# Maximum number of 2×2 squares that can be fit inside a right isosceles triangle

What is the maximum number of squares of size 2×2 units that can be fit in a right-angled isosceles triangle of a given base (in units).

A side of the square must be parallel to the base of the triangle.

Examples:

Input : 8 Output : 6 Please refer below diagram for explanation.

Input : 7 Output : 3

Since the triangle is isosceles, the given base would also be equal to the height. Now in the diagonal part, we would always need an extra length of 2 units in both height and base of the triangle to accommodate a triangle. (The CF and AM segment of the triangle in the image. The part that does not contribute to any square). In the remaining length of base, we can construct length / 2 squares. Since each square is of 2 units, same would be the case of height, there is no need to calculate that again.

So, for each level of given length we can construct “(length-2)/2” squares. This gives us a base of “(length-2)” above it. Continuing this process to get the no of squares for all available “length-2” height, we can calculate the squares.

while length > 2 answer += (length - 2 )/2 length = length - 2

**For more effective way, we can use the formula of sum of AP n * ( n + 1 ) / 2, where n = length – 2**

## C++

`// C++ program to count number of 2 x 2` `// squares in a right isosceles triangle` `#include<bits/stdc++.h>` `using` `namespace` `std;` `int` `numberOfSquares(` `int` `base)` `{` ` ` `// removing the extra part we would` ` ` `// always need` ` ` `base = (base - 2);` ` ` `// Since each square has base of` ` ` `// length of 2` ` ` `base = ` `floor` `(base / 2);` ` ` `return` `base * (base + 1)/2;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `base = 8;` ` ` `cout << numberOfSquares(base);` ` ` `return` `0;` `}` `// This code is improved by heroichitesh.` |

## Java

`// Java program to count number of 2 x 2` `// squares in a right isosceles triangle` `class` `Squares` `{` ` ` `public` `static` `int` `numberOfSquares(` `int` `base)` ` ` `{` ` ` `// removing the extra part` ` ` `// we would always need` ` ` `base = (base - ` `2` `);` ` ` ` ` `// Since each square has` ` ` `// base of length of 2` ` ` `base = Math.floorDiv(base, ` `2` `);` ` ` ` ` `return` `base * (base + ` `1` `)/` `2` `;` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `main(String args[])` ` ` `{` ` ` ` ` `int` `base = ` `8` `;` ` ` `System.out.println(numberOfSquares(base));` ` ` `}` `}` `// This code is contributed by Anshika Goyal and improved by heroichitesh.` |

## Python3

`# Python3 program to count number` `# of 2 x 2 squares in a right` `# isosceles triangle` `def` `numberOfSquares(base):` ` ` ` ` `# removing the extra part we would` ` ` `# always need` ` ` `base ` `=` `(base ` `-` `2` `)` ` ` ` ` `# Since each square has base of` ` ` `# length of 2` ` ` `base ` `=` `base ` `/` `/` `2` ` ` ` ` `return` `base ` `*` `(base ` `+` `1` `) ` `/` `2` ` ` `# Driver code` `base ` `=` `8` `print` `(numberOfSquares(base))` `# This code is contributed by Anant Agarwal and improved by heroichitesh.` |

## C#

`// C# program to count number of 2 x 2` `// squares in a right isosceles triangle` `using` `System;` `class` `GFG {` ` ` ` ` `public` `static` `int` `numberOfSquares(` `int` `_base)` ` ` `{` ` ` ` ` `// removing the extra part` ` ` `// we would always need` ` ` `_base = (_base - 2);` ` ` ` ` `// Since each square has` ` ` `// base of length of 2` ` ` `_base = _base / 2;` ` ` ` ` `return` `_base * (_base + 1)/2;` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` ` ` `int` `_base = 8;` ` ` `Console.WriteLine(numberOfSquares(_base));` ` ` `}` `}` `// This code is contributed by anuj_67.` |

## PHP

`<?php` `// PHP program to count number of 2 x 2` `// squares in a right isosceles triangle` ` ` `function` `numberOfSquares( ` `$base` `)` ` ` `{` ` ` ` ` `// removing the extra` ` ` `// part we would` ` ` `// always need` ` ` `$base` `= (` `$base` `- 2);` ` ` ` ` `// Since each square` ` ` `// has base of` ` ` `// length of 2` ` ` `$base` `= intdiv(` `$base` `, 2);` ` ` ` ` `return` `$base` `* (` `$base` `+ 1)/2;` ` ` `}` `// Driver code` `$base` `= 8;` `echo` `numberOfSquares(` `$base` `);` `// This code is contributed by anuj_67 and improved by heroichitesh.` `?>` |

## Javascript

`<script>` ` ` `// Program to count number of 2 x 2` ` ` `// squares in a right isosceles triangle` ` ` ` ` `function` `numberOfSquares(base)` ` ` `{` ` ` ` ` `// Removing the extra part we would` ` ` `// always need` ` ` `base = (base - 2);` ` ` ` ` `// Since each square has base of` ` ` `// length of 2` ` ` `base = Math.floor(base / 2);` ` ` ` ` `return` `base * (base + 1) / 2;` ` ` `}` ` ` ` ` `// Driver code` ` ` `let base = 8;` ` ` `document.write(numberOfSquares(base));` ` ` ` ` `// This code is contributed by Mayank Tyagi` `</script>` |

**Output: **

6

**Time complexity :** O(1)

**Auxiliary Space : **O(1)

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