Max square size for unknown number inside rectangle

Question

If I have a set of tiles (squares) which can be any number and they are to fill a container (rectangle) of an unknown size how do I work out the maximum size of the tiles wi

Answer 1

I've managed to come up with a 'relatively' optimal solution. Partially based on Zac's pseudocode answer.

        //total number of tiles
        var tile_count : Number = numberOfSlides;
        //height of rectangle
        var b : Number = unscaledHeight;
        //width of rectanlge
        var a : Number = unscaledWidth;

        //divide the area but the number of tiles to get the max area a tile could cover
        //this optimal size for a tile will more often than not make the tiles overlap, but
        //a tile can never be bigger than this size
        var maxSize : Number = Math.sqrt((b * a) / tile_count);
        //find the number of whole tiles that can fit into the height
        var numberOfPossibleWholeTilesH : Number = Math.floor(b / maxSize);
        //find the number of whole tiles that can fit into the width
        var numberOfPossibleWholeTilesW : Number = Math.floor(a / maxSize);
        //works out how many whole tiles this configuration can hold
        var total : Number = numberOfPossibleWholeTilesH * numberOfPossibleWholeTilesW;

        //if the number of number of whole tiles that the max size tile ends up with is less than the require number of 
        //tiles, make the maxSize smaller and recaluate
        while(total < tile_count){
            maxSize--;
            numberOfPossibleWholeTilesH = Math.floor(b / maxSize);
            numberOfPossibleWholeTilesW = Math.floor(a / maxSize);
            total = numberOfPossibleWholeTilesH * numberOfPossibleWholeTilesW;
        }

        return maxSize;

What this does is to work out the total area of the rectanlge, then divide it by the required number of tiles. As each tile is a square I can SQRT this so that I get the max size of the optimal tile.

With this optimal size I then check to see how many WHOLE tiles I can fit into the width & height. Multiply these together and if it is less than the required number of tiles then I reduce the optimal size and perform the checking again until all of the tiles fit the rectanlge.

I could optimise this further by doing something like reduce the optimal size by -2 insted of -1 each time and then if all the tiles fit increase by 1 just to make sure that I've not missed a valid size. or I could jump back more than -2, say -10 then if they all tiles fit increase by 5, then if the don't fit reduce by -2 etc until I get an optimal fit.

Check out http://kennethsutherland.com/flex/stackover/SlideSorterOK.html for my example. Thanks for all the various info.