For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisﬁes $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
1. Given the polynomial \( p(x) = ax^3 + bx^2 + cx + d \), we know that \( p(5) + p(25) = 1906 \). We need to find the minimum possible value for \( |p(15)| \).

2. First, we express \( p(5) \) and \( p(25) \) in terms of \( a, b, c, \) and \( d \):
   \[
   p(5) = 125a + 25b + 5c + d
   \]
   \[
   p(25) = 15625a + 625b + 25c + d
   \]

3. Adding these two expressions, we get:
   \[
   p(5) + p(25) = (125a + 25b + 5c + d) + (15625a + 625b + 25c + d)
   \]
   \[
   p(5) + p(25) = 15750a + 650b + 30c + 2d
   \]

4. Given \( p(5) + p(25) = 1906 \), we have:
   \[
   15750a + 650b + 30c + 2d = 1906
   \]

5. We want to minimize \( |p(15)| \), where:
   \[
   p(15) = 3375a + 225b + 15c + d
   \]

6. Let \( p(15) = n \). Then:
   \[
   3375a + 225b + 15c + d = n
   \]

7. We can express \( d \) from the equation \( 3375a + 225b + 15c + d = n \):
   \[
   d = n - 3375a - 225b - 15c
   \]

8. Substitute \( d \) into the equation \( 15750a + 650b + 30c + 2d = 1906 \):
   \[
   15750a + 650b + 30c + 2(n - 3375a - 225b - 15c) = 1906
   \]
   \[
   15750a + 650b + 30c + 2n - 6750a - 450b - 30c = 1906
   \]
   \[
   9000a + 200b + 2n = 1906
   \]
   \[
   2n = 1906 - 9000a - 200b
   \]
   \[
   n = \frac{1906 - 9000a - 200b}{2}
   \]

9. For \( n \) to be an integer, \( 1906 - 9000a - 200b \) must be even. We need to find the smallest \( |n| \).

10. Simplify the congruence:
    \[
    2n \equiv 1906 \pmod{200}
    \]
    \[
    2n \equiv 106 \pmod{200}
    \]
    \[
    n \equiv 53 \pmod{100}
    \]

11. The smallest \( |n| \) satisfying this congruence is \( n = -47 \).

12. Verify if \( n = -47 \) can be achieved:
    \[
    -47 = \frac{1906 - 9000a - 200b}{2}
    \]
    \[
    -94 = 1906 - 9000a - 200b
    \]
    \[
    9000a + 200b = 2000
    \]
    \[
    45a + b = 10
    \]

13. Choose \( a = 1 \) and \( b = -35 \):
    \[
    45(1) + (-35) = 10
    \]

14. Substitute \( a = 1 \) and \( b = -35 \) into the original polynomial:
    \[
    15750(1) + 650(-35) + 30c + 2d = 1906
    \]
    \[
    15750 - 22750 + 30c + 2d = 1906
    \]
    \[
    -7000 + 30c + 2d = 1906
    \]
    \[
    30c + 2d = 8906
    \]

15. Choose \( c = -296 \) and \( d = -13 \):
    \[
    30(-296) + 2(-13) = -8880 - 26 = -8906
    \]

16. Verify \( p(15) \):
    \[
    p(15) = 3375(1) + 225(-35) + 15(-296) + (-13)
    \]
    \[
    p(15) = 3375 - 7875 - 4440 - 13
    \]
    \[
    p(15) = -47
    \]

The final answer is \( \boxed{47} \).