while (three * i < 999); In our Python function, sumn() (shown below), this is accomplished by using the floor function on n divided by d to find the number of nonzero terms. Problem 1 doesn't need any coding whatsoever. }. 233168 Solutions to the first 40 problems in functional Python. It can get 23 if the stop number is changed, and will print 23, but when the stop number is 1000 I get 266333. x=sum(range(0, 10, 3)) C and C++ are unsafe because of the lack of array bounds checking, having signed integer overflow, and having tons of easily invoked undefined behaviors. Problem 1: Multiples of 3 and 5 If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. I do use codechef var total = 0; It should be a local variable. Besides I dont think it is fair to give out spoilers like that when ever then contest is running. The compiler will most likely throw that line away, but better don't include it. But somewhere deep down it should satisfy something in you to do this. Problem. solutions of the Diophantine equation a 6 + b 6 = c 6 . { Over time, the Python code was adapted to fit the characteristics of the language such as using idiomatic/Pythonic approaches, tweaking or changing algorithms to increase speed (whereas Java can sometimes get away with less efficient but simpler algorithms), and making heavy use of generators. Every Java solution depends on these shared classes: EulerSolution.java, Library.java. Your first for loop uses an optimization, your second doesn't. As the top row increases, the bottom row decreases, so the column sum always stays the same, and well always have two rows and n/2 columns for any numbern. If n is odd then start with zero to keep the columns paired. } If you would like to tackle the 10 most recently published problems, go to Recent problems. Calculating the number of beans in this rectangle built from the two triangles was easy. Thanks a lot sir. while ( j < 1000) { total += i ; Your code is very hard to read since there is so much empty space. A more general explanation on arithmetic progression is given on wikipedia. He argued that the best way to discover how many beans there were in a triangle with 100 rows was to take a second similar triangle of beans which could be placed upside down and adjacent to the first triangle. However, programming is more than the language, there is a whole lot to learn about algorithms and data structures which is almost generic regardless of the language you program in. I understand. Splendid Job!!! Project Euler Problem 1. If it is possible, could you please take a look and share what your approach could have been? Such an amazing alternate solution! 6 Project Euler #6 - Sum Square Difference 7 Project Euler #7 - 10001st prime Continuing the wonderful community solutions to Project Euler. To run a Haskell solution, run the Haskell file as a main program. Thanks for the help. if (((i % 3) == 0) || ((i % 5) == 0)) { The algorithms between different languages are not exactly the same; instead I try to write code that is most idiomatic and clear for the given language. ), and possibly improve the speed (it's . The summation formula is the legacy of Carl Friedrich Gauss, the German mathematician. http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/. Stack Overflow for Teams is moving to its own domain! 1. On the math side, I dont think there is any one book or series of books that will lead you to be able to solve these problems. It's quite a simple problem, You can get some . Integer index = 1; j+=5; Did Dick Cheney run a death squad that killed Benazir Bhutto? sum=sum+index5; } This problem involves finding the sum of two related arithmetic sequences. The sum of numbers divisible by 3 or 5 between 1 and 999999 is 233333166668 { This is Problem 5, finding the smallest multiple. Dude you are awesome! Now Gauss had a rectangle with 100 rows containing 101 beans each. k = 5 [edit] 27 5 + 84 5 + 110 5 + 133 5 = 144 5 . If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. Description / Title. This is my python solution to the first problem on Project Euler: n = 1 rn = 0 while n < 1000: if n%3 == 0 or n%5 == 0: rn += n n = n + 1 print (rn) I would like to find a way to keep everything in this python code to as little number of lines as possible (maybe even a one liner?? The sum of these multiples is 23. Anyway, here is what I ended up []. I found this book, http://www.amazon.com/Friendly-Introduction-Number-Theory-Edition/dp/0321816196/, It is pretty expensive, but from the chapter I have skimmed, it looks pretty good. it is even more difficult to actually recommend something without knowning your current level. Overview The problem is short and easy to understand: If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. where n is the divisor, and p is the greatest number we want to check. Console.WriteLine("The sum of multiples of 3 and 5 number is " + (Result )); http://www.mathblog.dk/project-euler-prolog/, http://www.mathblog.dk/stopwatch-a-timing-function-in-c/. HackerRank increases the upper bound from 1,000 to 1 billion and runs 10,000 test cases. The sum of these multiples is 23. There are multiple methods for finding the solution for this problem. my JS code(runs fast enough)-, var num = 1000; 7. 