The easiest way I can think of is pulling those tiles in as a view - limit it to the 6 most recently submitted tiles, and add a notice within the /node/add form to inform the user that creating a new tile will replace the oldest tile.
You could also add an extra field as a "Tile Display" - either a boolean (so an editor could go through and select the 6 they want to appear), or a number (set up a single-value number field, max value = 5, all must be unique. That way the user can go in and add a new tile, and if they want it to display first, they'd set the field to '0'; if '0' already exists, there will be a warning, and they'll have to go to the current 0-index tile to remove the value).
Your other option is to find some way to process the forms so that publishing a new tile forces the user to choose an old tile to un-publish - but that would be difficult to set up, and most likely confusing (at least for most content editors I've interacted with).