Pythagoras

Project Euler 94: Investigating almost equilateral triangles with integral sides and area.

Project Euler 94: Investigating almost equilateral triangles with integral sides and area.

I can realy feel that the problems become harder and harder. Problem 94 of Project Euler took me a good while to figure out. The problem is easy to understand and reads

It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the
5-5-6 has an area of 12 square units.

We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than one unit.

Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000).

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Posted by Kristian in Project Euler, 25 comments
Project Euler 91: Find the number of right angle triangles in the quadrant

Project Euler 91: Find the number of right angle triangles in the quadrant

In Problem 91 of Project Euler we are dealing with a geometry problem which reads

The points P (x1, y1) and Q (x2, y2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form ΔOPQ.

There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between 0 and 2 inclusive; that is,
x1, y1, x2, y2  2.

Given that 0 ≤ x1, y1, x2, y2  50, how many right triangles can be formed?

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Posted by Kristian in Project Euler, 1 comment
Project Euler 75: Find the number of different lengths of wire that can form a right angle triangle in only one way.

Project Euler 75: Find the number of different lengths of wire that can form a right angle triangle in only one way.

Assuming we have solved the problems from one end, we are now at Problem 75 of Project Euler which should unlock the 75 problem achievement as well as the Pythagorean Triplet achievement. The problem description reads

It turns out that 12 cm is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.

12 cm: (3,4,5)
24 cm: (6,8,10)
30 cm: (5,12,13)
36 cm: (9,12,15)
40 cm: (8,15,17)
48 cm: (12,16,20)

In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found; for example, using 120 cm it is possible to form exactly three different integer sided right angle triangles.

120 cm: (30,40,50), (20,48,52), (24,45,51)

Given that L is the length of the wire, for how many values of L ≤ 1,500,000 can exactly one integer sided right angle triangle be formed?

Note: This problem has been changed recently, please check that you are using the right parameters.

I have already noted that we should unlock the Pythagorean triplet achievement, and that is for a reason. In order to get a good solution we need to revisit the theory we used in Problem 9 about Pythagorean triples. Continue reading →

Posted by Kristian in Project Euler, 18 comments

Project Euler 39: If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p ≤ 1000, has the most solutions?

Even though the title nor the description of Problem 39 of Project Euler mentions Pythagorean Triplets, this is the topic we are revisiting. The problem description reads

If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.

{20,48,52}, {24,45,51}, {30,40,50}

For which value of p ≤ 1000, is the number of solutions maximised?

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Posted by Kristian in Project Euler, 5 comments

A 1000 Pythagorean triplets – Problem 9

Today’s problem in Project Euler, is related to probably THE most well known theorem of all times.  Pythagoras theorem stating that Geometric interpretation of Pythagoras

The square of the hypotenuse is the sum of squares of the two other sides.

It can also be stated in a more geometrical way as

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle

Which can also be shown in a graphical sense, on the figure to the right, where the blue area equals the orange area.

But enough with the background info, the problem reads

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.

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Posted by Kristian in Project Euler, 36 comments