I'm having trouble with going through a list. I made this code which gives me a list of numbers which are evenly divided by the sum of their digits. For example, consider the number 123: 123/6 = 20.5, so it will not be in the list. One the other hand 280 will be on the list, because 280/10 = 28.
let inaHelper x = (floor(x)`mod`10)+ (floor(x/10)`mod`10)+(floor(x/100)`mod`10)
let ina = [x | x <- [1..999] , x `mod` (inaHelper x) == 0 ]
[1,2,3,4,5,6,7,8,9,10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,80,81,84,90,100,102,108,110,111,112,114,117,120,126,132,133,135,140,144,150,152,153,156,162,171,180,190,192,195,198,200,201,204,207,209,210,216,220,222,224,225,228,230,234,240,243,247,252,261,264,266,270,280,285,288,300,306,308,312,315,320,322,324,330,333,336,342,351,360,364,370,372,375,378,392,396,399,400,402,405,407,408,410,414,420,423,432,440,441,444,448,450,460,465,468,476,480,481,486,500,504,506,510,511,512,513,516,518,522,531,540,550,552,555,558,576,588,592,594,600,603,605,612,621,624,629,630,640,644,645,648,660,666,684,690,700,702,704,711,715,720,730,732,735,736,738,756,770,774,777,780,782,792,800,801,803,804,810,820,825,828,832,840,846,864,870,874,880,882,888,900,902,910,912,915,918,935,936,954,960,966,972,990,999]
But my problem now is: from the list above I only want the numbers that will not have a "neighbour" within a gap of 5 units. For example, the number 300 will be in the new list because it's neighbors (288 and 306) are not within the 5 unit gap.
I came up it this code:
let rare = [ x | x <- [ina] , ((x-5) >= [ina-1]) && ((x+5) <= [ina+1]) ]
I'm a beginner, can someone help?
An easy, though not very efficient, way would be to make a helper function which checks whether there is an element of a list within a certain range:
hasElemInRange :: (Int,Int) -> [Int] -> Bool
hasElemInRange (lo, hi) xs = -- left as exercise
(hint: you can use the any function)
and then you can include it in your list comprehension:
let rare = [ x | x <- ina, hasElemInRange (x-5,x+5) ina ]
Another idiom that you might consider is zipping a list with its own tail. So you can do:
ghci> let xs = [1,2,3,4,5,6,7]
ghci> zip3 xs (tail xs) (tail (tail xs))
[(1,2,3),(2,3,4),(3,4,5),(4,5,6),(5,6,7)]
Which will give you each element of the list with its "context", the element just before and after. Maybe you can figure out how to use that for what you need.
I'm really new to programming and Haskell in particular (so new that I actually don't know if this is a stupid question or not). But I was watching the lecture given by Eric Meijer (http://channel9.msdn.com/Series/C9-Lectures-Erik-Meijer-Functional-Programming-Fundamentals) and i was fascinated by the program written by Dr. Graham Hutton in lecture 11; The countdown problem.
My question is:
Is there a way of "filtering" the list of solutions by the length (number of elements), so that the list of solutions are restricted to the solutions that only uses (for example) three of the source numbers? In other words, I would like to change the question from "given the numbers [1,2,3,4,5,6,8,9] construct 18 using the operators..." to "given the numbers [..] which three numbers can be used to construct..."
In my futile attempts, I've been trying to put a kind restriction on his function subbags (which returns all permutations and subsequences of a list)
subbags :: [a] -> [[a]]
subbags xs = [zs | ys <- subs xs, zs <- perms ys]
So that I get all the permutations and subsequences that only contain three of the source numbers. Is this possible? If so, how?
Like I said, I have no idea if this is even a legitimate question - but I have gone from curious to obsessed, so any form of help or hint would be greatly appreciated!
The simplest way would be to just select from the candidates three times
[ (x, y, z) | x <- xs, y <- xs, z <- xs ]
although this assumes that repeat use of a single number is OK.
If it's not, we'll have to get smarter. In a simpler scenario we'd like to pick just two candidates:
[ (x, y) | x <- xs, y <- ys, aboveDiagonal (x, y) ]
in other words, if we think of this as a cartesian product turning a list into a grid of possibilities, we'd like to only consider the values "above the diagonal", where repeats don't happen. We can express this by zipping the coordinates along with the values
[ (x, y) | (i, x) <- zip [1..] xs
, (j, y) <- zip [1..] xs
, i < j
]
which can be extended back out to the n=3 scenario
[ (x, y, z) | (i, x) <- zip [1..] xs
, (j, y) <- zip [1..] xs
, (k, z) <- zip [1..] xs
, i < j
, j < k
]
Ultimately, however, this method is inefficient since it still has to scan through all of the possible pairs and then prune the repeats. We can be a bit smarter by only enumerating the above diagonal values to begin with. Returning to n=2 we'll write this as
choose2 :: [a] -> [(a, a)]
choose2 [] = []
choose2 (a:as) = map (a,) as ++ choose2 as
In other words, we pick first all of the pairs where the head of the list comes first and a value in the tail of the list comes second—this captures one edge of the upper triangle—and then we recurse by adding all of the upper diagonal values of the list of candidates sans the head.
