Has anyone seen a programming puzzle similar to this? - language-agnostic

Question

"Suppose you want to build a solid panel out of rows of 4×1 and 6×1 Lego blocks. For structural strength, the spaces between the blocks must never line up in adjacent rows. As an example, the 18×3 panel shown below is not acceptable, because the spaces between the blocks in the top two rows line up.
There are 2 ways to build a 10×1 panel, 2 ways to build a 10×2 panel, 8 ways to build an 18×3 panel, and 7958 ways to build a 36×5 panel.
How many different ways are there to build a 64×10 panel? The answer will fit in a 64-bit signed integer. Write a program to calculate the answer. Your program should run very quickly – certainly, it should not take longer than one minute, even on an older machine. Let us know the value your program computes, how long it took your program to calculate that value, and on what kind of machine you ran it. Include the program’s source code as an attachment.
"
I was recently given a programming puzzle and have been racking my brains trying to solve it. I wrote some code using c++ and I know the number is huge...my program ran for a few hours before I decided just to stop it because the requirement was 1 minute of run time even on a slow computer. Has anyone seen a puzzle similar to this? It has been a few weeks and I can't hand this in anymore, but this has really been bugging me that I couldn't solve it correctly. Any suggestions on algorithms to use? Or maybe possible ways to solve it that are "outside the box". What i resorted to was making a program that built each possible "layer" of 4x1 and 6x1 blocks to make a 64x1 layer. That turned out to be about 3300 different layers. Then I had my program run through and stack them into all possible 10 layer high walls that have no cracks that line up...as you can see this solution would take a long, long, long time. So obviously brute force does not seem to be effective in solving this within the time constraint. Any suggestions/insight would be greatly appreciated.

Answer 1

The main insight is this: when determining what's in row 3, you don't care about what's in row 1, just what's in row 2.
So let's call how to build a 64x1 layer a "row scenario". You say that there are about 3300 row scenarios. That's not so bad.
Let's compute a function:
f(s, r) = the number of ways to put row scenario number "s" into row "r", and legally fill all the rows above "r".
(I'm counting with row "1" at the top, and row "10" at the bottom)
STOP READING NOW IF YOU WANT TO AVOID SPOILERS.
Now clearly (numbering our rows from 1 to 10):
f(s, 1) = 1
for all values of "s".
Also, and this is where the insight comes in, (Using Mathematica-ish notation)
f(s, r) = Sum[ f(i, r-1) * fits(s, i) , {i, 1, 3328} ]
where "fits" is a function that takes two scenario numbers and returns "1" if you can legally stack those two rows on top of each other and "0" if you can't. This uses the insight because the number of legal ways to place scenario depends only on the number of ways to place scenarios above it that are compatible according to "fits".
Now, fits can be precomputed and stored in a 3328 by 3328 array of bytes. That's only about 10 Meg of memory. (Less if you get fancy and store it as a bit array)
The answer then is obviously just
Sum[ f(i, 10) , {i, 1, 3328} ]

Answer 2

Here is my answer. It's Haskell, among other things, you get bignums for free.
EDIT: It now actually solves the problem in a reasonable amount of time.
MORE EDITS: With a sparse matrix it takes a half a second on my computer.
You compute each possible way to tile a row. Let's say there are N ways to tile a row. Make an NxN matrix. Element i,j is 1 if row i can appear next to row j, 0 otherwise. Start with a vector containing N 1s. Multiply the matrix by the vector a number of times equal to the height of the wall minus 1, then sum the resulting vector.
module Main where
import Data.Array.Unboxed
import Data.List
import System.Environment
import Text.Printf
import qualified Data.Foldable as F
import Data.Word
import Data.Bits
-- This records the index of the holes in a bit field
type Row = Word64
-- This generates the possible rows for given block sizes and row length
genRows :: [Int] -> Int -> [Row]
genRows xs n = map (permToRow 0 1) $ concatMap comboPerms $ combos xs n
where
combos [] 0 = return []
combos [] _ = [] -- failure
combos (x:xs) n =
do c <- [0..(n `div` x)]
rest <- combos xs (n - x*c)
return (if c > 0 then (x, c):rest else rest)
comboPerms [] = return []
comboPerms bs =
do (b, brest) <- choose bs
rest <- comboPerms brest
return (b:rest)
choose bs = map (\(x, _) -> (x, remove x bs)) bs
remove x (bc#(y, c):bs) =
if x == y
then if c > 1
then (x, c - 1):bs
else bs
else bc:(remove x bs)
remove _ [] = error "no item to remove"
permToRow a _ [] = a
permToRow a _ [_] = a
permToRow a n (c:cs) =
permToRow (a .|. m) m cs where m = n `shiftL` c
-- Test if two rows of blocks are compatible
-- i.e. they do not have a hole in common
rowCompat :: Row -> Row -> Bool
rowCompat x y = x .&. y == 0
-- It's a sparse matrix with boolean entries
type Matrix = Array Int [Int]
type Vector = UArray Int Word64
-- Creates a matrix of row compatibilities
compatMatrix :: [Row] -> Matrix
compatMatrix rows = listArray (1, n) $ map elts [1..n] where
elts :: Int -> [Int]
elts i = [j | j <- [1..n], rowCompat (arows ! i) (arows ! j)]
arows = listArray (1, n) rows :: UArray Int Row
n = length rows
-- Multiply matrix by vector, O(N^2)
mulMatVec :: Matrix -> Vector -> Vector
mulMatVec m v = array (bounds v)
[(i, sum [v ! j | j <- m ! i]) | i <- [1..n]]
where n = snd $ bounds v
initVec :: Int -> Vector
initVec n = array (1, n) $ zip [1..n] (repeat 1)
main = do
args <- getArgs
if length args < 3
then putStrLn "usage: blocks WIDTH HEIGHT [BLOCKSIZE...]"
else do
let (width:height:sizes) = map read args :: [Int]
printf "Width: %i\nHeight %i\nBlock lengths: %s\n" width height
$ intercalate ", " $ map show sizes
let rows = genRows sizes width
let rowc = length rows
printf "Row tilings: %i\n" rowc
if null rows
then return ()
else do
let m = compatMatrix rows
printf "Matrix density: %i/%i\n"
(sum (map length (elems m))) (rowc^2)
printf "Wall tilings: %i\n" $ sum $ elems
$ iterate (mulMatVec m) (initVec (length rows))
!! (height - 1)
And the results...
$ time ./a.out 64 10 4 6
Width: 64
Height 10
Block lengths: 4, 6
Row tilings: 3329
Matrix density: 37120/11082241
Wall tilings: 806844323190414
real 0m0.451s
user 0m0.423s
sys 0m0.012s
Okay, 500 ms, I can live with that.

Answer 3

I solved a similar problem for a programming contest tiling a long hallway with tiles of various shapes. I used dynamic programming: given any panel, there is a way to construct it by laying down one row at a time. Each row can have finitely many shapes at its end. So for each number of rows, for each shape, I compute how many ways there are to make that row. (For the bottom row, there is exactly one way to make each shape.) Then the shape of each row determines the number of shapes that the next row can take (i.e. never line up the spaces). This number is finite for each row and in fact because you have only two sizes of bricks, it is going to be small. So you wind up spending constant time per row and the program finishes quickly.
To represent a shape I would just make a list of 4's and 6's, then use this list as a key in a table to store the number of ways to make that shape in row i, for each i.