## Knights On A Keypad

### January 20, 2012

Today’s exercise is an interview question that appeared on Stack Overflow a few years ago:

The numbers on a telephone keypad are arranged thus:

`1 2 3`

4 5 6

7 8 9

0Starting from the digit 1, and choosing successive digits as a knight moves in chess, determine how many different paths can be formed of length

n. There is no need to make a list of the paths, only to count them.A knight moves two steps either horizontally or vertically followed by one step in the perpendicular direction; thus, from the digit 1 on the keypad a knight can move to digits 6 or 8, and from the digit 4 on the keypad a knight can move to digits 3, 9 or 0. A path may visit the same digit more than once.

Your task is to write a function that determines the number of paths of length *n* that a knight can trace on a keyboard starting from digit 1. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Pages: 1 2

If one takes “path” to mean a sequence of distinct nodes, the number of different paths is small and plain recursion is well fast enough. One keeps track of visited nodes in order to not step on them again.

The total numbers of paths of any length from each node are as follows.

There is no way out of (or in to) 5, and 0 stands out as its own kind, and 4 and 6 are symmetric to each other. Counting paths of length 8 or 9 is already overkill: there can be none.

Jussi: A path may visit the same digit more than once. That’s explicitly stated in the last sentence of the problem.

Somehow I managed to miss that statement. I see it now. Sorry about the noise. (Though I like to think of variations.)

Construct an adjacency matrix that represents the possible destinations from each square on the keypad. This is a 10×10 matrix of ones and zeros, where the i,j entry contains a 1 if and only if there is a knight’s move from square i to square j (let’s index from zero just for fun). Taking powers of this matrix gives a matrix containing in each i,j entry the number of distinct paths (walks, if you want to use the graph-theoretic term) from the ith square to the jth square. Since we’re only multiplying 10×10 matrices, I’d expect this to be relatively fast, even for some large n. However, if you want to get fancy with the cheese-whiz, we could diagonalize the adjacency matrix (it is symmetric by construction, so by the spectral theorem it is diagonalizable), take powers of that, and then revert it to its original form using the change of basis we got from our diagonalization process. The fact that this works is simple matrix algebra. Taking powers of an nxn diagonal matrix is linear in n, so the speed of the algorithm boils down to the speed of the decomposition, which is essentially constant (it’s only a 10×10 matrix).

Unfortunately I don’t have the time to code this up right now, but it’d be a 5 minute affair in Mathematica or any language that has the diagonalization thing built in.

Here’s my mathematica solution, using the idea from my previous comment, and Mathematica’s JordanDecomposition routine (which anyone could implement as an exercise, I’m sure).

And running it on n=10 gives the 9th matrix power (to agree with your definition of path length), and provides the correct answer:

So we can compute the right value for n=1,000,000, but the number has 359,503 digits, so it won’t fit in this comment. The nice thing is, while the dynamic approach is O(n^2), this approach is O(n)! The first result of the Timing call gives the CPU time spent on the function, which is a mere 0.076 seconds. Beat that :)

J2kun: Well, why don’t

youimplement JordanDecomposition as an exercise. If you like, email me privately, and I’ll help you write it as a guest author for a future exercise.By the way, the dynamic programming solution is

O(n), notO(n^{2}); the matrix isnalong one dimension but 9 on the other dimension. Your matrix algorithm does require much less space, however,O(1) compared toO(n).Yes of course, it’s O(n). I just saw a big table in your solution and had that knee-jerk, witch-hunting reaction, but I was mistaken.

At the risk of sounding pretentious, here are a few more notes on why my algorithm is neat: First, it counts all paths between all pairs of squares on the keypad, so it gives us much more information; and second, the same algorithm applies to counting paths in any graph. All we need to do to generalize the algorithm is change the input matrix, and we still have the lightning-fast implementation.

I’ll get back to you about writing JordanDecomposition. It’s significantly less interesting than what you can use it for, and if I remember correctly it basically reduces to row-reduction. Plus, for a general matrix, JordanDecomposition is frowned upon because it’s numerically unstable. So people usually avoid it.

You can use the matrix approach to get a closed-form expression (Maple code below).

This formula involves the golden ratio (just as for the Fibonacci numbers).

phi := (sqrt(5)+1)/2; # golden ratio

q := ;

H := ;

Phi := ;

c := 2^(n/2)/40 * q^+ . H . Phi;

simplify(eval(c, n=9)); # => 1424

You can use the matrix approach to get a closed-form expression (Maple code below).

