# Shuffle a pack of cards and answer the query

Given a pack of 2^N cards (0 … 2^N – 1), shuffle it in N steps. At step k (0 < k < N) we divide the deck into 2k equal-sized decks. Each one of those decks is reordered by having all the cards that lie on even positions first, followed by all cards that lie on odd positions (the order is preserved in each one of the two subsequences). Now, we are given a key (index). We have to answer the card on that position (0-based indexing). Examples:

Input : N = 3 (Size = 2^N), Key = 3 Output : 6 Explanation : Pack : 0 1 2 3 4 5 6 7 Shuffle 1 : 0 2 4 6|1 3 5 7 Shuffle 2 : 0 4|2 6|1 5|3 7 Card at index 3 : 6

**Method 1 :** We can simply simulate the whole process and find the exact order of the cards after all the N shuffles are done.

Time Complexity : O(N * 2^N)

**Method 2 :**

Let us try to find the binary representation of Key and the final answer and try to spot some observations based on it.

Let N = 3

Below is the table :

Key ANS

000 000

001 100

010 010

011 110

100 001

101 101

110 011

111 111

It is clearly visible that the answer is the reverse of binary representation of Key.

## C++

`// C++ program to find the card at given index ` `// after N shuffles ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// function to find card at given index ` `void` `shuffle(` `int` `N, ` `int` `key) ` `{ ` ` ` ` ` `// Answer will be reversal of N bits from MSB ` ` ` `unsigned ` `int` `NO_OF_BITS = N; ` ` ` `unsigned ` `int` `reverse_num = 0, temp; ` ` ` ` ` `// Calculating the reverse binary representation ` ` ` `for` `(` `int` `i = 0; i < NO_OF_BITS; i++) { ` ` ` `temp = (key & (1 << i)); ` ` ` `if` `(temp) ` ` ` `reverse_num |= (1 << ((NO_OF_BITS - 1) - i)); ` ` ` `} ` ` ` ` ` `// Printing the result ` ` ` `cout << reverse_num; ` `} ` ` ` `// driver code ` `int` `main() ` `{ ` ` ` `// No. of Shuffle Steps ` ` ` `int` `N = 3; ` ` ` ` ` `// Key position ` ` ` `unsigned ` `int` `key = 3; ` ` ` ` ` `shuffle(N, key); ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find the card at given index ` `// after N shuffles ` `class` `GFG { ` ` ` ` ` `// function to find card at given index ` ` ` `static` `void` `shuffle(` `int` `N, ` `int` `key) ` ` ` `{ ` ` ` ` ` `// Answer will be reversal of N bits from MSB ` ` ` `int` `NO_OF_BITS = N; ` ` ` `int` `reverse_num = ` `0` `, temp; ` ` ` ` ` `// Calculating the reverse binary representation ` ` ` `for` `(` `int` `i = ` `0` `; i < NO_OF_BITS; i++) { ` ` ` `temp = (key & (` `1` `<< i)); ` ` ` `if` `(temp>` `0` `) ` ` ` `reverse_num |= (` `1` `<< ((NO_OF_BITS - ` `1` `) - i)); ` ` ` `} ` ` ` ` ` `// Printing the result ` ` ` `System.out.print(reverse_num); ` ` ` `} ` ` ` ` ` `//Driver code ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{ ` ` ` ` ` `// No. of Shuffle Steps ` ` ` `int` `N = ` `3` `; ` ` ` ` ` `// Key position ` ` ` `int` `key = ` `3` `; ` ` ` ` ` `shuffle(N, key); ` ` ` `} ` `} ` ` ` `// This code is contributed by Anant Agarwal. ` |

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## Python3

`# Python3 program to find the card ` `# at given index after N shuffles ` ` ` `# Function to find card at given index ` `def` `shuffle(N, key): ` ` ` ` ` `# Answer will be reversal ` ` ` `# of N bits from MSB ` ` ` `NO_OF_BITS ` `=` `N ` ` ` `reverse_num ` `=` `0` ` ` ` ` `# Calculating the reverse binary representation ` ` ` `for` `i ` `in` `range` `(NO_OF_BITS): ` ` ` `temp ` `=` `(key & (` `1` `<< i)) ` ` ` `if` `(temp): ` ` ` `reverse_num |` `=` `(` `1` `<< ((NO_OF_BITS ` `-` `1` `) ` `-` `i)) ` ` ` ` ` `# Printing the result ` ` ` `print` `(reverse_num) ` ` ` `# Driver code ` ` ` `# No. of Shuffle Steps ` `N ` `=` `3` ` ` `# Key position ` `key ` `=` `3` `shuffle(N, key) ` ` ` `# This code is contributed by Anant Agarwal. ` |

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## C#

`// C# program to find the card at given index ` `// after N shuffles ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// function to find card at given index ` ` ` `static` `void` `shuffle(` `int` `N, ` `int` `key) ` ` ` `{ ` ` ` ` ` `// Answer will be reversal of N bits from MSB ` ` ` `int` `NO_OF_BITS = N; ` ` ` `int` `reverse_num = 0, temp; ` ` ` ` ` `// Calculating the reverse binary representation ` ` ` `for` `(` `int` `i = 0; i < NO_OF_BITS; i++) { ` ` ` `temp = (key & (1 << i)); ` ` ` `if` `(temp > 0) ` ` ` `reverse_num |= (1 << ((NO_OF_BITS - 1) - i)); ` ` ` `} ` ` ` ` ` `// Printing the result ` ` ` `Console.Write(reverse_num); ` ` ` `} ` ` ` ` ` `//Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` ` ` `// No. of Shuffle Steps ` ` ` `int` `N = 3; ` ` ` ` ` `// Key position ` ` ` `int` `key = 3; ` ` ` ` ` `shuffle(N, key); ` ` ` `} ` `} ` ` ` `// This code is contributed by Anant Agarwal. ` |

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Output:

6

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