Show HN: Chessboardify – Make the grid a chessboard | Hacker News Reader

Show HN: Chessboardify – Make the grid a chessboard

chessboardifygame.xyz

95 pointsmapehe33 comments

Neat variation on LightsOut (http://chenglou.github.io/react-lights-out/). I had some trouble since the rules for which lights toggle are slightly different with this one:

Classic lights out only toggles the light you press and the 4 lights in the 4 cardinal directions. I was surprised how baked into my brain that was and how much trouble I had getting my brain to adapt.

Thats exactly how I tried solving at first.

I had a technical interview just last week wherein I was asked to write a program to determine whether a given lights out board could be solved or not! Hadn't ever seen the game before then.

My only complaint is that the alert box prevents seeing the final board. It'd be nice to get to actually see my beautiful chessboard creations.

Dismiss the alert. Tada.

Then a new round is started and the final board is removed.

That's how it behaves on my device, actually, and that indeed is how it's supposed to work. Thanks for pointing this out!

I did a little update which should address this! Thanks again!

I wanted to sketch this idea of a puzzle game that I came up with while on a walk. It turned out to roll pretty well though it couldn't be any more minimal.

Simple things are pretty powerful sometimes, don't you think?

Like Lights Out, this game can be solved using Gaussian elimination over Z/2 (integers modulo 2). This means the order of the tiles you press do not matter, and that you can solve the board by forming the desired pattern row by row.

This is interesting but I don't really understand because I've tried doing what you're saying. Maybe a side note: I'm very interested in learning algorithms but they are so inaccessible to me.

Have you taken linear algebra? It gives a structure for solving systems of linear equations and the parent post is noting that the game state is a linear function of which squares you've pressed, when considering the states of a square as either one or zero and doing arithmetic mod 2. So the standard techniques of linear algebra can solve this system, such as Gaussian elimination. However, you still have to write down the large matrix that encodes what each square does (an n x n board will need an n^2 x n^2 matrix which shows how each square affects all the others once pressed). Then you solve that system.

For this particular game there is probably a more intuitive way to directly solve it.

Sorry how does it follow that you can form the pattern row by row? When you write this down as a linear algebra problem you kinda forget about the structure of the board, and the rows of the resulting matrix you apply Gauss elimination to don't correspond to the rows of the game board. So you must have meant something else that I'm missing, but I'm confused how you would solve just the bottom row without touching any others.

For every boards are two chessboard patterns we can solve for. For example, for a 3x3 board we can have:

  101        010
  010   or   101
  101        010

where 1 is a white square and 0 is a black square.

We can represent any N×N board as a N² vector. So, we can represent one of the chessboard pattern as:

  010
  101 = 010101010 = y
  010

At the beginning of the game, the board is at some random initial pattern.

  110
  111 = 110111010 = y0
  010

The goal of the game is to find the necessary moves to form the chessboard pattern. Hence, we want to solve the equation

  (y - y0) = Mx

where M is a N²×N² matrix representing the effect of every possible moves. Pressing on a tile is equation to addition by 1 modulo 2.

  1 + 1 = 0 (mod 2)
  0 + 1 = 1 (mod 2)

For example, pressing the top left tile is the same as adding this effect vector to the board vector:

110110000

If we do this for all 9 tiles of the 3x3 board, we find the matrix M to be:

  110110000
  111111000
  011011000
  110110110
  111111111 = M
  011011011
  000110110
  000111111
  000011011

where the i-th row is the effect of pressing the i-th tile. Now, we can just solve for x (assuming M is not singular). Re-using our above example, we find

  (y - y0) = Mx
  (010101010 - 110111010) = Mx
  100010000 = Mx
  111100100 = x   (by Gaussian elimination)

This means we can press the following tiles (in any order) on the board to solve it:

  111
  100
  100

If you are interested, I wrote some quick demo code that shows how this could be implemented in Python: https://gist.github.com/avassalotti/7452318c9e9b8dec637bc7c0...

