In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$.
1. Let the number of people initially in the room be \(144\).
2. Let \(n\) be the number of additional people who join the room.
3. Each of these \(n\) additional people is carrying \(k\) coins.
4. When these coins are shared among all \(n + 144\) people, each person has \(2\) coins.

We need to find the minimum possible value of \(2n + k\).

First, we set up the equation based on the given information:
\[
2(n + 144) = nk
\]

Simplify the equation:
\[
2n + 288 = nk
\]

Rearrange to isolate \(n\):
\[
nk - 2n = 288
\]

Factor out \(n\):
\[
n(k - 2) = 288
\]

Thus, we have:
\[
n = \frac{288}{k - 2}
\]

To find the minimum value of \(2n + k\), we need to consider the possible values of \(k\) such that \(n\) is an integer. This means \(k - 2\) must be a divisor of \(288\).

The divisors of \(288\) are:
\[
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288
\]

Thus, the possible values of \(k\) are:
\[
3, 4, 5, 6, 8, 10, 11, 14, 18, 20, 26, 34, 38, 50, 74, 98, 146, 290
\]

For each \(k\), we calculate \(n\) and then \(2n + k\):

- For \(k = 3\):
  \[
  n = \frac{288}{3 - 2} = 288
  \]
  \[
  2n + k = 2(288) + 3 = 579
  \]

- For \(k = 4\):
  \[
  n = \frac{288}{4 - 2} = 144
  \]
  \[
  2n + k = 2(144) + 4 = 292
  \]

- For \(k = 5\):
  \[
  n = \frac{288}{5 - 2} = 96
  \]
  \[
  2n + k = 2(96) + 5 = 197
  \]

- For \(k = 6\):
  \[
  n = \frac{288}{6 - 2} = 72
  \]
  \[
  2n + k = 2(72) + 6 = 150
  \]

- For \(k = 8\):
  \[
  n = \frac{288}{8 - 2} = 48
  \]
  \[
  2n + k = 2(48) + 8 = 104
  \]

- For \(k = 10\):
  \[
  n = \frac{288}{10 - 2} = 36
  \]
  \[
  2n + k = 2(36) + 10 = 82
  \]

- For \(k = 11\):
  \[
  n = \frac{288}{11 - 2} = 32
  \]
  \[
  2n + k = 2(32) + 11 = 75
  \]

- For \(k = 14\):
  \[
  n = \frac{288}{14 - 2} = 24
  \]
  \[
  2n + k = 2(24) + 14 = 62
  \]

- For \(k = 18\):
  \[
  n = \frac{288}{18 - 2} = 18
  \]
  \[
  2n + k = 2(18) + 18 = 54
  \]

- For \(k = 20\):
  \[
  n = \frac{288}{20 - 2} = 16
  \]
  \[
  2n + k = 2(16) + 20 = 52
  \]

- For \(k = 26\):
  \[
  n = \frac{288}{26 - 2} = 12
  \]
  \[
  2n + k = 2(12) + 26 = 50
  \]

- For \(k = 34\):
  \[
  n = \frac{288}{34 - 2} = 9
  \]
  \[
  2n + k = 2(9) + 34 = 52
  \]

- For \(k = 38\):
  \[
  n = \frac{288}{38 - 2} = 8
  \]
  \[
  2n + k = 2(8) + 38 = 54
  \]

- For \(k = 50\):
  \[
  n = \frac{288}{50 - 2} = 6
  \]
  \[
  2n + k = 2(6) + 50 = 62
  \]

- For \(k = 74\):
  \[
  n = \frac{288}{74 - 2} = 4
  \]
  \[
  2n + k = 2(4) + 74 = 82
  \]

- For \(k = 98\):
  \[
  n = \frac{288}{98 - 2} = 3
  \]
  \[
  2n + k = 2(3) + 98 = 104
  \]

- For \(k = 146\):
  \[
  n = \frac{288}{146 - 2} = 2
  \]
  \[
  2n + k = 2(2) + 146 = 150
  \]

- For \(k = 290\):
  \[
  n = \frac{288}{290 - 2} = 1
  \]
  \[
  2n + k = 2(1) + 290 = 292
  \]

The minimum value of \(2n + k\) is \(50\) when \(k = 26\) and \(n = 12\).

The final answer is \(\boxed{50}\)