Integer Break

Integer Break

1. What is this problem about?

The Integer Break interview question asks you to take a positive integer $n$ and break it into the sum of at least two positive integers. Your goal is to maximize the product of those integers. For example, if $n=10$, you could break it into $3+3+4$, and the product $3 \times 3 \times 4 = 36$ is the maximum.

2. Why is this asked in interviews?

Companies like Google, Microsoft, and TikTok use the Integer Break coding problem to test a candidate's mathematical intuition and dynamic programming skills. It evaluations whether you can recognize a mathematical pattern (why 3 is the magic number) or implement an optimal substructure approach. It’s a core Math interview pattern.

3. Algorithmic pattern used

This problem can be solved with Dynamic Programming or Math.

• DP: dp[i] is the max product for integer i. dp[i] = max(j * (i - j), j * dp[i - j]) for all $1 \leq j < i$.
• Math: It can be proven that to maximize the product, you should use as many 3s as possible. If the remainder is 1, you turn one 3 into a 4 ($3+1 \to 2+2$). If the remainder is 2, you just include it.

4. Example explanation

$n = 10$.

• $10 / 3 = 3$ remainder 1.
• Use two 3s and one 4. $3 \times 3 \times 4 = 36$. $n = 8$.
• $8 / 3 = 2$ remainder 2.
• Use two 3s and one 2. $3 \times 3 \times 2 = 18$.

5. Common mistakes candidates make

• Not breaking enough: Forgetting the rule that you must break the number into at least two parts. For $n=2$, the answer is $1 \times 1 = 1$, not 2.
• Using 1s: Including 1 in the product, which never increases the value (except for $n=2$ and $n=3$).
• Complexity: Writing an exponential recursive solution without memoization.

6. Interview preparation tip

Learn the "Number 3" rule for product maximization. Any number larger than 4 is better represented as a sum containing 2s and 3s. This mathematical property is useful for many optimization problems in Math interview patterns.