An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again?
1. **Identify the periodicity and break pattern:**
   - The event occurs for \( x \) consecutive years and then there is a break for \( y \) consecutive years.
   - Given occurrences: 1964, 1986, 1996, 2008.
   - Given non-occurrences: 1976, 1993, 2006, 2013.

2. **Determine constraints on \( x \) and \( y \):**
   - Since the event occurred in 2008 but not in 2006 and 2013, we have \( x < 7 \).
   - Since the event occurred in 1986 and 1996 but not in 1993, we have \( y < 10 \).
   - Therefore, \( x + y < 17 \).

3. **Key observation:**
   - If the event occurs/doesn't occur in year \( k \), it also occurs/doesn't occur in year \( k + x + y \).

4. **Case-checking for \( x + y \):**
   - Calculate differences between given years:
     - \( 1976 - 1964 = 12 \)
     - \( 1993 - 1986 = 7 \)
     - \( 2006 - 1996 = 10 \)
     - \( 2013 - 2008 = 5 \)
   - From these differences, we deduce that \( x + y \) could be 11 or 13.

5. **Eliminate \( x + y = 13 \):**
   - If \( x + y = 13 \), the event would have occurred in 1990 and 1996.
   - This implies \( y < 5 \), which would mean \( x > 7 \), which is impossible since \( x < 7 \).

6. **Conclude \( x + y = 11 \):**
   - If \( x + y = 11 \), check the non-occurrence years:
     - \( 2006 - 11 = 1995 \) (event did not occur in 1995)
     - \( 1976 + 22 = 1998 \) (event did not occur in 1998)
   - This implies \( x < 3 \).

7. **Determine \( x \) and \( y \):**
   - The event occurred in 1987, so \( x = 2 \).
   - Therefore, \( y = 11 - 2 = 9 \).

8. **Predict the next occurrence:**
   - The event occurred in 2008.
   - The next occurrence will be \( 2008 + x + y = 2008 + 11 = 2018 \).

The final answer is \( \boxed{2018} \).