A11293. Yet Another Coin Problem

You have $5$ different types of coins, each with a value equal to one of the first $5$ triangular numbers: $1$ , $3$ , $6$ , $10$ , and $15$ . These coin types are available in abundance. Your goal is to find the minimum number of these coins required such that their total value sums up to exactly $n$ .

We can show that the answer always exists.

The first line contains one integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. The description of the test cases follows.

The first line of each test case contains an integer $n$ ( $1 \leq n \leq 10^9$ ) — the target value.

For each test case, output a single number — the minimum number of coins required.

In the first test case, for $n = 1$ , the answer is $1$ since only one $1$ value coin is sufficient. $1 = 1 \cdot 1$ .

In the fourth test case, for $n = 5$ , the answer is $3$ , which can be achieved using two $1$ value coins and one $3$ value coin. $5 = 2 \cdot 1 + 1 \cdot 3$ .

In the seventh test case, for $n = 12$ , the answer is $2$ , which can be achieved using two $6$ value coins.

In the ninth test case, for $n = 16$ , the answer is $2$ , which can be achieved using one $1$ value coin and one $15$ value coin or using one $10$ value coin and one $6$ value coin. $16 = 1 \cdot 1 + 1 \cdot 15 = 1 \cdot 6 + 1 \cdot 10$ .