<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:
    def minimumBoxes(self, n: int) -> int:
        res = 0
        k = 0
        if n <= 3:
            return n
            
        for i in range(1, 10 ** 5):
            if (i) * (i + 1) * (i + 2) > 6 * n:
                k = i - 1
                break
        triangular_sum = (k) * (k + 1) * (k + 2) // 6
        triangular_num = (k) * (k + 1) // 2
        res += triangular_num
        boxes = n - triangular_sum
        l = 0
        for i in range(10 ** 5):
            if i* (i + 1) // 2 >= boxes:
                l = i
                break
        res += l
        return res

            
        return res
            
16400