I thought about it for 15 mins, but couldn’t think of any mathematical tricks. I thought of lots of minor tricks, like comparing the number to the result and not adding any more multiplications if it’s over, things that would cut 10%-20% here and there, but nothing which fundamentally changes big O running time.
For reference, here’s my solution for part 2 in smalltalk. I just generated every possible permutation and tested it. Part 1 is similar, mainly I just used bit magic to avoid generating permutations.
(even if you haven’t used it, smalltalk is fairly readable, everything is left to right, except in parens)
day7p2: in
| input |
input := in lines collect: [ :l | (l splitOn: '\:|\s' asRegex) reject: #isEmpty thenCollect: #asInteger ].
^ (input select: [ :line |
(#(1 2 3) permutationsWithRepetitionsOfSize: line size - 2)
anySatisfy: [ :num | (self d7addmulcat: line ops: num) = (line at: 1)]
]) sum: #first.
d7addmulcat: nums ops: ops
| final |
final := nums at: 2.
ops withIndexDo: [ :op :i |
op = 1 ifTrue: [ final := final * (nums at: i + 2) ].
op = 2 ifTrue: [ final := final + (nums at: i + 2) ].
op = 3 ifTrue: [ final := (final asString, (nums at: i+2) asString) asInteger ]
].
^ final
Probably the easiest optimization (which admittedly I didn’t think of myself) is to work backwards: you can eliminate multiplication and concatenation early if you start with the answer and check terms from the right.
Call it optimization and people will assume it is magic. Instead I call this a simple algebra challenge.(With part two having that quirky concatenation operation)
It is like solving for x. Let’s take an example:
3267: 81 40 27If you change it to equations:
81 (+ or *) 40 (+ or *) 27 = 3267Which effectively is:
81 (+ or *) 40 = x (+ or *) 27 = 3267So what is the x that either add or multiply would need to create 3267? Or solve for x for only two equations.
x + 27 = 3267->x = 3267 - 27 = 3240x * 27 = 3267->x = 3267 / 27 = 121(Special note since multiplying and adding will never make a float then a division that will have a remainder(or float) is impossible for x, we can easily remove multiplying as a possible choice)
Now with our example we go plug in x:
81 (+ or *) 40 = (3240 or 121) (+ or *) 27 = 3267So now we see if either 40 or 81 can divide or subtract from x to get the other number. Since it is easier to just program to use the next number in the list, we will use 40.
81 + 40 = 3240->x = 3240 - 40 = 3200 != 8181 * 40 = 3240->x = 3240 / 40 = 8181 + 40 = 121->x = 121 - 40 = 8181 * 40 = 121->x = 121 / 40 = 3.025 != 81This particular example from the website has two solutions.
For the concatenation operation, you simply check if the number ends with the number in the list. If not then a concatenation cannot occur, and if true remove the number from the end and continue in the list.
Call it pruning tree paths but it is simply algebraic.
I don’t entirely see how, you still need every possible combination of the left side to see what they would become. Plus addition and multiplication are order independent anyway
The point is to prune away search space as early as possible. If the rightmost operand is 5, say, and the answer ends in a 7, then the rightmost operator cannot be anything other than plus. This is a deduction you can’t make going left to right. Remember that in this problem the usual order of operations does not apply.
I actually did this. It did not end up being any faster than the brute-force solution since it seems to only catch the easy cases
Perhaps your implementation continues the search longer than necessary? I got curious and tried it myself. Runtime went from 0.599s (brute force) to 0.017s.