Minimum tiles of sizes in powers of two to cover whole area

Last Updated : 25 May, 2022

Given an area of N X M. You have the infinite number of tiles of size 2ⁱ X 2ⁱ, where i = 0, 1, 2,… so on. The task is to find the minimum number of tiles required to fill the given area with tiles.
Examples:

Input : N = 5, M = 6.
Output : 9
Area of 5 X 6 can be covered with minimum 9 tiles.
6 tiles of 1 X 1, 2 tiles of 2 X 2, 1 tile of 4 X 4.

Input : N = 10, M = 5.
Output : 14

The idea is to divide the given area into the nearest 2ⁱ X 2ⁱ.
Let’s divide the problem into cases:
Case 1: if N is odd and M is even, fill the row or column with M number of 1 X 1 tiles. Then count the minimum number of tiles for N/2 X M/2 size of the area. Similarly, if M is odd and N is even, add N to our answer and find a minimum number of tiles for the N/2 X M/2 area.
Case 2: If N and M both are odd, fill one row and one column, so add N + M – 1 to the answer and find the minimum number of tiles required to fill the N/2 X M/2 area.
Case 3: If N and M both are even, calculate the minimum number of tiles required to fill the area with N/2 X M/2 area. Because halving both the dimensions doesn’t change the number of tiles required.
Below is the implementation of this approach:

C++

<div id="highlighter_190951" class="syntaxhighlighter nogutter  "><table border="0" cellpadding="0" cellspacing="0"><tbody><tr><td class="code"><div class="container"><div class="line number1 index0 alt2"><code class="preprocessor">#include<bits/stdc++.h></code></div><div class="line number2 index1 alt1"><code class="keyword bold">using</code> <code class="keyword bold">namespace</code> <code class="plain">std;</code></div><div class="line number3 index2 alt2"> </div><div class="line number4 index3 alt1"><code class="color1 bold">int</code> <code class="plain">minTiles(</code><code class="color1 bold">int</code> <code class="plain">n, </code><code class="color1 bold">int</code> <code class="plain">m)</code></div><div class="line number5 index4 alt2"><code class="plain">{</code></div><div class="line number6 index5 alt1"><code class="undefined spaces">  </code><code class="comments">// base case, when area is 0.</code></div><div class="line number7 index6 alt2"><code class="undefined spaces">  </code><code class="keyword bold">if</code> <code class="plain">(n == 0 || m == 0)</code></div><div class="line number8 index7 alt1"><code class="undefined spaces">    </code><code class="keyword bold">return</code> <code class="plain">0;</code></div><div class="line number9 index8 alt2"> </div><div class="line number10 index9 alt1"><code class="undefined spaces">  </code><code class="comments">// If n and m both are even, calculate tiles for n/2 x m/2</code></div><div class="line number11 index10 alt2"><code class="undefined spaces">  </code><code class="comments">// Halving both dimensions doesn't change the number of tiles</code></div><div class="line number12 index11 alt1"><code class="undefined spaces">  </code><code class="keyword bold">else</code> <code class="keyword bold">if</code> <code class="plain">(n%2 == 0 && m%2 == 0)</code></div><div class="line number13 index12 alt2"><code class="undefined spaces">    </code><code class="keyword bold">return</code> <code class="plain">minTiles(n/2, m/2);</code></div><div class="line number14 index13 alt1"><code class="undefined spaces">  </code> </div><div class="line number15 index14 alt2"><code class="undefined spaces">  </code><code class="comments">// If n is even and m is odd</code></div><div class="line number16 index15 alt1"><code class="undefined spaces">  </code><code class="comments">// Use a row of 1x1 tiles</code></div><div class="line number17 index16 alt2"><code class="undefined spaces">  </code><code class="keyword bold">else</code> <code class="keyword bold">if</code> <code class="plain">(n%2 == 0 && m%2 == 1)</code></div><div class="line number18 index17 alt1"><code class="undefined spaces">    </code><code class="keyword bold">return</code> <code class="plain">(n + minTiles(n/2, m/2));</code></div><div class="line number19 index18 alt2"> </div><div class="line number20 index19 alt1"><code class="undefined spaces">  </code><code class="comments">// If n is odd and m is even</code></div><div class="line number21 index20 alt2"><code class="undefined spaces">  </code><code class="comments">// Use a column of 1x1 tiles</code></div><div class="line number22 index21 alt1"><code class="undefined spaces">  </code><code class="keyword bold">else</code> <code class="keyword bold">if</code> <code class="plain">(n%2 == 1 && m%2 == 0)</code></div><div class="line number23 index22 alt2"><code class="undefined spaces">    </code><code class="keyword bold">return</code> <code class="plain">(m + minTiles(n/2, m/2));</code></div><div class="line number24 index23 alt1"> </div><div class="line number25 index24 alt2"><code class="undefined spaces">  </code><code class="comments">// If n and m are odd</code></div><div class="line number26 index25 alt1"><code class="undefined spaces">  </code><code class="comments">// add row + column number of tiles</code></div><div class="line number27 index26 alt2"><code class="undefined spaces">  </code><code class="keyword bold">else</code></div><div class="line number28 index27 alt1"><code class="undefined spaces">    </code><code class="keyword bold">return</code> <code class="plain">(n + m - 1 + minTiles(n/2, m/2)); </code></div><div class="line number29 index28 alt2"><code class="plain">}</code></div><div class="line number30 index29 alt1"> </div><div class="line number31 index30 alt2"><code class="comments">// Driven Program</code></div><div class="line number32 index31 alt1"><code class="color1 bold">int</code> <code class="plain">main()</code></div><div class="line number33 index32 alt2"><code class="plain">{</code></div><div class="line number34 index33 alt1"><code class="undefined spaces">  </code><code class="color1 bold">int</code> <code class="plain">n = 5, m = 6;</code></div><div class="line number35 index34 alt2"> </div><div class="line number36 index35 alt1"><code class="undefined spaces">  </code><code class="plain">cout << minTiles(n, m) << endl;</code></div><div class="line number37 index36 alt2"><code class="undefined spaces">  </code><code class="keyword bold">return</code> <code class="plain">0;</code></div><div class="line number38 index37 alt1"><code class="plain">}</code></div></div></td></tr></tbody></table></div>