This method can be straightforwardly extended to the n=3 case by using the n=2 case as a building block
choose3 :: [a] -> [(a, a, a)]
choose3 [] = []
choose3 (a:as) = map (\(y, z) -> (a, y, z)) (choose2 as) ++ choose3 as
which also provides a direct generalization to the fully general n dimensional solution
choose :: Int -> [a] -> [[a]]
choose 0 as = [[]] -- there's one way to choose 0 elements
choose _ [] = [] -- there are 0 ways to choose (n>0) elements of none
choose 1 as = map (:[]) as -- there are n ways to choose 1 element of n
choose n (a:as) = map (a:) (choose (n-1) as) ++ choose n as
I like this solution, which does not require the list elements to be an instance of Eq:
import Data.List (tails)
triples ls = [[x,y,z] | (x:xs) <- tails ls,
(y:ys) <- tails xs,
z <- ys]
This returns only subsequences, not permutations, though.
I want to iterate 2 (or 3) infinite lists and find the "smallest" pair that satisfies a condition, like so:
until pred [(a,b,c) | a<-as, b<-bs, c<-cs]
where pred (a,b,c) = a*a + b*b == c*c
as = [1..]
bs = [1..]
cs = [1..]
The above wouldn't get very far, as a == b == 1 throughout the run of the program.
Is there a nice way to dovetail the problem, e.g. build the infinite sequence [(1,1,1),(1,2,1),(2,1,1),(2,1,2),(2,2,1),(2,2,2),(2,2,3),(2,3,2),..] ?
Bonus: is it possible to generalize to n-tuples?
There's a monad for that, Omega.
Prelude> let as = each [1..]
Prelude> let x = liftA3 (,,) as as as
Prelude> let x' = mfilter (\(a,b,c) -> a*a + b*b == c*c) x
Prelude> take 10 $ runOmega x'
[(3,4,5),(4,3,5),(6,8,10),(8,6,10),(5,12,13),(12,5,13),(9,12,15),(12,9,15),(8,15,17),(15,8,17)]
Using it's applicative features, you can generalize to arbitrary tuples:
quadrupels = (,,,) <$> as <*> as <*> as <*> as -- or call it liftA4
But: this alone does not eliminate duplication, of course. It only gives you proper diagonalization. Maybe you could use monad comprehensions together with an approach like Thomas's, or just another mfilter pass (restricting to b /= c, in this case).
List comprehensions are great (and concise) ways to solve such problems. First, you know you want all combinations of (a,b,c) that might satisfy a^2 + b^2 = c^2 - a helpful observation is that (considering only positive numbers) it will always be the case that a <= c && b <= c.
To generate our list of candidates we can thus say c ranges from 1 to infinity while a and b range from one to c.
[(a,b,c) | c <- [1..], a <- [1..c], b <- [1..c]]
To get to the solution we just need to add your desired equation as a guard:
[(a,b,c) | c <- [1..], a <- [1..c], b <- [1..c], a*a+b*b == c*c]
This is inefficient, but the output is correct:
[(3,4,5),(4,3,5),(6,8,10),(8,6,10),(5,12,13),(12,5,13),(9,12,15)...
There are more principled methods than blind testing that can solve this problem.
{- It depends on what is "smallest". But here is a solution for a concept of "smallest" if tuples were compared first by their max. number and then by their total sum. (You can just copy and paste my whole answer into a file as I write the text in comments.)
We will need nub later. -}
import Data.List (nub)
{- Just for illustration: the easy case with 2-tuples. -}
-- all the two-tuples where 'snd' is 'n'
tuples n = [(i, n) | i <- [1..n]]
-- all the two-tuples where 'snd' is in '1..n'
tuplesUpTo n = concat [tuples i | i <- [1..n]]
{-
To get all results, you will need to insert the flip of each tuple into the stream. But let's do that later and generalize first.
Building tuples of arbitrary length is somewhat difficult, so we will work on lists. I call them 'kList's, if they have a length 'k'.
