<p>You have a cubic storeroom where the width, length, and height of the room are all equal to <code>n</code> units. You are asked to place <code>n</code> boxes in this room where each box is a cube of unit side length. There are however some rules to placing the boxes:</p>

<ul>
	<li>You can place the boxes anywhere on the floor.</li>
	<li>If box <code>x</code> is placed on top of the box <code>y</code>, then each side of the four vertical sides of the box <code>y</code> <strong>must</strong> either be adjacent to another box or to a wall.</li>
</ul>

<p>Given an integer <code>n</code>, return<em> the <strong>minimum</strong> possible number of boxes touching the floor.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2021/01/04/3-boxes.png" style="width: 135px; height: 143px;" /></p>

<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> 3
<strong>Explanation:</strong> The figure above is for the placement of the three boxes.
These boxes are placed in the corner of the room, where the corner is on the left side.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2021/01/04/4-boxes.png" style="width: 135px; height: 179px;" /></p>

<pre>
<strong>Input:</strong> n = 4
<strong>Output:</strong> 3
<strong>Explanation:</strong> The figure above is for the placement of the four boxes.
These boxes are placed in the corner of the room, where the corner is on the left side.
</pre>

<p><strong class="example">Example 3:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2021/01/04/10-boxes.png" style="width: 271px; height: 257px;" /></p>

<pre>
<strong>Input:</strong> n = 10
<strong>Output:</strong> 6
<strong>Explanation:</strong> The figure above is for the placement of the ten boxes.
These boxes are placed in the corner of the room, where the corner is on the back side.</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution {
 public:
  int minimumBoxes(int n) {
    int nBoxes = 0;
    int nextTouchings = 0;   // j
    int currLevelBoxes = 0;  // 1 + 2 + ... + j

    // Find the minimum j s.t. `nBoxes` = 1 + (1 + 2) + ... + (1 + 2 + ... + j)
    // >= n.
    while (nBoxes < n) {
      ++nextTouchings;
      currLevelBoxes += nextTouchings;
      nBoxes += currLevelBoxes;
    }

    // If nBoxes = n, the answer is `currLevelBoxes` = 1 + 2 + ... + j.
    if (nBoxes == n)
      return currLevelBoxes;

    // Otherwise, need to remove the boxes in the current level and rebuild it.
    nBoxes -= currLevelBoxes;
    currLevelBoxes -= nextTouchings;
    nextTouchings = 0;

    while (nBoxes < n) {
      ++nextTouchings;
      nBoxes += nextTouchings;
    }

    return currLevelBoxes + nextTouchings;
  }
};
7200