Java

// Java code for Minimum tiles of 
// sizes in powers of two to cover 
// whole area
 
class GFG {
     
    static int minTiles(int n, int m)
    {
    // base case, when area is 0.
    if (n == 0 || m == 0)
        return 0;
     
    // If n and m both are even, 
    // calculate tiles for n/2 x m/2
    // Halving both dimensions doesn't 
    // change the number of tiles
    else if (n % 2  == 0 && m % 2 == 0)
        return minTiles(n / 2, m / 2);
         
    // If n is even and m is odd
    // Use a row of 1x1 tiles
    else if (n % 2 == 0 && m % 2 == 1)
        return (n + minTiles(n / 2, m / 2));
     
    // If n is odd and m is even
    // Use a column of 1x1 tiles
    else if (n % 2 == 1 && m % 2 == 0)
        return (m + minTiles(n / 2, m / 2));
     
    // If n and m are odd
    // add row + column number of tiles
    else
        return (n + m - 1 + minTiles(n / 2, m / 2)); 
    }
         
    // Driver code
    public static void main (String[] args)
    {
            int n = 5, m = 6;
            System.out.println(minTiles(n, m));
    }
}
 
// This code is contributed by Anant Agarwal.

Python3

def minTiles(n, m):
     
    # base case, when area is 0.
    if n == 0 or m == 0:
        return 0
 
    # If n and m both are even, calculate
    # tiles for n/2 x m/2
    # Halving both dimensions doesn't
    # change the number of tiles
    elif n%2 == 0 and m%2 == 0:
        return minTiles(int(n/2), int(m/2))
 
    # If n is even and m is odd
    # Use a row of 1x1 tiles
    elif n % 2 == 0 and m % 2 == 1:
        return (n + minTiles(int(n/2), int(m/2)))
 
    # If n is odd and m is even
    # Use a column of 1x1 tiles
    elif n % 2 == 1 and m % 2 == 0:
        return (m + minTiles(int(n/2), int(m/2)))
 
    # If n and m are odd add
    # row + column number of tiles
    else:
        return (n + m - 1 + minTiles(int(n/2), int(m/2))) 
 
