Count Square Sum Triples

Count Square Sum Triples

What is this problem about?

The "Count Square Sum Triples interview question" is a mathematical enumeration task. You are given an integer $n$. You need to find the number of triples $(a, b, c)$ such that $1 \leq a, b, c \leq n$ and $a^2 + b^2 = c^2$. These are also known as Pythagorean triples. Note that $(3, 4, 5)$ and $(4, 3, 5)$ are considered two distinct triples.

Why is this asked in interviews?

Companies like Meta and Qualtrics use the "Count Square Sum Triples coding problem" as an introductory technical question. It tests a candidate's ability to implement nested loops efficiently and apply basic mathematical properties (like square roots) to avoid unnecessary computations. It’s a test of "Math interview pattern" and "Enumeration" basics.

Algorithmic pattern used

This problem follows the Nested Loop with Early Exit or Two Sum Square pattern.

1. Nested Loops: Iterate through all possible values of $a$ from 1 to $n$.
2. Inner Loop: For each $a$, iterate through $b$ from 1 to $n$.
3. Condition: Calculate $a^2 + b^2$. Let this be $S$.
4. Check Square: Determine if $S$ is a perfect square ($c^2$) and if its square root $c$ is less than or equal to $n$.
5. Optimization: Since the equation is symmetric, you can iterate $b$ from $a$ to $n$ and multiply results by 2, or just use the full range for simplicity given the small constraints ($n \leq 250$).

Example explanation

$n = 5$

• $a=3, b=4$: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. Since $5 \leq 5$, this is a triple.
• $a=4, b=3$: $4^2 + 3^2 = 25 = 5^2$. This is another triple. Result: 2. $n = 10$
• Triples: (3,4,5), (4,3,5), (6,8,10), (8,6,10). Result: 4.

Common mistakes candidates make

• Integer Precision: Using sqrt and then squaring back to check for equality without handling potential floating-point precision issues (though not usually an issue for small integers).
• Over-counting: Including triples where $c > n$.
• Redundant Calculation: Recalculating $a^2$ inside the inner loop instead of once in the outer loop.

Interview preparation tip

Always look for the upper bound of your search space. In this problem, because $a^2 + b^2 = c^2$ and $c \leq n$, then $a$ and $b$ must also be less than $n$. Small mathematical insights can significantly clean up your "Enumeration interview pattern" code.