Add some solutions and hints for fibonacci exercises

src/functions_exercises.md

@@ -12,6 +12,22 @@
 
    For testing you can `assert fib(8) == 21;`.
 
+   <details>
+   <summary>Solution</summary>
+
+   ```spicy
+   function fib(n: uint64): uint64 {
+       if (n < 2)
+           return n;
+
+       # This runs iff above `if` condition was false, but in this case could also be written
+       # as an `else` branch.
+       return fib(n - 1) + fib(n - 2);
+   }
+   ```
+
+   </details>
+
 1. Add memoization to your `fib` function. For that change its signature to
 
    ```spicy
@@ -27,5 +43,51 @@
 
    For testing you can `assert fib(64, m_fib) == 10610209857723;`.
 
+   <details>
+   <summary>Solution</summary>
+
+   ```spicy
+   function fib(n: uint64, inout cache: map<uint64, uint64>): uint64 {
+       # If the value is already in the cache we do not need to compute it.
+       if (n in cache)
+           return cache[n];
+
+       # Value was not in the cache. Compute its value and store it.
+       local result = 0;
+
+       if (n < 2)
+           result = n;
+
+       else
+           # Here we want an `else` branch for sure. We need to pass the cache
+           # down to other invocations. Since the passing happens by reference all
+           # invocations share a cache.
+           result = fib(n - 1, cache) + fib(n - 2, cache);
+
+       # Persist result in cache.
+       cache[n] = result;
+
+       # Return the result.
+       return result;
+   }
+   ```
+
+   </details>
+
 1. Try modifying your `fib` functions so users do not have to provide the cache
    themselves.
+
+   <details>
+   <summary>Hint</summary>
+
+   You want to store the cache somewhere yourself and provide users with a
+   wrapped function which implicitly uses your cache.
+
+   There are two places to put the cache:
+
+   - Construct the cache as a local variable inside the body of your wrapper
+     function. With this different invocations of the wrapper function would
+     not share the same cache which can be useful in certain scenarios.
+   - Alternatively one could store the cache in a `global`.
+
+   </details>