# Driven Program
n = 5
m = 6
print (minTiles(n, m))
 
# This code is contributed
# by Shreyanshi Arun.

C#

<div id="highlighter_324618" class="syntaxhighlighter nogutter  "><table border="0" cellpadding="0" cellspacing="0"><tbody><tr><td class="code"><div class="container"><div class="line number1 index0 alt2"><code class="comments">// C# code for Minimum tiles of</code></div><div class="line number2 index1 alt1"><code class="comments">// sizes in powers of two to cover</code></div><div class="line number3 index2 alt2"><code class="comments">// whole area</code></div><div class="line number4 index3 alt1"><code class="keyword">using</code> <code class="plain">System;</code></div><div class="line number5 index4 alt2"> </div><div class="line number6 index5 alt1"><code class="keyword">class</code> <code class="plain">GFG {</code></div><div class="line number7 index6 alt2"> </div><div class="line number8 index7 alt1"><code class="undefined spaces">    </code><code class="keyword">static</code> <code class="keyword">int</code> <code class="plain">minTiles(</code><code class="keyword">int</code> <code class="plain">n, </code><code class="keyword">int</code> <code class="plain">m)</code></div><div class="line number9 index8 alt2"><code class="undefined spaces">    </code><code class="plain">{</code></div><div class="line number10 index9 alt1"><code class="undefined spaces">        </code> </div><div class="line number11 index10 alt2"><code class="undefined spaces">        </code><code class="comments">// base case, when area is 0.</code></div><div class="line number12 index11 alt1"><code class="undefined spaces">        </code><code class="keyword">if</code> <code class="plain">(n == 0 || m == 0)</code></div><div class="line number13 index12 alt2"><code class="undefined spaces">            </code><code class="keyword">return</code> <code class="plain">0;</code></div><div class="line number14 index13 alt1"> </div><div class="line number15 index14 alt2"><code class="undefined spaces">        </code><code class="comments">// If n and m both are even,</code></div><div class="line number16 index15 alt1"><code class="undefined spaces">        </code><code class="comments">// calculate tiles for n/2 x m/2</code></div><div class="line number17 index16 alt2"><code class="undefined spaces">        </code><code class="comments">// Halving both dimensions doesn't</code></div><div class="line number18 index17 alt1"><code class="undefined spaces">        </code><code class="comments">// change the number of tiles</code></div><div class="line number19 index18 alt2"><code class="undefined spaces">        </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(n % 2 == 0 && m % 2 == 0)</code></div><div class="line number20 index19 alt1"><code class="undefined spaces">            </code><code class="keyword">return</code> <code class="plain">minTiles(n / 2, m / 2);</code></div><div class="line number21 index20 alt2"> </div><div class="line number22 index21 alt1"><code class="undefined spaces">        </code><code class="comments">// If n is even and m is odd</code></div><div class="line number23 index22 alt2"><code class="undefined spaces">        </code><code class="comments">// Use a row of 1x1 tiles</code></div><div class="line number24 index23 alt1"><code class="undefined spaces">        </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(n % 2 == 0 && m % 2 == 1)</code></div><div class="line number25 index24 alt2"><code class="undefined spaces">            </code><code class="keyword">return</code> <code class="plain">(n + minTiles(n / 2, m / 2));</code></div><div class="line number26 index25 alt1"> </div><div class="line number27 index26 alt2"><code class="undefined spaces">        </code><code class="comments">// If n is odd and m is even</code></div><div class="line number28 index27 alt1"><code class="undefined spaces">        </code><code class="comments">// Use a column of 1x1 tiles</code></div><div class="line number29 index28 alt2"><code class="undefined spaces">        </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(n % 2 == 1 && m % 2 == 0)</code></div><div class="line number30 index29 alt1"><code class="undefined spaces">            </code><code class="keyword">return</code> <code class="plain">(m + minTiles(n / 2, m / 2));</code></div><div class="line number31 index30 alt2"> </div><div class="line number32 index31 alt1"><code class="undefined spaces">        </code><code class="comments">// If n and m are odd</code></div><div class="line number33 index32 alt2"><code class="undefined spaces">        </code><code class="comments">// add row + column number of tiles</code></div><div class="line number34 index33 alt1"><code class="undefined spaces">        </code><code class="keyword">else</code></div><div class="line number35 index34 alt2"><code class="undefined spaces">            </code><code class="keyword">return</code> <code class="plain">(n + m - 1 + minTiles(n / 2, m / 2));</code></div><div class="line number36 index35 alt1"><code class="undefined spaces">    </code><code class="plain">}</code></div><div class="line number37 index36 alt2"> </div><div class="line number38 index37 alt1"><code class="undefined spaces">    </code><code class="comments">// Driver code</code></div><div class="line number39 index38 alt2"><code class="undefined spaces">    </code><code class="keyword">public</code> <code class="keyword">static</code> <code class="keyword">void</code> <code class="plain">Main()</code></div><div class="line number40 index39 alt1"><code class="undefined spaces">    </code><code class="plain">{</code></div><div class="line number41 index40 alt2"><code class="undefined spaces">        </code><code class="keyword">int</code> <code class="plain">n = 5, m = 6;</code></div><div class="line number42 index41 alt1"><code class="undefined spaces">        </code> </div><div class="line number43 index42 alt2"><code class="undefined spaces">        </code><code class="plain">Console.WriteLine(minTiles(n, m));</code></div><div class="line number44 index43 alt1"><code class="undefined spaces">    </code><code class="plain">}</code></div><div class="line number45 index44 alt2"><code class="plain">}</code></div><div class="line number46 index45 alt1"> </div><div class="line number47 index46 alt2"><code class="comments">// This code is contributed by vt_m.</code></div></div></td></tr></tbody></table></div>