-}
-- just copied from the tuples case, only we need a base case for k=1 and
-- we can combine all results utilizing the list monad.
kLists 1 n = [[n]]
kLists k n = do
rest <- kLists (k-1) n
add <- [1..head rest]
return (add:rest)
-- same as above. all the klists with length k and max number of n
kListsUpTo k n = concat [kLists k i | i <- [1..n]]
-- we can do that unbounded as well, creating an infinite list.
kListsInf k = concat [kLists k i | i <- [1..]]
{-
The next step is rotating these lists around, because until now the largest number is always in the last place. So we just look at all rotations to get all the results. Using nub here is admittedly awkward, you can improve that. But without it, lists where all elements are the same are repeated k times.
-}
rotate n l = let (init, end) = splitAt n l
in end ++ init
rotations k l = nub [rotate i l | i <- [0..k-1]]
rotatedKListsInf k = concatMap (rotations k) $ kListsInf k
{- What remains is to convert these lists into tuples. This is a bit awkward, because every n-tuple is a separate type. But it's straightforward, of course. -}
kListToTuple2 [x,y] = (x,y)
kListToTuple3 [x,y,z] = (x,y,z)
kListToTuple4 [x,y,z,t] = (x,y,z,t)
kListToTuple5 [x,y,z,t,u] = (x,y,z,t,u)
kListToTuple6 [x,y,z,t,u,v] = (x,y,z,t,u,v)
{- Some tests:
*Main> take 30 . map kListToTuple2 $ rotatedKListsInf 2
[(1,1),(1,2),(2,1),(2,2),(1,3),(3,1),(2,3),(3,2),(3,3),(1,4),(4,1),(2,4),(4,2),(3,4),
(4,3),(4,4),(1,5),(5,1),(2,5),(5,2),(3,5),(5,3),(4,5),(5,4),(5,5),(1,6),(6,1),
(2,6), (6,2), (3,6)]
*Main> take 30 . map kListToTuple3 $ rotatedKListsInf 3
[(1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,2,2),(2,2,1),(2,1,2),(2,2,2),(1,1,3),(1,3,1),
(3,1,1),(1,2,3),(2,3,1),(3,1,2),(2,2,3),(2,3,2),(3,2,2),(1,3,3),(3,3,1),(3,1,3),
(2,3,3),(3,3,2),(3,2,3),(3,3,3),(1,1,4),(1,4,1),(4,1,1),(1,2,4),(2,4,1),(4,1,2)]
Edit:
I realized there is a bug: Just rotating the ordered lists isn't enough of course. The solution must be somewhere along the lines of having
rest <- concat . map (rotations (k-1)) $ kLists (k-1) n
in kLists, but then some issues with repeated outputs arise. You can figure that out, I guess. ;-)
-}
It really depends on what you mean by "smallest", but I assume you want to find a tuple of numbers with respect to its maximal element - so (2,2) is less than (1,3) (while standard Haskell ordering is lexicographic).
There is package data-ordlist, which is aimed precisely at working with ordered lists. It's function mergeAll (and mergeAllBy) allows you to combine a 2-dimensional matrix ordered in each direction into an ordered list.
First let's create a desired comparing function on tuples:
import Data.List (find)
import Data.List.Ordered
compare2 :: (Ord a) => (a, a) -> (a, a) -> Ordering
compare2 x y = compare (max2 x, x) (max2 y, y)
where
max2 :: Ord a => (a, a) -> a
max2 (x, y) = max x y
Then using mergeAll we create a function that takes a comparator, a combining function (which must be monotonic in both arguments) and two sorted lists. It combines all possible elements from the two lists using the function and produces a result sorted list:
mergeWith :: (b -> b -> Ordering) -> (a -> a -> b) -> [a] -> [a] -> [b]
mergeWith cmp f xs ys = mergeAllBy cmp $ map (\x -> map (f x) xs) ys
With this function, it's very simple to produce tuples ordered according to their maximum:
incPairs :: [(Int,Int)]
incPairs = mergeWith compare2 (,) [1..] [1..]
Its first 10 elements are:
> take 10 incPairs
[(1,1),(1,2),(2,1),(2,2),(1,3),(2,3),(3,1),(3,2),(3,3),(1,4)]
and when we (for example) look for the first pair whose sum of squares is equal to 65:
find (\(x,y) -> x^2+y^2 == 65) incPairs
we get the correct result (4,7) (as opposed to (1,8) if lexicographic ordering were used).
This answer is for a more general problem for a unknown predicate. If the predicate is known, more efficient solutions are possible, like others have listed solutions based on knowledge that you don't need to iterate for all Ints for a given c.
