
return
# if the last level (limit) is reached
if level = = minimum:
return
# if total can be divided by D, recurse
# by dividing the total by D
if total % D = = 0 :
minLevel(total / D, N, D, level + 1 )
# recurse for total + D
minLevel(total + D, N, D, level + 1 )
# if total - D is greater than 0, recurse for total - D
if total - D > 0 :
minLevel(total - D, N, D, level + 1 )
# recurse for total * D
minLevel(total * D, N, D, level + 1 )
# function to generate the minimum expression def generate(total, N, D, level):
# if total is equal to N
if total = = N:
return True
# if the last level is reached
if level = = minimum:
return False
# if the total is not already seen or if
# the total is seen with more level
# then mark total as seen with current level
if seen.get(total) is None or seen.get(total) > = level:
seen[total] = level
divide = - 1
# if total is divisible by D
if total % D = = 0 :
divide = total / D
# if the number (divide) is not seen, mark as seen
if seen.get(divide) is None :
seen[divide] = level + 1
addition = total + D
# if the number (addition) is not seen, mark as seen
if seen.get(addition) is None :
seen[addition] = level + 1
subtraction = - 1
# if D can be subtracted from total
if total - D > 0 :
subtraction = total - D
# if the number (subtraction) is not seen, mark as seen
if seen.get(subtraction) is None :
seen[subtraction] = level + 1
multiplication = total * D
# if the number (multiplication) is not seen, mark as seen
if seen.get(multiplication) is None :
seen[multiplication] = level + 1
# recurse by dividing the total if possible and store the operator
if divide ! = - 1 and generate(divide, N, D, level + 1 ):
operators.append( '/' )
return True
# recurse by adding D to total and store the operator
if generate(addition, N, D, level + 1 ):
operators.append( '+' )
return True
# recurse by subtracting D from total
# if possible and store the operator
if subtraction ! = - 1 and generate(subtraction, N, D, level + 1 ):
operators.append( '-' )
return True
# recurse by multiplying D by total and store the operator
if generate(multiplication, N, D, level + 1 ):
operators.append( '*' )
return True
# expression is not found yet
return False
# function to print the expression def printExpression(N, D):
# find the minimum number of operands
minLevel(D, N, D, 1 )
# generate expression if possible
if generate(D, N, D, 1 ):
expression = ''
# if stack is not empty, concatenate D and
# operator popped from stack
if len (operators) > 0 :
expression = str (D) + operators.pop()
# until stack is empty, concatenate the
# operator popped with parenthesis for precedence
while len (operators) > 0 :
popped = operators.pop()
if popped = = '/' or popped = = '*' :
expression = '(' + expression + str (D) + ')' + popped
else :
expression = expression + str (D) + popped
expression = expression + str (D)
print ( "Expression: " + expression)
# not possible within 10 digits.
else :
print ( "Expression not found!" )
# Driver's code if __name__ = = '__main__' :
# N = 7, D = 4
minimum = LIMIT
printExpression( 7 , 4 )
minimum = LIMIT
printExpression( 100 , 7 )
minimum = LIMIT
printExpression( 200 , 9 )
# This code is contributed by thecodingpanda |
using System;
using System.Collections.Generic;
// C# program to generate minimum expression containing // only the given digit D that evaluates to number N. public class Program
{ // limit of Digits in the expression
const int LIMIT = 10;
static int minimum = LIMIT;
// map that store if a number is seen or not
static Dictionary< int , int > seen = new Dictionary< int , int >();
// stack for storing operators
static List< char > operators = new List< char >();
// Driver's Code
public static void Main()
{
minimum = LIMIT;
// print the Expression if possible
printExpression(7, 4);
minimum = LIMIT;
// print expression for N =100, D =7
printExpression(100, 7);
minimum = LIMIT;
// print expression for N =200, D =9
printExpression(200, 9);
}
// function to find minimum levels in the recursive tree
static void minLevel( int total, int N, int D, int level)
{
// if total is equal to given N
if (total == N)
{
// store if level is minimum
minimum = Math.Min(minimum, level);
return ;
}
// if the last level is reached
if (level == minimum)
{
return ;
}
// if total can be divided by D.
// recurse by dividing the total by D
if (total % D == 0)
{
minLevel(total / D, N, D, level + 1);
}
// recurse for total + D
minLevel(total + D, N, D, level + 1);
// if total - D is greater than 0
if (total - D > 0)
{
// recurse for total - D
minLevel(total - D, N, D, level + 1);
}
// recurse for total multiply D
minLevel(total * D, N, D, level + 1);
}
// function to generate the minimum expression
static bool generate( int total, int N, int D, int level)
{
// if total is equal to N
if (total == N)
{
return true ;
}
// if the last level is reached
if (level == minimum)
{
return false ;
}
// if total is seen at level greater than current
// level or if we haven't seen total before. Mark
// the total as seen at current level
if (!seen.ContainsKey(total) || seen[total] >= level)
{
seen[total] = level;
int divide = -1;
// if total is divisible by D
if (total % D == 0)
{
divide = total / D;
// if divide isn't seen before
// mark it as seen
if (!seen.ContainsKey(divide))
{
seen[divide] = level + 1;
}
}
int addition = total + D;
// if addition isn't seen before
// mark it as seen
if (!seen.ContainsKey(addition))
{
seen[addition] = level + 1;
}
int subtraction = -1;
// if D can be subtracted from total
if (total - D > 0)
{
subtraction = total - D;
// if subtraction isn't seen before
// mark it as seen
if (!seen.ContainsKey(subtraction))
{
seen[subtraction] = level + 1;
}
}
int multiplication = total * D;
// if multiply isn't seen before
// mark it as seen
if (!seen.ContainsKey(multiplication))
{
seen[multiplication] = level + 1;
}
// recurse by dividing the total if possible
if (divide != -1 && generate(divide, N, D, level + 1))
{
// store the operator.
