# Find optimal weights which can be used to weigh all the weights in the range [1, X]

Given an integer X, the task is to find an optimal set of weights {w1, w2, w3, …, wn} such that we can weigh/determine all the weights from 1 to X using a two-sided weighing balance pan. Note that all the weights must be unique and n should be as minimum as possible.

Examples:

Input: X = 7
Output: 1 3 9

Weights Left Side Right Side
1 1 1
2 2 + 1 3
3 3 3
4 4 1 + 3
5 5 + 1 + 3 9
6 6 + 3 9
7 7 + 3 1 + 9

Input: X = 20
Output: 1 3 9 27

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

Approach:

1. One optimal approach is to use weights which are a powers of 3 i.e. {1, 3, 9, 27, 81, 243, …}
2. It can be proved through induction that using {1, 3, 9, 27, 81, …, pow(3, n)}, we can balance all the weights from 1 to (pow(3, n + 1) – 1) / 2.
3. So, find the values of n such that all the values from 1 to X can be balanced and print the results.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the optimal weights ` `void` `findWeights(``int` `X) ` `{ ` `    ``int` `sum = 0; ` ` `  `    ``// Number of weights required ` `    ``int` `power = 0; ` ` `  `    ``// Finding the value of required powers of 3 ` `    ``while` `(sum < X) { ` `        ``sum = ``pow``(3, power + 1) - 1; ` `        ``sum /= 2; ` `        ``power++; ` `    ``} ` ` `  `    ``// Optimal Weights are powers of 3 ` `    ``int` `ans = 1; ` `    ``for` `(``int` `i = 1; i <= power; i++) { ` `        ``cout << ans << ``" "``; ` `        ``ans = ans * 3; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `X = 2; ` ` `  `    ``findWeights(X); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach ` `import` `java.util.*;  ` ` `  `public` `class` `GFG  ` `{  ` `     `  `    ``// Function to find the optimal weights  ` `    ``static` `void` `findWeights(``int` `X)  ` `    ``{  ` `        ``int` `sum = ``0``;  ` ` `  `        ``// Number of weights required  ` `        ``int` `power = ``0``;  ` `        ``int` `number = ``3``; ` `         `  `        ``// Finding the value of required powers of 3  ` `        ``while` `(sum < X) ` `        ``{  ` `            ``sum = number - ``1``;  ` `            ``sum /= ``2``;  ` `            ``power++; ` `            ``number *= ``3``; ` `        ``}  ` ` `  `        ``// Optimal Weights are powers of 3  ` `        ``int` `ans = ``1``;  ` `        ``for` `(``int` `i = ``1``; i <= power; i++) ` `        ``{  ` `            ``System.out.print(ans + ``" "``);  ` `            ``ans = ans * ``3``;  ` `        ``}  ` `    ``}  ` ` `  `     `  `    ``// Driver code  ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{  ` `        ``int` `X = ``2``;  ` ` `  `        ``findWeights(X);  ` ` `  `    ``}  ` `}  ` ` `  `// This code is contributed by Sam007.  `

## Python3

 `# Python3 implementation of the approach ` ` `  `# Function to find the optimal weights ` `def` `findWeights(X): ` ` `  `    ``sum` `=` `0` ` `  `    ``# Number of weights required ` `    ``power ``=` `0` ` `  `    ``# Finding the value of required powers of 3 ` `    ``while` `(``sum` `< X): ` `        ``sum` `=` `pow``(``3``, power ``+` `1``) ``-` `1` `        ``sum` `/``/``=` `2` `        ``power ``+``=` `1` ` `  `    ``# Optimal Weights are powers of 3 ` `    ``ans ``=` `1` `    ``for` `i ``in` `range``(``1``, power ``+` `1``): ` `        ``print``(ans, end ``=` `" "``) ` `        ``ans ``=` `ans ``*` `3` ` `  `# Driver code ` `X ``=` `2` ` `  `findWeights(X) ` ` `  `# This code is contributed by Mohit Kumar `

## C#

 `// C# implementation of the approach ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `    ``// Function to find the optimal weights  ` `    ``static` `void` `findWeights(``int` `X)  ` `    ``{  ` `        ``int` `sum = 0;  ` ` `  `        ``// Number of weights required  ` `        ``int` `power = 0;  ` `        ``int` `number = 3; ` `         `  `        ``// Finding the value of required powers of 3  ` `        ``while` `(sum < X) ` `        ``{  ` `            ``sum = number - 1;  ` `            ``sum /= 2;  ` `            ``power++; ` `            ``number *= 3; ` `        ``}  ` ` `  `        ``// Optimal Weights are powers of 3  ` `        ``int` `ans = 1;  ` `        ``for` `(``int` `i = 1; i <= power; i++) ` `        ``{  ` `            ``Console.Write(ans + ``" "``);  ` `            ``ans = ans * 3;  ` `        ``}  ` `    ``}  ` ` `  `    ``// Driver code  ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `X = 2;  ` `        ``findWeights(X);  ` `    ``}  ` `}  ` ` `  `// This code is contributed by ajit. `

Output:

```1 3
```

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