This is a very nice exposition of the strategy, but I still don't see how it answers my more specific question, about whether you can solve the problem row by row (is the grid).

Sorry, it turns that was misunderstanding from my part. It turns out we cannot solve a board row by row without revisiting the previous rows. The board below is a counterexample:

  101
  010
  100

The first and second rows are solved. However we cannot solve the last row without re-doing the first and second rows. The two solutions of this board shows this:

  solution #1:
  110
  100
  000

  solution #2:
  110
  110
  000

Actually, seeing my mistake made me challenge my assumption that a singular matrix over the reals might not be over integers modulo 2. This is likely wrong too. I don't know much about abstract algebra (and I am not a mathematician). Wikipedia (https://en.wikipedia.org/wiki/Determinant#Square_matrices_ov...) states "the reduction modulo m of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m."

The move matrix M for all boards of size 3n + 2 appears to be singular. This means these boards may have no solution or a large number of solutions.

Gotcha, thanks.

FYI, a singular matrix taking values in {0,1} that is singular over R if and only if it is also singular over Z/2, for exactly the reason you provide, that the determinant commutes with modding by 2.

Cool! I was inspired to write an implementation of the game in K: http://johnearnest.github.io/ok/ike/ike.html?gist=0007b9afbe...

I used to play something similar all the time on my Merlin back in 1978. https://en.wikipedia.org/wiki/Merlin_(game)

Am i the only one who finds it more fun to try to get all the squares the same color, rather than into a chessboard?

Neat game.

Got an excel solver for the 3x3

I'd share i'd via google spreadsheet if I could do it anonymously without creating a throwaway account

(All this in 3x3 grids, with the numbers mod 2)

- 1. Add the current light up state to the desired light state (chessboard)

- 2. Create a grid that's lit up as if from button presses from previous grid (using the game rules)

- 3. Flip the positions of the previous grid across the center. This grid is the grid of buttons to push

No idea how it works though, and it doesn't seem to work in the 4x4 case either

I just created a solver for the 4x4 case, also in google sheets... Who would have though it could handle ~15,000 rows with decent performance!

Works by calculating via formulae the result of any sequence, 6 moves or less. The bit patterns of each individual move are XORed together for each possible combination of moves.

The difference between the current state and the target is calculated, again via XOR, and then looked up in this data sheet via the query functionality, to give the sequence of moves which resulted in that delta bit pattern.

I did it in one move ;). Pretty neat game.

http://imgur.com/f5sryfs

Good job!

I think it's pretty amazing how good you get in this game once you play it a little more. At first it's simply impossible, but you start to recognize the patterns pretty quickly and you can usually beat the 3x3 ones in less than 10 moves.

You should never have to use more than 9 moves for 3x3. Hitting the same square twice has no net effect, even if the two moves on the same square are separated by other moves.

You can actually knock that down to 8 moves (maybe less?) as the max required for a solution. You never have to press the middle square, since it flips every square, inverting the board. Inversion doesn't bring you closer to the checkered pattern of the chessboard. (An inverted chessboard... is a chessboard!)

I just posted this on Facebook: https://www.facebook.com/chessboardify/?fref=nf

There is a lot of math involved in this game and you can quite quickly note some less obvious rules like this one. Maybe there are some very nontrivial, but interesting things out there. I'm not good enough in combinatorics to tell... :D

8 moves are sometimes required. If you have a chessboard 3x3 grid with the middle square the wrong color, you have to flip all squares except the middle.

Actually, a chessboard has a light colored square on the bottom right. This game accepts either a chessboard or an inverse chessboard.

Is it me, or bigger boards are much easier than 3x3? You can pretty much build those row by row.

I think it really depends on the size. Often the 5x5 feels like a breeze, but then 6x6 is a lot tougher. Probably there is some mathematical reason for this related to the divisibility of the grid side length.

What are you using Socket.IO for on the page?

The puzzles are generated and the solutions are verified on the server. I'm passing data back and forth with that.

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