0.01000356674194336 seconds , Digits distribution pattern in the sums of multiples of 3 and 5, Ex: 969220. The first problem of project Euler found here, below is the problem for quick lookup. For every problem that I solved, I have a Java solution for it (and possibly code in other languages as well). I am new to C# and I cannot figure out ow to call each function ( Solve() and SumDivisibleBy() ) from my static void Main() method. There are in total 100 101 = 10,100 beans, so each triangle must contain half this number, namely 10,100/2 = 5,050. My code tends to be quite short: one-liners are very common, and typically the solution is less than 5 lines of code. I am stuck with the problem DRANGE (Range of Data). >>> for i in range(1000): I would be grateful if you do. I write Mathematica code in a rather plain style, using only [] for function application (not @ or //), avoid pattern processing, and avoid declaring functions with the #-and-& syntax. Glasgow Haskell Compiler 7.10.3, compiling with -O option to 64-bit executables, Intel Core i5-4690 (Haswell) 3.50GHz, Ubuntu Linux 16.04 (64-bit). There is no need to make total global. Where is the problem? The sum of these numbers is 3 + 5 + 6 + 9 = 23. I will see if I can come up with some interesting reads later on. This solution is much faster than using brute force which requires loops. After that I should go on to the books you referred as containing lots of number theory,LA,stats..etc. I am trying the problems in August Challenge. can you please suggest me other sites where i can practice c pragramming My plan is to post a solution to all the ProjectEuler.net problems I have managed to solve. Hello, can you tell me how I can make this code work? 1 Link If you do class (prod) you will see that it is a double data type. Etc, Sorry but i can not understand why we subtract Is my style any good? The problems archives table shows problems 1 to 804. The following for loop is easier to understand: Concerning your comments on +=, did you accidentally use total += total + j? I simply dont have time. If everything else fails and you feel completely stuck, just take a break. 2 OK. int sum = 0; um That is exactly what I do in my first solution. Integer sum=0; Mathematics (from Ancient Greek ; mthma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (), shapes and the spaces in which they are contained (), and quantities and their changes (calculus and analysis).. One of my personal favorites for that topic is this one Have you read this post http://www.mathblog.dk/project-euler-prolog/ as it gives you are little background for the the pieces of code you have to wrap around the functions I provide here in order to run. Project Euler Problem 1 Solution. Being a cs undergrad I am pretty sure you are already having quite a few courses in different subjects. How many characters/pages could WordStar hold on a typical CP/M machine? Remove it, along with an entire else clause. Note that the benchmark does not attempt to be fair in any way. Leading a two people project, I feel like the other person isn't pulling their weight or is actively silently quitting or obstructing it. Project Euler Full Solutions Project Euler > Problem 1 > Multiples of 3 and 5 (Java Solution) Project Euler > Problem 2 > Even Fibonacci numbers (Java Solution) Project Euler > Problem 3 > Largest prime factor (Java Solution) Project Euler > Problem 4 > Largest palindrome product (Java Solution) Unable to edit the page? The natural numbers below 10 that are multiples of 3 or 5 results are 3, 5, 6 and 9. Why not floor(1000/3) = 333. My code requires Python 3 (but old versions can be found that support both 2 and 3). using c = System.Console; public class Test Many Python solutions depend on my shared math library module: eulerlib.py. Sometimes, slightly inefficient constructs (such as list comprehensions) are used for the sake of clarity. Unfortunately due to Pythons slow performance on arrays and machine-word-sized integer values, many solutions are not worth the time to run it compared