# Check if item can be measured using a scale and some weights

Given some weights of masses a0, a1, a2, …, a100, a being an integer, and a weighing scale where weights can be put on both the sides of the scale. Check whether a particular item of weight W can be measured using these weights and scale.

Constraints: 2 ≤ W ≤ 109

Examples:

Input : a = 2, W = 5
Output : YES
Explanation : Weights of masses (20 = 1) and (22 = 4) can be placed on one side of the scale and the item can be placed on other side, i.e. 1 + 4 = 5

Input : a = 4, W = 11
Output : YES
Explanation : Weights of masses (40 = 1) and (41 = 4) and the item can be placed on one side and a weight of mass (42 = 16) can be placed on the other side, i.e. 1 + 4 + 11 = 16

Input : a = 4, W = 7
Output : NO

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• Firstly, it can be carefully observed that for a = 2 or a = 3, the answer always exists.
• Maintain an array containing weights of masses and only include that weights that are less than 109
• Now, the problem can be solved recursively by either including current weight to the side containing
item or including current weight to the opposite side containing item or by not using that weight at all.
• Below is the implementation of above approach .

## C++

 `// CPP Program to check whether an item ` `// can be measured using some weight and ` `// a weighing scale. ` `#include ` `using` `namespace` `std; ` ` `  `// Variable to denote that answer has ` `// been found ` `int` `found = 0; ` ` `  `void` `solve(``int` `idx, ``int` `itemWt, ``int` `wts[], ` `           ``int` `N) ` `{ ` `    ``if` `(found) ` `        ``return``; ` ` `  `    ``// Item has been measured ` `    ``if` `(itemWt == 0) { ` `        ``found = 1; ` `        ``return``; ` `    ``} ` ` `  `    ``// If the index of current weight ` `    ``// is greater than totalWts ` `    ``if` `(idx > N) ` `        ``return``; ` ` `  `    ``// Current weight is not included ` `    ``// on either side ` `    ``solve(idx + 1, itemWt, wts, N); ` ` `  `    ``// Current weight is included on the ` `    ``// side containing item ` `    ``solve(idx + 1, itemWt + wts[idx], wts, ` `          ``N); ` ` `  `    ``// Current weight is included on the ` `    ``// side opposite to the side ` `    ``// containing item ` `    ``solve(idx + 1, itemWt - wts[idx], wts, ` `          ``N); ` `} ` ` `  `// This function computes the required array ` `// of weights using a ` `bool` `checkItem(``int` `a, ``int` `W) ` `{ ` `    ``// If the a is 2 or 3, answer always ` `    ``// exists ` `    ``if` `(a == 2 || a == 3) ` `        ``return` `1; ` ` `  `    ``int` `wts; ``// weights array ` `    ``int` `totalWts = 0; ``// feasible weights ` `    ``wts = 1; ` `    ``for` `(``int` `i = 1;; i++) { ` `        ``wts[i] = wts[i - 1] * a; ` `        ``totalWts++; ` ` `  `        ``// if the current weight ` `        ``// becomes greater than 1e9 ` `        ``// break from the loop ` `        ``if` `(wts[i] > 1e9) ` `            ``break``; ` `    ``} ` `    ``solve(0, W, wts, totalWts); ` `    ``if` `(found) ` `        ``return` `1; ` ` `  `    ``// Item can't be measured ` `    ``return` `0; ` `} ` ` `  `// Driver Code to test above functions ` `int` `main() ` `{ ` `    ``int` `a = 2, W = 5; ` `    ``if` `(checkItem(a, W)) ` `        ``cout << ``"YES"` `<< endl; ` `    ``else` `        ``cout << ``"NO"` `<< endl; ` ` `  `    ``a = 4, W = 11, found = 0; ` `    ``if` `(checkItem(a, W)) ` `        ``cout << ``"YES"` `<< endl; ` `    ``else` `        ``cout << ``"NO"` `<< endl; ` ` `  `    ``a = 4, W = 7, found = 0; ` `    ``if` `(checkItem(a, W)) ` `        ``cout << ``"YES"` `<< endl; ` `    ``else` `        ``cout << ``"NO"` `<< endl; ` `    ``return` `0; ` `} `

