Project Euler # 18

Project Euler # 18

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

3
7 4
4 6
8 5 9 3

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23

NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)

Brute Force

The obvious solution is to brute force the solution by trying every possible route, through recursion. However, since at each junction the path branches into two, there will be 214 = 16384 routes. With a 100 rows, there will be 299 rows, which is way too much to brute force!

Non – Obvious method

Logic:
Let us understand this with the help of this small triangle:

3
7 4
4 6
8 5 9 3

Each element can lead to the element directly below, or the element directly below and to the right.

Assuming a given element is a point on the solution path, the next element has to be the greatest of the two possible elements. Thus add the greatest number below a given number to it and iterate upwards. This is demonstrated on the small triangle.

Original:

3
7 4
4 6
8 5 9 3

Add bottom row elements to the top.

8 > 5

9 > 5

9 > 3

3
7 4
10 13 15
8 5 9 3

Add the new row to the row above it.

10 < 13

13 < 15

3
20 19
10 13 15
8 5 9 3

And again, add the new row, this time to the topmost row.

20  > 19

23
20 19
10 13 15
8 5 9 3

The topmost element is the largest sum!

Source in C++

Data file containing triangle (Program reads from this file)

Note: You can use this to solve problem 67 as well. Remember to change array sizes, iteration limits and the data file!

Minesweeper

When Vista came out, Microsoft did away with the classic Minesweeper game that they distributed with Windows XP. So, for my own entertainment and that of the general public, I decided to create my own implementation 😛

 

Board Screenshot

Its written in C++, with openGL for graphics, and the SOIL library for texture loading.

Menu Screenshot

Windows Installer

Source

Basketball

My very first sports game! A simple basketball game. Get the ball into the basket. No nonsense, really. The physics for this game was relatively simple. And fun. But then again, physics is always fun.

The key here is to setup a good aiming system. The game builds itself around that. One of the best solutions is to represent exactly half the parabolic path of the ball. The player will still have to judge the path of the ball, but the portion that we show on the screen would give him an idea of the velocity and angle.

Our interest is in calculating the path of the ball, which is determined by two parameters:

  • The velocity
  • The launch angle

Given that the mouse pointer represents the highest point on the parabolic path, both launch velocity and launch angle can be calculated. It’s really simple (should you have taken physics in high school).

Parabolic path of the ball

 

In the above equations R is the range of the projectile and H is the maximum height to which it rises. It’s easy to see why, for a given position of the mouse, R and H are constants that can be calculated. Rearranging and solving these equations will give us the velocity and angle which is what we need to describe the motion. The velocity will then be resolved into components, for convenience.

The game depends on a few libraries like SOIL for texture loading, freeglut for openGL and irrKlang for audio.

Coming up next is a version with highscore tables! Gotta learn a little php for that 😛

Edit: High score table is live but a little buggy.

Available on gumroad:
Basketball

Source

Tower Defense

First tower defense game 😀 Of course, it doesn’t have all the bells and whistles of commercial games. But it’s simple enough to play a couple of times.

If this project taught me anything, it’s that some things are easier with Vectors. The key to any good tower defense game is predicting the path of the enemy in order to hit!

Diagrammatic representation

Simply aiming at the enemy when you’re about to fire is no good. He isn’t likely to hang around there for two long!
The problem and its givens are listed:

We know,

  • The position of the enemy(which moves) at the time of firing
  • The position of the gun (which doesn’t move)
  • The absolute velocity (speed) of the bullet. (The direction is what we need to determine).
  • The direction and magnitude of the velocity of the enemy

My approach was as follows:

In order to hit the enemy, the final position of the bullet and the enemy must be the same. Also, in time t, the displacement of the bullet and the enemy must be equal. This is written vectorially as:

Which on expanding,


And on squaring and rearranging,

Where cos θ is given by,

Solving this equation using the quadratic formula, and selecting the positive root (since time can’t be negative, in this discussion), will give us the time after which the bullet and the enemy will collide.

