 # Jezen Thomas

At a previous company off-site, a colleague shared a mathematical brain teaser with the team:

With the numbers 123456789, make them add up to 100. They must stay in the same order but you can use addition, subtraction, multiplication, division, brackets etc. All numbers must be used exactly once.

I’m admittedly not great at arithmetic, and I didn’t make much progress when trying to brute-force a solution in my head. Naturally I decided to reach for my big hammer: Haskell.

The first idea I had to attack this problem was to enumerate all possible expressions you would have as a result of combining the different permitted mathematical operators, slotted between each of the digits.

Whenever I need to generate all the combinations of things, I usually reach for Haskell’s list comprehensions. For example, if you wanted to generate all possible pairs of the numbers 1, 2, and 3, you might write the following list comprehension:

[ (x, y) | x <- [1..3], y <- [1..3] ]

-- evaluates to:
--   [(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)]

Applying this approach to the maths riddle is essentially the same:

let ops = [ '+', '-', '/', '*' ]
in [ [ '1', a, '2', b, '3', c, '4', d, '5', e, '6', f, '7', g, '8', h, '9' ]
| a <- ops, b <- ops
, c <- ops, d <- ops
, e <- ops, f <- ops
, g <- ops, h <- ops
]

Each of the digits and mathematical operators are represented as character values, because Haskell lists are homogenous — all values must be of the same type. The let binding cleans up what would otherwise be quite a bit of tedious repetition.

Evaluating this expression generates a whole bunch of mathematical expressions encoded in strings — 65,536 expressions to be exact. The expressions look like this:

[ "1*2*3*4/5-6-7*8-9"
, "1*2*3*4/5-6-7*8/9"
, "1*2*3*4/5-6-7*8*9"
, "1*2*3*4/5-6/7+8+9"
-- etc.

Now that we have a collection of expressions, we need to evaluate each of them and find those which evaluate to 100. Evaluating string values as code is something that even a decade ago in the JavaScript world would have been frowned upon. JavaScript provides the eval() function for this, but conventional wisdom states that eval() is evil, and should never be used.

I beg to differ.

But even still, eval() is just this gross hacky thing that only exists in languages like JavaScript, right? Surely you can’t do the same thing in a pure, ivory-tower language like Haskell, right?! As it turns out, you can!

If you add the hint package, you can do something like this:

import Language.Haskell.Interpreter

f = runInterpreter $do setImports ["Prelude"] eval "3 + 5" -- evaluates to Right "8" Pretty cool, if not somewhat heretical. Essentially all that’s left to do is plug the 65,536 mathematical expression strings into the machinery that evaluates them, and filter the results to only those where the result is 100. Here’s what I came up with: import Control.Monad (forM) import Language.Haskell.Interpreter expressions :: [String] expressions = let ops = [ '+', '-', '/', '*' ] in [ [ '1', a, '2', b, '3', c, '4', d, '5', e, '6', f, '7', g, '8', h, '9' ] | a <- ops, b <- ops , c <- ops, d <- ops , e <- ops, f <- ops , g <- ops, h <- ops ] result = runInterpreter$ do
setImports ["Prelude"]
exprs <- forM expressions evaluate
pure $filter (\(_, a) -> a == "100")$ fromRight [] exprs
where
evaluate expr = eval expr >>= \a -> pure (expr, a)

The above is an expanded version of what I originally wrote. When I was playing with this, I actually wrote it as a one-liner directly in GHCi, which is a similar experience to composing a Unix command line. Is my approach an efficient way to find the possible answers? No. Do I care enough to optimise my approach? Also no. Who said Haskell can’t do quick and dirty scripting work?

Running this produces 14 possible answers, and that’s without enumerating all those possible answers that would result from changing the order of operations with brackets. After submitting my answers, my colleague rejected my approach because I “used a program”, and “that’s cheating.”

$$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 * 9$$ $$1 + 2 + 3 - 4 * 5 + 6 * 7 + 8 * 9$$ $$1 + 2 - 3 * 4 + 5 * 6 + 7 + 8 * 9$$ $$1 + 2 - 3 * 4 - 5 + 6 * 7 + 8 * 9$$ $$1 + 2 * 3 + 4 * 5 - 6 + 7 + 8 * 9$$ $$1 - 2 + 3 * 4 * 5 + 6 * 7 + 8 - 9$$ $$1 - 2 + 3 * 4 * 5 - 6 + 7 * 8 - 9$$ $$1 - 2 * 3 + 4 * 5 + 6 + 7 + 8 * 9$$ $$1 - 2 * 3 - 4 + 5 * 6 + 7 + 8 * 9$$ $$1 - 2 * 3 - 4 - 5 + 6 * 7 + 8 * 9$$ $$1 * 2 * 3 + 4 + 5 + 6 + 7 + 8 * 9$$ $$1 * 2 * 3 - 4 * 5 + 6 * 7 + 8 * 9$$ $$1 * 2 * 3 * 4 + 5 + 6 + 7 * 8 + 9$$ $$1 * 2 * 3 * 4 + 5 + 6 - 7 + 8 * 9$$