This formula involves the golden ratio (just as for the Fibonacci numbers).

I took the brute force matrix approach:

it appears to run relatively quickly:

My solution that requires constant memory although it does not seem to be as fast as the others above.

or here:

http://codepad.org/VY2TvQjU

[…] a perfect brain teaser for a Friday night. Let’s find how many paths of a certain length a chess knight on a phone […]

Simple, with memoization.

P!

In CL, more idiomatic.

A solution in Haskell:

Here is my DP solution. It uses constant memory and takes linear time:

Here is my DP solution. It uses constant memory and is pretty fast:

Am I missing something or is the solution posted at the top of page 2 really counting the number of unique digits of length n starting at 1? I’m just going by the ‘graph’ data structure. I’d imagine that a path of length=1 would include two digits and so there should be 4 unique paths of length=1 from the number 1 (1 to 6 via 2, 1 to 6 via 4, 1 to 8 via 2, 1 to 8 via 4).

If I’m reading it correctly, to solve the problem as written I’d imagine the graph array would need to look more like this:

$movesA = array(

0=>array(4,6),

1=>array(6,6,8,8),

2=>array(7,7,9,9),

3=>array(4,4,8,8),

4=>array(3,3,9,0),

5=>NULL,

6=>array(1,1,7,7,0),

7=>array(2,2,6,6),

8=>array(1,1,3,3),

9=>array(4,4,2,2),

);

Then, instead of returning 1 if n==1 it should be modified to return 1 if n==0. Counting this way the number of unique 10-segment paths that can be drawn starting from the 1 key is 1267976.

Does that seem right?

Jeff: No, that’s not right.

First, a path with two digits has length 2, not length 1.

Second, there are two paths of length 2 starting from digit 1: 16 and 18.

Third, your successor array is incorrect. There are only two successors of 1 — 6 and 8 — not four successors. Your successor lists for 2, 3, 4, 6, 7, 8 and 9 are similarly incorrect.

@programmingpraxis And that’s certainly the problem that most of the solutions are solving, and where I would have gone if the problem had been stated something like, “determine how many different numbers can be formed of length n.”

Either way, they’re both great puzzles. Thanks for the fun excercise!

My solution in Scala, time and space in O(n):

val neighbours = Map(

0 -> List(4,6),

1 -> List(6,8),

2 -> List(7,9),

3 -> List(4,8),

4 -> List(0,3,9),

5 -> List(),

6 -> List(0,1,7),

7 -> List(2,6),

8 -> List(1,3),

9 -> List(2,4)

)

def count(n: Int): Int = {

// memoization table (DP)

val T = Array.ofDim[Int](n+1, 10)

// initialize first row

for (key <- 0 to 9)

T(0)(key) = 1

// fill in other rows

for (n <- 1 to n; k <- 0 to 9) {

T(n)(k) = neighbours(k) map { T(n-1)(_) } sum

}

`T(n)(1)`

}

Solution in Clojure: http://georgepabst.com/106505810

Am a M.C.A Student i cant understand the problem will you please help me to understand the problem?

My doubt is how to find length and the value 1424 is been calculated?

please help me awaiting for your reply

by sathish

A did one for a couple other pieces as well just to see what it looked like:

In the spirit of beating a dead horse, let me point out you can get a further speed improvement over the memoized solution by taking the transition matrix approach and exploiting the keypad symmetry to reduce the number of states. Follow this line of thought and you can reduce the problem to 2×2 matrix multiplication, which makes the code heap-access free and tiny, hence roughly 100x faster on my laptop. I timed the C# code below at 10 nanoseconds.

static long knights(int digits)

{

long V6;

long V8;

int start;

if (digits == 1)

return 1;

if( 0 == digits % 2 )

{ //even case; start with second move

V6= 1;

V8= 1;

start = 2;

}

else

{ //odd case; start with third move

V6= 3;

V8= 2;

start = 3;

}

//multiply 2×2 transition matrix

for (int i = start; i < digits; i += 2)

{

long newV6 = 2 * V8 + 4 * V6;

long newV8 = 2 * V8 + 2 * V6;

V6 = newV6;

V8 = newV8;

}

return V6 + V8;

}