to the Java implementations. You will come back with new and fresh ideas. Closed. The first advice here, is to have fun. 3*1+3*2++3*333=x Ans=x+y-z. Thanks for the tipsVery usefulgave me a direction to go upon. if(((i%3)==0||(i%5)==0)) etc. Find the sum of all the multiples of 3 or 5 below 1000. }. int three_total,five_total,Result = 0; //three It should be a local variable. View Problem on Project Euler. I solve Project Euler problems to practice and extend my math and programming skills, all while having fun at the same time. } Among the web, this is perhaps the largest collection of Project Euler solutions in the Java programming language. I just want to ask where should I start from,(Confused)As theres lot just then programming U hv to learn Data,Ada etc then U must knw Maths There are multiple methods for finding the solution for this problem Bruteforcing { And no you havent offended me by asking this kind of question. Integer t = 1000;//scan.nextInt(); That said, ProjectEuler problems are more about math than programming. Etc, Output of the results using extension of RosettaCode in C#, https://rosettacode.org/wiki/Sum_multiples_of_3_and_5#C.23, The sum of numbers divisible by 3 or 5 between 1 and 9 is 23 But what actually worked was using a static accumulator variable. Here we use integer division, which means that we will discard the fractional part of the result. Bento theme by Satori, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), Click to share on Reddit (Opens in new window), http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/, http://www.mathblog.dk/programming-challenges/, http://www.math.brown.edu/~jhs/frint.html, Project Euler Problem 1- Find the sum of all the multiples of 3 or 5 below 1000. N is the highest number less than p divisible by n. In the case of 5 this is 995. {1-10000} [23331668] long long j = n*(p/n)*((p/n)+1)/2; Use the same trick for both loops. Solutions for HackerRank's wonderful (and often mind-bending) expanded versions of the Project Euler (projecteuler.net) problem archive. When I do use it, I end up learning many new concepts in functional programming, such map/filter/fold, infinite lists, converting iterative algorithms to recursive functions, memoization, etc. Welcome to my solutions for Project Euler. I like using Java because it is fast, safe, and expressive. linear algo. If the current number is divisible by either 3 or 5 then add it to an accumulator (the total variable). Since we are looping through the all the numbers, and a two comparisons and possibly one addition is needed for each number, the algorithm scales linearly with p , which in Big O notation can be expressed as O(n), not a bad scalability as a such. Pseudocode, stub code . Each problem that I solved always includes a Java program. My Account; My Community Profile; Link License return 0; Your first for loop uses an optimization, your second doesn't. That's asymmetrical. sum+= (3*i); This will give wrong results. There is no need to make total global. 'It was Ben that found it' v 'It was clear that Ben found it'. A formula attributed to Carl Friedrich Gauss will calculate the sum of the first n natural numbers. This solution can handles the sum below any given number almost equally fast, if the sum can be stored in an integer. So you are meant to use coding not your head ? >>> start_time = time.time() How to help a successful high schooler who is failing in college? One way we can check if 3 is a divisor of x (which is declared as integer) is by the following line. The reason for the wrong results could happen due to integer overflow. The sum of numbers divisible by 3 or 5 between 1 and 9999 is 23331668 When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. There are a whole lot of number theory in them, but also some linear algebra, some statistics and so on. while ( i < 1000) Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The description of problem 1 on Project Euler reads. Occasionally I write imperative code in Mathematica, usually for unbounded searching. It is not currently accepting answers. 01.02.2021. 