Day 1

We generalize the problem to finding n integers that sum up to t. As everything in Haskell, this function will be recursive in nature and we will need to provide it 4 arguments.

1. n
2. t
3. A list of elements included in our sum
4. The remaining elements

Thus we get something which looks like

try_sums :: Integer -> Integer -> [Integer] -> [Integer] -> Maybe Integer
try_sums t 0 as _ = if sum as == t then Just (product as) else Nothing       -- Base 1
try_sums _ _ _ [] = Nothing                                                  -- Base 2
try_sums t n as (b:bs) = try_sums t (n - 1) (b:as) bs <|> try_sums t n as bs -- Recursive

which has the following cases:

1. This base case is exactly as we imagine it. If there are no more elements to include into the sum list, we check the sum against t and return the product if it is correct.
2. This is the next case we are interested in. Because the first case matches n = 0, this case matches n != 0 and no more numbers, which should fail.
3. The main recursive case is pretty simple. Now that we know that there are elements in the remaining list, we either include it or we don’t, and we ask that Alternative picks the first success.