# Smallest expression to represent a number using single digit

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 where n is the number of levels in the recursive tree (or we can say the number of times we want D to appear at-most in the expression which in our case is 10).

**Note**: We use the above approach two times. First to find the minimum level and then to find the expression that is possible at that level. So, we have two passes in this approach. We can get the expression in one go, but you’ll need to scratch your head for that.

## C++

`// 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;` `}` |

## Python3

`# 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` |

**Output:**

Expression: (4+4+4)/4+4 Expression: (((7+7)*7)*7+7+7)/7 Expression not found!

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