operators.Add( '/' );
return true ;
}
// recurse by adding D to total
if (generate(addition, N, D, level + 1))
{
operators.Add( '+' );
return true ;
}
// recurse by subtracting D from total
if (subtraction != -1 && generate(subtraction, N, D, level + 1))
{
operators.Add( '-' );
return true ;
}
// recurse by multiplying D by total
if (generate(multiplication, N, D, level + 1))
{
operators.Add( '*' );
return true ;
}
}
// expression is not found yet
return false ;
}
// function to print the expression
static void printExpression( int N, int D)
{
// find minimum level
minLevel(D, N, D, 1);
// generate expression if possible
if (generate(D, N, D, 1))
{
string expression = "" ;
// if stack is not empty
if (operators.Count > 0)
{
// concatenate D and operator at top of
// stack
expression = D + operators[operators.Count - 1].ToString();
operators.RemoveAt(operators.Count - 1);
}
// until stack is empty
// concatenate the operator with parenthesis for
// precedence
while (operators.Count > 0)
{
char popped = operators[operators.Count - 1];
operators.RemoveAt(operators.Count - 1);
if (popped == '/' || popped == '*' ) {
expression = '(' + expression + D.ToString() + ')' + popped;
} else {
expression = expression + D.ToString() + popped;
}
}
expression = expression + D.ToString();
// print the expression
Console.WriteLine( "Expression: " + expression);
}
else {
// not possible within 10 digits.
Console.WriteLine( "Expression not found!" );
}
}
} |
// Javascript Program to generate minimum expression containing // only given digit D that evaluates to number N. // limit of Digits in the expression const LIMIT = 10; // map that store if a number is seen or not const seen = new Map();
// stack for storing operators const operators = []; let minimum = LIMIT; // function to find minimum levels in the recursive tree function minLevel(total, N, D, level) {
// if total is equal to given N
if (total === N) {
// store if level is minimum
minimum = Math.min(minimum, level);
return ;
}
// if the last level is reached
if (level === minimum) return ;
// if total can be divided by D.
// recurse by dividing the total by D
if (total % D === 0) minLevel(total / D, N, D, level + 1);
minLevel(total + D, N, D, level + 1);
// if total - D is greater than 0
if (total - D > 0) minLevel(total - D, N, D, level + 1);
minLevel(total * D, N, D, level + 1);
} // function to generate the minimum expression function generate(total, N, D, level) {
// if total is equal to N
if (total === N) return true ;
// if the last level is reached
if (level === minimum) return false ;
// if total is seen at level greater than current
// level or if we haven't seen total before. Mark
// the total as seen at current level
if (
(seen.has(total) && seen.get(total) >= level) ||
!seen.has(total)
) {
seen.set(total, level);
let divide = Infinity;
// if total is divisible by D
if (total % D === 0) {
divide = total / D;
// if divide isn't seen before
// mark it as seen
if (seen.has(divide)) seen.set(divide, level + 1);
}
const addition = total + D;
// if addition isn't seen before
// mark it as seen
if (seen.has(addition)) seen.set(addition, level + 1);
let subtraction = Infinity;
// if D can be subtracted from total
if (total - D > 0) {
subtraction = total - D;
// if subtraction isn't seen before
// mark it as seen
if (seen.has(subtraction)) seen.set(subtraction, level + 1);
}
const multiply = total * D;
if (seen.has(multiply)) seen.set(multiply, level + 1);
// recurse by dividing the total if possible
if (divide !== Infinity)
if (generate(divide, N, D, level + 1)) {
// store the operator.
operators.push( '/' );
return true ;
}
// recurse by adding D to total
if (generate(addition, N, D, level + 1)) {
operators.push( '+' );
return true ;
}
// recurse by subtracting D from total
if (subtraction !== Infinity)
if (generate(subtraction, N, D, level + 1)) {
operators.push( '-' );
return true ;
}
// recurse by multiplying D by total
if (generate(multiply, N, D, level + 1)) {
// store the operator.
operators.push( '*' );
return true ;
}
}
// expression is not found yet
return false ;
} // function to print the expression function printExpression(N, D) {
// find minimum level
minLevel(D, N, D, 1);
// generate expression if possible
if (generate(D, N, D, 1)) {
let expression = "" ;
// if stack is not empty
if (operators.length > 0) {
// concatenate D and operator at top of
// stack
expression = D.toString() + operators.pop();
}
// until stack is empty
// concatenate the operator with parenthesis for
// precedence
while (operators.length > 0) {
if (operators[operators.length - 1] === '/' || operators[operators.length - 1] === '*' ) {
expression = "(" + expression + D.toString() + ")" + operators.pop();
} else {
expression = expression + D.toString() + operators.pop();
}
}
expression = expression + D.toString();
// print the expression
console.log( "Expression: " + expression);
}
// not possible within 10 digits.
else {
console.log( "Expression not found!" );
}
} // Driver's Code printExpression(7, 4); minimum = LIMIT; printExpression(100, 7); minimum = LIMIT; printExpression(200, 9); // This code id contributed by Shivhack999 |
Expression: (4+4+4)/4+4 Expression: (((7+7)*7)*7+7+7)/7 Expression not found!