PHP

<div id="highlighter_613659" class="syntaxhighlighter nogutter  "><table border="0" cellpadding="0" cellspacing="0"><tbody><tr><td class="code"><div class="container"><div class="line number1 index0 alt2"><code class="plain"><?php</code></div><div class="line number2 index1 alt1"><code class="comments">// PHP program for Minimum tiles of </code></div><div class="line number3 index2 alt2"><code class="comments">// sizes in powers of two to cover</code></div><div class="line number4 index3 alt1"><code class="comments">// whole area</code></div><div class="line number5 index4 alt2"> </div><div class="line number6 index5 alt1"><code class="keyword">function</code> <code class="plain">minTiles(</code><code class="variable">$n</code><code class="plain">, </code><code class="variable">$m</code><code class="plain">)</code></div><div class="line number7 index6 alt2"><code class="plain">{</code></div><div class="line number8 index7 alt1"><code class="undefined spaces">    </code> </div><div class="line number9 index8 alt2"><code class="undefined spaces">    </code><code class="comments">// base case, when area is 0.</code></div><div class="line number10 index9 alt1"><code class="undefined spaces">    </code><code class="keyword">if</code> <code class="plain">(</code><code class="variable">$n</code> <code class="plain">== 0 </code><code class="keyword">or</code> <code class="variable">$m</code> <code class="plain">== 0)</code></div><div class="line number11 index10 alt2"><code class="undefined spaces">        </code><code class="keyword">return</code> <code class="plain">0;</code></div><div class="line number12 index11 alt1"><code class="undefined spaces">    </code> </div><div class="line number13 index12 alt2"><code class="undefined spaces">    </code><code class="comments">// If n and m both are even,</code></div><div class="line number14 index13 alt1"><code class="undefined spaces">    </code><code class="comments">// calculate tiles for n/2 x m/2</code></div><div class="line number15 index14 alt2"><code class="undefined spaces">    </code><code class="comments">// Halving both dimensions doesn't </code></div><div class="line number16 index15 alt1"><code class="undefined spaces">    </code><code class="comments">// change the number of tiles</code></div><div class="line number17 index16 alt2"><code class="undefined spaces">    </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(</code><code class="variable">$n</code> <code class="plain">% 2 == 0 </code><code class="keyword">and</code></div><div class="line number18 index17 alt1"><code class="undefined spaces">                </code><code class="variable">$m</code> <code class="plain">% 2 == 0)</code></div><div class="line number19 index18 alt2"><code class="undefined spaces">        </code><code class="keyword">return</code> <code class="plain">minTiles(</code><code class="variable">$n</code> <code class="plain">/ 2, </code><code class="variable">$m</code> <code class="plain">/ 2);</code></div><div class="line number20 index19 alt1"><code class="undefined spaces">    </code> </div><div class="line number21 index20 alt2"><code class="undefined spaces">    </code><code class="comments">// If n is even and m is odd</code></div><div class="line number22 index21 alt1"><code class="undefined spaces">    </code><code class="comments">// Use a row of 1x1 tiles</code></div><div class="line number23 index22 alt2"><code class="undefined spaces">    </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(</code><code class="variable">$n</code> <code class="plain">% 2 == 0 </code><code class="keyword">and</code> <code class="variable">$m</code> <code class="plain">% 2 == 1)</code></div><div class="line number24 index23 alt1"><code class="undefined spaces">        </code><code class="keyword">return</code> <code class="functions">floor</code><code class="plain">(</code><code class="variable">$n</code> <code class="plain">+ minTiles(</code><code class="variable">$n</code> <code class="plain">/ 2,</code></div><div class="line number25 index24 alt2"><code class="undefined spaces">                                   </code><code class="variable">$m</code> <code class="plain">/ 2));</code></div><div class="line number26 index25 alt1"><code class="undefined spaces">    </code> </div><div