## Java

 `// Java Program to check whether an item ` `// can be measured using some weight and ` `// a weighing scale. ` ` `  `class` `GFG { ` ` `  `// Variable to denote that answer has ` `// been found ` `    ``static` `int` `found = ``0``; ` ` `  `    ``static` `void` `solve(``int` `idx, ``int` `itemWt, ``int` `wts[], ``int` `N) { ` `        ``if` `(found == ``1``) { ` `            ``return``; ` `        ``} ` ` `  `        ``// Item has been measured ` `        ``if` `(itemWt == ``0``) { ` `            ``found = ``1``; ` `            ``return``; ` `        ``} ` ` `  `        ``// If the index of current weight ` `        ``// is greater than totalWts ` `        ``if` `(idx > N) { ` `            ``return``; ` `        ``} ` ` `  `        ``// Current weight is not included ` `        ``// on either side ` `        ``solve(idx + ``1``, itemWt, wts, N); ` ` `  `        ``// Current weight is included on the ` `        ``// side containing item ` `        ``solve(idx + ``1``, itemWt + wts[idx], wts, ` `                ``N); ` ` `  `        ``// Current weight is included on the ` `        ``// side opposite to the side ` `        ``// containing item ` `        ``solve(idx + ``1``, itemWt - wts[idx], wts, ` `                ``N); ` `    ``} ` ` `  `// This function computes the required array ` `// of weights using a ` `    ``static` `boolean` `checkItem(``int` `a, ``int` `W) { ` `        ``// If the a is 2 or 3, answer always ` `        ``// exists ` `        ``if` `(a == ``2` `|| a == ``3``) { ` `            ``return` `true``; ` `        ``} ` ` `  `        ``int` `wts[] = ``new` `int``[``100``]; ``// weights array ` `        ``int` `totalWts = ``0``; ``// feasible weights ` `        ``wts[``0``] = ``1``; ` `        ``for` `(``int` `i = ``1``;; i++) { ` `            ``wts[i] = wts[i - ``1``] * a; ` `            ``totalWts++; ` ` `  `            ``// if the current weight ` `            ``// becomes greater than 1e9 ` `            ``// break from the loop ` `            ``if` `(wts[i] > 1e9) { ` `                ``break``; ` `            ``} ` `        ``} ` `        ``solve(``0``, W, wts, totalWts); ` `        ``if` `(found == ``1``) { ` `            ``return` `true``; ` `        ``} ` ` `  `        ``// Item can't be measured ` `        ``return` `false``; ` `    ``} ` ` `  `// Driver Code to test above functions ` `    ``public` `static` `void` `main(String[] args) { ` ` `  `        ``int` `a = ``2``, W = ``5``; ` `        ``if` `(checkItem(a, W)) { ` `            ``System.out.println(``"YES"``); ` `        ``} ``else` `{ ` `            ``System.out.println(``"NO"``); ` `        ``} ` ` `  `        ``a = ``4``; W = ``11``;found = ``0``; ` `        ``if` `(checkItem(a, W)) { ` `            ``System.out.println(``"YES"``); ` `        ``} ``else` `{ ` `            ``System.out.println(``"NO"``); ` `        ``} ` ` `  `        ``a = ``4``; W = ``7``; found = ``0``; ` `        ``if` `(checkItem(a, W)) { ` `            ``System.out.println(``"YES"``); ` `        ``} ``else` `{ ` `            ``System.out.println(``"NO"``); ` `        ``} ` `    ``} ` `} ` ` `  `//this code contributed by Rajput-Ji `

## C#

 `     `  `// C# Program to check whether an item ` `// can be measured using some weight and ` `// a weighing scale. ` `using` `System;  ` `public` `class` `GFG { ` `  `  `// Variable to denote that answer has ` `// been found ` `    ``static` `int` `found = 0; ` `  `  `    ``static` `void` `solve(``int` `idx, ``int` `itemWt, ``int` `[]wts, ``int` `N) { ` `        ``if` `(found == 1) { ` `            ``return``; ` `        ``} ` `  `  `        ``// Item has been measured ` `        ``if` `(itemWt == 0) { ` `            ``found = 1; ` `            ``return``; ` `        ``} ` `  `  `        ``// If the index of current weight ` `        ``// is greater than totalWts ` `        ``if` `(idx > N) { ` `            ``return``; ` `        ``} ` `  `  `        ``// Current weight is not included ` `        ``// on either side ` `        ``solve(idx + 1, itemWt, wts, N); ` `  `  `        ``// Current weight is included on the ` `        ``// side containing item ` `        ``solve(idx + 1, itemWt + wts[idx], wts, ` `                ``N); ` `  `  `        ``// Current weight is included on the ` `        ``// side opposite to the side ` `        ``// containing item ` `        ``solve(idx + 1, itemWt - wts[idx], wts, ` `                ``N); ` `    ``} ` `  `  `// This function computes the required array ` `// of weights using a ` `    ``static` `bool` `checkItem(``int` `a, ``int` `W) { ` `        ``// If the a is 2 or 3, answer always ` `        ``// exists ` `        ``if` `(a == 2 || a == 3) { ` `            ``return` `true``; ` `        ``} ` `  `  `        ``int` `[]wts = ``new` `int``; ``// weights array ` `        ``int` `totalWts = 0; ``// feasible weights ` `        ``wts = 1; ` `        ``for` `(``int` `i = 1;; i++) { ` `            ``wts[i] = wts[i - 1] * a; ` `            ``totalWts++; ` `  `  `            ``// if the current weight ` `            ``// becomes greater than 1e9 ` `            ``// break from the loop ` `            ``if` `(wts[i] > 1e9) { ` `                ``break``; ` `            ``} ` `        ``} ` `        ``solve(0, W, wts, totalWts); ` `        ``if` `(found == 1) { ` `            ``return` `true``; ` `        ``} ` `  `  `        ``// Item can't be measured ` `        ``return` `false``; ` `    ``} ` `  `  `// Driver Code to test above functions ` `    ``public` `static` `void` `Main() { ` `  `  `        ``int` `a = 2, W = 5; ` `        ``if` `(checkItem(a, W)) { ` `            ``Console.WriteLine(``"YES"``); ` `        ``} ``else` `{ ` `            ``Console.WriteLine(``"NO"``); ` `        ``} ` `  `  `        ``a = 4; W = 11;found = 0; ` `        ``if` `(checkItem(a, W)) { ` `            ``Console.WriteLine(``"YES"``); ` `        ``} ``else` `{ ` `            ``Console.WriteLine(``"NO"``); ` `        ``} ` `  `  `        ``a = 4; W = 7; found = 0; ` `        ``if` `(checkItem(a, W)) { ` `            ``Console.WriteLine(``"YES"``); ` `        ``} ``else` `{ ` `            ``Console.WriteLine(``"NO"``); ` `        ``} ` `    ``} ` `} ` `  `  `//this code contributed by Rajput-Ji `

Output:

```YES
YES
NO
```

Time Complexity: O(3N), where N cannot be more than 20 since 420 is greater than 109

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Improved By : Rajput-Ji