When dealing with infinite lists, you need to perform breadth-first search for solution. The list comprehension only affords depth-first search, that is why you never arrive at a solution in your original code.
counters 0 xs = [[]]
counters n xs = concat $ foldr f [] gens where
gens = [[x:t | t <- counters (n-1) xs] | x <- xs]
f ys n = cat ys ([]:n)
cat (y:ys) (x:xs) = (y:x): cat ys xs
cat [] xs = xs
cat xs [] = [xs]
main = print $ take 10 $ filter p $ counters 3 [1..] where
p [a,b,c] = a*a + b*b == c*c
counters generates all possible counters for values from the specified range of digits, including a infinite range.
First, we obtain a list of generators of valid combinations of counters - for each permitted digit, combine it with all permitted combinations for counters of smaller size. This may result in a generator that produces a infinite number of combinations. So, we need to borrow from each generator evenly.
So gens is a list of generators. Think of this as a list of all counters starting with one digit: gens !! 0 is a list of all counters starting with 1, gens !! 1 is a list of all counters starting with 2, etc.
In order to borrow from each generator evenly, we could transpose the list of generators - that way we would get a list of first elements of the generators, followed by a list of second elements of the generators, etc.
Since the list of generators may be infinite, we cannot afford to transpose the list of generators, because we may never get to look at the second element of any generator (for a infinite number of digits we'd have a infinite number of generators). So, we enumerate the elements from the generators "diagonally" - take first element from the first generator; then take the second element from the first generator and the first from the second generator; then take the third element from the first generator, the second from the second, and the first element from the third generator, etc. This can be done by folding the list of generators with a function f, which zips together two lists - one list is the generator, the other is the already-zipped generators -, the beginning of one of them being offset by one step by adding []: to the head. This is almost zipWith (:) ys ([]:n) - the difference is that if n or ys is shorter than the other one, we don't drop the remainder of the other list. Note that folding with zipWith (:) ys n would be a transpose.
For this answer I will take "smallest" to refer to the sum of the numbers in the tuple.
To list all possible pairs in order, you can first list all of the pairs with a sum of 2, then all pairs with a sum of 3 and so on. In code
pairsWithSum n = [(i, n-i) | i <- [1..n-1]]
xs = concatMap pairsWithSum [2..]
Haskell doesn't have facilities for dealing with n-tuples without using Template Haskell, so to generalize this you will have to switch to lists.
ntuplesWithSum 1 s = [[s]]
ntuplesWithSum n s = concatMap (\i -> map (i:) (ntuplesWithSum (n-1) (s-i))) [1..s-n+1]
nums n = concatMap (ntuplesWithSum n) [n..]
Here's another solution, with probably another slightly different idea of "smallest". My order is just "all tuples with max element N come before all tuples with max element N+1". I wrote the versions for pairs and triples:
gen2_step :: Int -> [(Int, Int)]
gen2_step s = [(x, y) | x <- [1..s], y <- [1..s], (x == s || y == s)]
gen2 :: Int -> [(Int, Int)]
gen2 n = concatMap gen2_step [1..n]
gen2inf :: [(Int, Int)]
gen2inf = concatMap gen2_step [1..]
gen3_step :: Int -> [(Int, Int, Int)]
gen3_step s = [(x, y, z) | x <- [1..s], y <- [1..s], z <- [1..s], (x == s || y == s || z == s)]
gen3 :: Int -> [(Int, Int, Int)]
gen3 n = concatMap gen3_step [1..n]
gen3inf :: [(Int, Int, Int)]
gen3inf = concatMap gen3_step [1..]
You can't really generalize it to N-tuples, though as long as you stay homogeneous, you may be able to generalize it if you use arrays. But I don't want to tie my brain into that knot.
I think this is the simplest solution if "smallest" is defined as x+y+z because after you find your first solution in the space of Integral valued pythagorean triangles, your next solutions from the infinite list are bigger.
take 1 [(x,y,z) | y <- [1..], x <- [1..y], z <- [1..x], z*z + x*x == y*y]
-> [(4,5,3)]
It has the nice property that it returns each symmetrically unique solution only once. x and z are also infinite, because y is infinite.
This does not work, because the sequence for x never finishes, and thus you never get a value for y, not to mention z. The rightmost generator is the innermost loop.
take 1 [(z,y,x)|z <- [1..],y <- [1..],x <- [1..],x*x + y*y == z*z]
Sry, it's quite a while since I did haskell, so I'm going to describe it with words.
As I pointed out in my comment. It is not possible to find the smallest anything in an infinite list, since there could always be a smaller one.
What you can do is, have a stream based approach that takes the lists and returns a list with only 'valid' elements, i. e. where the condition is met. Lets call this function triangle
You can then compute the triangle list to some extent with take n (triangle ...) and from this n elements you can find the minium.