class="line number27 index26 alt2"><code class="undefined spaces">    </code><code class="comments">// If n is odd and m is even</code></div><div class="line number28 index27 alt1"><code class="undefined spaces">    </code><code class="comments">// Use a column of 1x1 tiles</code></div><div class="line number29 index28 alt2"><code class="undefined spaces">    </code><code class="keyword">else</code> <code class="keyword">if</code> <code class="plain">(</code><code class="variable">$n</code> <code class="plain">% 2 == 1 </code><code class="keyword">and</code></div><div class="line number30 index29 alt1"><code class="undefined spaces">             </code><code class="variable">$m</code> <code class="plain">% 2 == 0)</code></div><div class="line number31 index30 alt2"><code class="undefined spaces">        </code><code class="keyword">return</code> <code class="plain">(</code><code class="variable">$m</code> <code class="plain">+ minTiles(</code><code class="variable">$n</code> <code class="plain">/ 2, </code></div><div class="line number32 index31 alt1"><code class="undefined spaces">                              </code><code class="variable">$m</code> <code class="plain">/ 2));</code></div><div class="line number33 index32 alt2"><code class="undefined spaces">    </code> </div><div class="line number34 index33 alt1"><code class="undefined spaces">    </code><code class="comments">// If n and m are odd</code></div><div class="line number35 index34 alt2"><code class="undefined spaces">    </code><code class="comments">// add row + column number of tiles</code></div><div class="line number36 index35 alt1"><code class="undefined spaces">    </code><code class="keyword">else</code></div><div class="line number37 index36 alt2"><code class="undefined spaces">        </code><code class="keyword">return</code> <code class="functions">floor</code><code class="plain">(</code><code class="variable">$n</code> <code class="plain">+ </code><code class="variable">$m</code> <code class="plain">- 1 + </code></div><div class="line number38 index37 alt1"><code class="undefined spaces">                    </code><code class="plain">minTiles(</code><code class="variable">$n</code> <code class="plain">/ 2, </code><code class="variable">$m</code> <code class="plain">/ 2)); </code></div><div class="line number39 index38 alt2"><code class="plain">}</code></div><div class="line number40 index39 alt1"> </div><div class="line number41 index40 alt2"><code class="comments">// Driver Code</code></div><div class="line number42 index41 alt1"><code class="variable">$n</code> <code class="plain">= 5; </code><code class="variable">$m</code> <code class="plain">= 6;</code></div><div class="line number43 index42 alt2"><code class="functions">echo</code> <code class="plain">minTiles(</code><code class="variable">$n</code><code class="plain">, </code><code class="variable">$m</code><code class="plain">);</code></div><div class="line number44 index43 alt1"> </div><div class="line number45 index44 alt2"><code class="comments">// This code is contributed by anuj_67.</code></div><div class="line number46 index45 alt1"><code class="plain">?></code></div></div></td></tr></tbody></table></div>

Javascript

<script>
// JavaScript code for Minimum tiles of 
// sizes in powers of two to cover 
// whole area
 
    function minTiles(n, m)
    {
    // base case, when area is 0.
    if (n == 0 || m == 0)
        return 0;
     
    // If n and m both are even, 
    // calculate tiles for n/2 x m/2
    // Halving both dimensions doesn't 
    // change the number of tiles
    else if (n % 2  == 0 && m % 2 == 0)
        return minTiles((n / 2),(m / 2));
         
    // If n is even and m is odd
    // Use a row of 1x1 tiles
    else if (n % 2 == 0 && m % 2 == 1)
        return (n + minTiles(Math.floor(n / 2), Math.floor(m / 2)));
     
    // If n is odd and m is even
    // Use a column of 1x1 tiles
    else if (n % 2 == 1 && m % 2 == 0)
        return (m + minTiles(Math.floor(n / 2), Math.floor(m / 2)));
     
    // If n and m are odd
    // add row + column number of tiles
    else
        return (n + m - 1 + minTiles(Math.floor(n / 2), Math.floor(m / 2))); 
    }
 
// Driver Code
 
    let n = 5, m = 6;
    document.write(Math.floor(minTiles(n, m)));
         
</script>