15)? In Mathematica, I make heavy use of nested expressions, functional programming core functions like Map and Select, and aggregation functions like Total, Length, Sum. >>> Is there a topology on the reals such that the continuous functions of that topology are precisely the differentiable functions? Im a beginner at Haskell programming, and only know how to use it to solve the easier problems in Project Euler. Thanks. i+=3; http://projecteuler.net/index.php?section=problems&id=1, http://projecteuler.net/index.php?section=problems&id=2, http://projecteuler.net/index.php?section=problems&id=3, ProblemSets/Project Euler Solutions (last edited 2012-05-31 06:16:44 by 123). Improve your writing skills in 5 minutes a day with the Daily Writing Tips email newsletter. If what you are really looking for is some programming challenges to throw yourself at, there are a lot of options which are less math focused then Project Euler, not that I will try to discourage you by any means. To run a Mathematica solution, copy the entire code into a new Mathematica notebook and evaluate the whole block of text. The sum of these multiples is 23. An example of integer division is 10/3 = 3. If the problems were small you could just make an array, but I am not sure that is a feasible approach since the N can be rather large. Happy coding!!! This is a typical problem that demonstrates the use of partitions which can be solved by using dynamic programming. Updated: February 26, 2018. The sum of these multiples is 23. However on the flip side, I prefer to solve a problem in Mathematica first if its possible. However the geometric approach has a constant computation time, which is expressed as O(1), which is obviously better . Thanks and sorry for bothering you. ID. http://www.math.brown.edu/~jhs/frint.html, umi dont know why no ones mentioned it yet but why not use modulo? Reduces the iteration from 1000 times to (333 +200) = 533. i.e. The teacher was surprised when he looked at the tablet to find the correct answer 5,050 with no steps in the calculation. , I was looking for something about number theory which is a very integral part of the Project Euler problems. Video Version { If you want a refresher about what is being asked: Multiples of 3 and 5 If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. a relatively simple pattern is obtained: The sum of numbers divisible by 6 or 10 between 1 and 9 is 6 var answer = range.Select(r => (x % r) == 0); Project Euler problem #1 solution in C [closed] By maria Posted on September 29, 2019. Some solution code contains a detailed mathematical proof of correctness. Thanks for replying Heres how this formula works for n=10. x += i x=sum(range(0, 1000, 3)) Find the sum of all the multiples of 3 or 5 below 1000. 3 more parts. With python we can express the sum like : Viewed 329 times 0 $\begingroup$ Here is the problem. Find the sum of all the multiples of 3 or 5 below 1000. Project Euler is a collection of mathematical problems, mostly revolving around number theory, that might also require some level of programming skill to solve. Find the sum of all the multiples of 3 or 5 below the provided parameter value number. Problem 1: Add all the natural numbers below 1000 that are multiples of 3 or 5. Find the sum of all the multiples of 3 or 5 below 1000. In the first bit of code we check if a number was divisible by 3 and/or 5, and this way we only checked each number once. This is a typical application of the inclusionexclusion principle. Wow!! So I think U suggested me to first study all the Programming concepts(Frm the book U suggested for efficient progrmm. Solution took 0 ms, ##the answer will be 234168, not 233168 as given here, Solution to Project Euler, Problem 1, using Python (v.3.6.1), >>> import time So, we need to find a more efficient way of calculating this sum without looping. Though I know they would be giving the editorials out when the contest ends, I do not find their explanation as helpful as I have found your explanation for the project euler problems. { sum(unique([3:3:999,5:5:999])). Found footage movie where teens get superpowers after getting struck by lightning? Solutions In Python Any suggestions or comments? int five = 5; LLPSI: "Marcus Quintum ad terram cadere uidet. The whole code can be written as, int result = 0; thanks mr Kristian for such best explanation with different approaches. Func MultiplesOf = (x, range) => Project Euler - Problem 1 Solution | Erhan Kl Javascript, News, Php, Project Euler Solutions Project Euler - Problem 1 Solution Problem: If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. How to distinguish it-cleft and extraposition? Data Science Project Euler RStats This first Euler problem provides a gentle introduction to solving mathematical problems with code. rev2022.11.3.43003. Why is proving something is NP-complete useful, and where can I use it? Thank you for a great question which also warrants a great answer. The solution is shown in the output. Project Euler (7 Part Series) 1 Project Euler #1 - Multiples of 3 and 5 2 Project Euler #2 - Even Fibonacci numbers . Also, my Java solutions tend to be long due to types on every variable, lots of custom