Recursion: Count Change. The number of ways to change amount A is equal to. The number of ways to change amount A using all but the largest coin, plus; the number of ways to change amount A - D using all kinds of coins, where D is the denomination of the largest kind of coin.

15 Puzzle Game

Tile Sliding Puzzle(Code in Python) ©Knightwing #defining cmp def cmp(a,b): return (ab)-(asliding puzzle game'). Given a puzzle board, return the least number of moves required so that the state of the board is solved. If it is impossible for the state of the board to be solved, return -1. Examples: Input: board = 1,2,3,4,0,5 Output: 1 Explanation: Swap the 0 and the 5 in one move. The aim of this challenge is to use HTML, CSS and Javascript to create an interactive sliding puzzle. Step 1: Creating 9 tiles from a single picture, using CSS Our aim is the use a single picture file called flower.png and to create 9 CSS classes for each tile of the picture as displayed below: We will create our tiles using DIV tags. We will use the flower.png picture file as the background. 15 Puzzle Game This game is the 15 Puzzle Game. In this game, there is a 4.4 board with 15 numbers and an empty square. The numbers are then shuffled randomly. The goal of the game is to move the numbers in such a way that the numbers are ordered again as shown in the picture below. The rules are simple. You can only move tiles into the empty.

This game is the 15 Puzzle Game. In this game, there is a 4*4 board with 15 numbers and an empty square. The numbers are then shuffled randomly. The goal of the game is to move the numbers in such a way that the numbers are ordered again as shown in the picture below. The rules are simple. You can only move tiles into the empty tile space.

First of all we need to make a class to represent this board:

Now we need a way to print this graph on the console of Python:

When you print the board method on the console, it should look like this:
An important thing to remember when you are writing __repr__ is that what you return MUST be a string.
Now, after we have a way to represent our board, we need to be able to do the basic things the player can do. I need to be able to move the empty block up, down, right, and left.
These 4 methods are the methods in the class that are going to be used to move the empty block. This is when the position of the empty block comes in handy.
We use a try and except to make sure we don't move off the board.
If you have an array and you take the index of [-1], you get the last element of that array. We don't want that to happen and therefore we put an if statement to make sure that the empty block is not at the top of the board.
If it passes these condition, then we can actually move the empty block. We subtract 1 from the row and make that index to be the empty block. We swap the empty block with the previous number that the new empty block has just replaced. We do this for all 4 directions.
After moving the board should look like this:
We have gotten the basic movements down. Let's make this a real game. In order to do that, we need a way to shuffle the whole board such that there is a game to solve.
This method will be used to mix the board randomly. I generate a random number between the range 1 to 5 including 1 and excluding 5 which gives me a total of 4 possible numbers. I associate these 4 numbers to each move (up, down, left, right).
Board after shuffled 100 times:
Now that we have a game to play, here comes the complicated part: to be able to write a function to solve this. The way we're going to solve this is using breadth first search which finds the shortest way to get to the goal.

Breadth First Search

There are three types of searches : Breadth First Search, Uniform Cost Search, and Depth First Search. In depth first search, you do not care about whether or not there are nodes of different levels. Depth first search always goes deeper down the node tree. Breadth First Search on the other hand explores the node with the most shallow level. As a result, breadth first search reaches the shortest path but depth first search is more efficient on memory. Uniform cost search is for weighted graphs when you go explore the next node that costs the least. For this program, the most applicable search is the Breath First Search since we want to find the shortest path to reach our goal.
Implementation of Breath First Search:

In this method called solve, we will output a list of strings that give you directions in order to solve the board ('up', 'down', 'left', 'right'). First thing you have to notice is that you MUST import Queue with the line:

Visualizing the Problem

In this program, each state that the board is in is a node. We are comparing node to node and checking until our current node is the same as the goal board, which means all the numbers are in order.

The Variables:

start stores the original list used for the board when we initialized this class and is the first node that is explored
pred is used in the method at the very end to create the directions to solve the board
visited stores all the nodes that we have already checked/visited and explored through
frontier is a Queue that stores our plan for which node to explore next.
NOTE : Frontier is a Queue which means first in first out

Parts

The first part of this method is the first if statement which checks whether or not our current node is the goal node that we are trying to achieve. If it is, we use our current dictionary and create a list of directions and return that list.

The Second Part:

The second part of this method is the second if statement which adds new nodes to our Queue such that we an explore them. First of all, we have to make sure that we are not visiting the same node twice so we check if our current node is in the visited array. Then there are 4 miniparts for each direction. These miniparts don't vary a lot. From one node of the board I can get to 4 new nodes by moving 4 different directions. I store these 4 possible nodes into the Queue such that I can explore them in the future. Then I add the directions to the dictionary such that, when I do find the solution, I can refer back to the dictionary 'pred' and create a list of directions.

NOTE : Some methods in the solve method are not shown for simplicity. At the bottom of this blog, you will be able to see the whole program with these small methods.

COMPLETE PROGRAM:

July 2017

Summary: Based on a Forth pentomino puzzle solver that my dad wrote and refined over the years, I wrote a version in Python. It uses a brute force (but clever) approach using code generation to find all 2339 tilings of the 12 free pentominoes on a 6x10 board.

Programming and pentominoes

My now-retired father was a full-time pastor and a hacker on the side. His enthusiasm for nerdy things like writing Forth compilers and low-level graphics libraries got me into programming when I was about 14. In fact, the first major program I wrote was Third, a self-hosting Forth compiler with the primitives written in 8086 assembly.

Over the years, Dad also wrote various incarnations of a program to find all solutions to the pentomino puzzle. Pentominoes are kind of like Tetris pieces, but with five squares instead of four. There are 12 “free” pentominoes, meaning free to rotate or be mirrored, 18 “one-sided” shapes if you include mirroring, and 63 shapes if you include mirroring and rotations.

Here’s a picture of the 18 shapes, with each of the 12 free pentominoes shown in a different color:

Each of these 12 pieces has a more-or-less standardized letter to name it, kind of corresponding to the piece’s shape. In the diagram above, the first row shows the I, F, L, P, and N pieces. The second row shows the T, U, V, W, X, Y, and Z pieces.

If you’re patient, you can fit these 12 five-square pentominoes exactly on a 6x10 board – or a 5x12 or 4x15 or 3x20 board. There are exactly 2339 solutions to the 6x10 puzzle (excluding rotations and reflections), but it’s usually hard for a person to find even one – it usually takes a newbie an hour or two.

Spoiler alert – here’s an example solution on a 6x10 board:

Dad’s solver

Dad started programming back in the punch card days, when you punched out your code on a stack of cards, and then submitted them and waited for your batch to run and produce results overnight on the university mainframe.

His first solution to the puzzle was slow, and didn’t finish in his 20-minute overnight timeslot. The algorithm was pretty much what you’d expect for

Slide Puzzle Python Code Examples