Velocity of the bullet is then given by:

Keep in mind that they’re all vectors, so you need to write appropriate classes/structures to deal with them.

Game Screenshot

Source

Executable (Windows) 

Lego NXT Line Follower

Building a line follower with the Lego NXT kit is easy!

Fine tuning it, however, requires some skill. One of the most common algorithm is PID, or Proportional Integral Derivative.

Proportional says: Fix myself according to my present error

Integral says: Fix myself according to the sum off my past errors

Derivative says: Fix myself based on what the error is going to be.

There is a very good treatment of PID over here:

NXC source code

Making Paint.

Recently, I took on a project that turned out to be real fun. I attempted to recreate paint. With OpenGL handling the drawing, it was a task to find a good UI library that integrated OpenGL code. An obvious solution would be to use Win32 controls but they’re cumbersome and difficult to use. GLUI turned out to be a disappointment as well, as it refused to work with Visual Studio without the dll. Finally, FLTK provided a much needed break. It’s pretty spiffy as far as GUI libraries go, is cross platform as well.

It’s also the first time I’ve split a project onto a more than 3 source files;
Screenshots:

 

Source

Release (Windows) 

Tetris

Roses are red,
Violets are Blue,
In Soviet Russia,
Tetris plays you

Tetris is a tile-matching puzzle video game originally designed and programmed byAlexey Pajitnov in the Soviet Union. It was released on June 6, 1984. Over the years, Tetris has been released on several platforms, from Nintendo’s NES to the iPhone.

I have attempted to remake Tetris, keeping most of the basic features intact. The game features the original soundtrack as well. 🙂
My version of Tetris is written in C++. I use OpenGL (GLFW Library) to draw, the SOIL library for texture loading, and the irrKlang Library for audio.

Download the C++ source

Download the release 

Note: You need to have the VC Redist installed on your PC

Project Euler – Problem 9

Problem 9 is sort of easy too! 🙂

The problem statement is as follows:

A Pythagorean triplet is a set of three natural numbers, a<b<c, for which,

a2 + b2 = c2

For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

Knowing a little maths can take you a long way in this problem.

Mathematical Discussion

(skip this it bores you. The problem can still be solved without it)

Knowing this can cut your run time from 0.291 seconds to 0.001 seconds (as measured by the ctime library), which is basically 99.6% faster!

Basically, Pythagorean Triplets are an expression for possible integral values of the sides of a right angled triangle. Armed with this little nugget of knowledge, we can effectively reduce the iterations that we have to go through to find the required triplet.

General Algorithm

  1. We know that a + b + c = 1000.
  2. Therefore, m2 n2 + 2mn + m2 + n2 = 1000. So, 2m2 + 2mn = 1000.
  3. We now attempt to get the answer by applying brute force on the above expression, rather than plugging in all possible values of a, b and c.
  4. I believe that the brute force method merits no discussion. It has been included in the source, for comparison with the above method.

Solution in C++ 

Project Euler – Problem 8

Problem 8 of Project Euler gets a little tricky! In order to save yourself some time, you must know a little bit of file input and output in whatever language you program in.
While the number is small enough to be attacked by brute force, it can be a pain to hand-code it into an array. Copy-paste the number into a text file or download it from here.

 

The problem, as stated on the website:

Find the greatest product of five consecutive digits in the 1000-digit number.

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

General Algorithm

  1. To solve this problem, we read each digit from the file, and push it onto vector <int> number. This is done till we reach the end of the file.
  2.  Once we have the number in our vector/array, we iterate through it, 5 digits at a time, and find the product. At the end we display the maximum product.

Solution in C++

Note: Syntax for File I/O varies differs based on language.

Project Euler – Problem 7

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

Problem 7 is pretty easy too!

General Algorithm

  1. Begin testing numbers to see if they are prime, starting from 2.
  2. Break the loop when you have found 10001 prime numbers.

It’s that simple. 102,892 people have solved this problem.

Run time: 0.062 s.  Do let me know if you manage to make an improvement.

Solution in C++