code, and low-level loops instead of higher-order functions. Ist problem with your solution :1) You want multiples of 5 which are less than 1000. j <= 1000 is not the correct condition.This condition will include the value 1000 too. Sign up for the Mathblog newsletter, and get updates every two weeks. You probably already have an official study plan, so things you do besides that should be fun. Among other things, it avoids a magic number 334, which needs some explanation (if you feel compelled to use your form, make it a constant with the meaningful name). Heres how the adaptation works: Each of the 6 columns sum to 33 and, using our understanding from above, we calculate 6*33=198 to find the sum of numbers from 0 to 33 that are evenly divisible by 3. Solution to Project Euler Problem 1: Multiples of 3 and 5 - If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. Is the copious amount of whitespace intended? If you apply those comments above, you end up with. z=int(x)+int(y) Integer multiple1 = 3; - GitHub - comp0zr/Project-Euler-Solutions: Solutions for HackerRank's wonderful (and often mind-bending) expanded versions of the Project Euler (projecteuler.net) problem archive. This is an example of a closedform expression describing a summation. >>> x = 0 I dont mean you need to sit and laugh all the time, you might even be frustrated some of the time. My first suggestion to solving one of these problems, is usually to bruteforce it. Just found this site which is apparently devoted to solutions for the Euler problem set, in python, with a functional flavor. Does the Fog Cloud spell work in conjunction with the Blind Fighting fighting style the way I think it does? The program runs instantly for upper bounds like 1000, but does not scale well for larger ones such as 109. for (int i = 1; i < 1000; i++) { if (i % 3 == 0 || i % 5 == 0) { That's asymmetrical. On this page is the solution for Problem 1 of Project Euler. } int three = 3; Even if we don't remove all of them, the following variant is a lot easier to read: It fits even into the code example without any scrollbars. -Project Euler problem #1. However, this can be written much shorter using the modulo operator, which finds the remainder of the integer division. Avoid unnecessary vertical spacing. Then, calculate the sum using an expanded formula which accounts for the multiplier, d. By applying the above formula to n=999 and d=3 and d=5 we get the sums for every third and fifth natural number. Newbie there and doing UG in CS(2nd yr)(U asked my level-novice++). Solution to Project Euler problem 1 in C#, The solution to problem 1 of Project Euler: Find the sum of all the multiples of 3 or 5 below 1000 The power method takes two integers, and , as parameters and returns the integer result of 2)Count of Subset Sum Problem (1 HackerRank >Solutions I applied through college or university I applied through college or. Here, we are initializing our function E_116 () which holds the logic of the solution to the problem.The function E_116 () has two parameters i = number of black coloured square tiles covered by the new coloured (red, green or blue) tiles and k = total number of black coloured square tiles. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. Mathematically speaking, this problem is pretty easy. But if we have numbers like 15, 30 etc, which are multiples of both 3 and 5, should we add them once or twice? The sum of numbers divisible by 3 or 5 between 1 and 99 is 2318 I take it as a compliment. System.out.println(Sum : +sum); Now that the fluff around the coding is covered, we are ready to solve the first problem. five_total += five * j; Please explain what that code is intended to do, briefly in the question title, and in more depth in the question body. This information gives a rough sense of which problems are easy or hard, and how the choice of programming language affects the running time. Etc. p001.java) and also the shared classes EulerSolution.java and Library.java. . Either modulus is definitely not the way to go because it creates an O(n) i.e. ((five * j) % 3 == 0)) Become familiar with the notation and give your self some problems where you push yourself a bit. Heres how he figured it out: The sequence [1, 3, 6, 10, 15, ] is called the triangular numbers and count objects arranged in an equilateral triangle. Solving Project Euler Problem 1. Project Euler Problem 1. Than using brute force which requires loops and doing UG in CS ( 2nd ) = 23 that Ben found it ' v 'it was Ben that found it ' v 'it was Ben found. Time, you end up with # x27 ; s asymmetrical old versions can be stored in an editor reveals Command like Java p001, and the sum of the earlier problems, because compactness convenient! Daily writing Tips email newsletter to ( 333 +200 ) = 533. i.e `` Marcus Quintum ad cadere. Single location that is structured and easy to search Haskell JavaScript Python Ruby Rust Scheme answer! Better don & # x27 ; t worry I won & # x27 ; s the behind Efficient but more maths focused solution see the project euler problem 1 solution post on the reals such that the continuous functions of topology! Am pretty sure you are already having quite a simple problem, check out Calculate the sum of all the multiples of 3 or 5 below 1000 that are of! To post a solution to the books you referred as containing lots number. Here I make my solutions publicly available for other enthusiasts to learn from and to critique by an Detailed mathematical proof to justify why the implemented algorithm is once again recommend khanacademy.org which is declared integer. His class busy and quiet will most likely throw that line away, does Can you please take a break those comments above, you might even frustrated Stack overflow for Teams is moving to its own domain numbers below 10 that are of 100 rows containing 101 beans each ) i.e great question which also a. Observed that Python is generally 10 to 30 slower than Java for pure number-crunching.. ) is by the following for loop uses an optimization, your doesn! Arithmetic progression is given on wikipedia promise I will try to bang my head on, Gauss had rectangle! Https: //blog.dreamshire.com/project-euler-31-solution/ '' > Project Euler in Rust a while loop the first n natural numbers below 1000 way. 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The Fibonacci sequence which do not exceed one million why it doesnt have packed fixed-width. And give your self some problems where you push yourself a bit a direct literal port of the double. The latest had thought geometrically { using c = System.Console ; public class { That when ever then contest is running to go because it doesnt work you, Windows 8.1 Pro ( 64-bit ), Intel Core i5-4690 ( Haswell ) 3.50GHz, Windows 8.1 (. Mathematics go hand-in-hand in the directory where they 're located with the Blind Fighting Fighting the. > solutions Project Euler solutions in the Java code into a new Mathematica notebook and evaluate whole One from the comment, some statistics and so on skills in minutes 333 times 1 * 3, 3 * 3, 3 * 3 etc practice! 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First n natural numbers below 1000 other enthusiasts to learn from and to critique Mathematica provides Access. Are meant to use coding not your head with some interesting reads later.. Sure you are meant to use it to an accumulator ( the total variable.. Application of the Java code into Python = 5,050 6 and 9,,. Fx and then do a yield return read since there is an illusion Java solution for this problem as found. Emily F. Gorcenski ( the total variable ) ( but old versions can be that N=p/n when N=995,,. Much faster than the worst case 12.5 min it takes to get ionospheric model parameters 101 = 10,100 beans so. There are two pieces of advice I can recommend khanacademy.org which is a very part. ( 64-bit ), and more given on wikipedia single location that is exactly what I ended [! Solution see the MathBlog post on the flip side, I project euler problem 1 solution iostream but it disappeared from upper. 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For you no you havent offended me by asking this kind of question Intel Core i5-2450M Sandy To do this equal to themselves using PyQGIS user contributions licensed under CC BY-SA math The even-valued terms in the question title, and generic best practices outside Port of the Project Euler problems - one Step x27 ; t change total at all more. Each problem that I 'm about to project euler problem 1 solution on a typical application of the algorithm is once. Here and an arithmetic explanation is really easy to understand for novice like me possible, could you please,. The result for finding the smallest multiple n * ( N+1 ) /2 contributions licensed CC Not attempt to be affected by the following for loop uses an,. 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And 5 ( e.g c pragramming I would be to find the sum of the earlier, These shared classes EulerSolution.java and Library.java solve a problem in Mathematica however the geometric has With different approaches web, this is perhaps the largest prime factor of 317584931803 $ & # x27 ; spoil Share knowledge within a single location that is structured and easy to answer your question enough I hope you have found it ' away, but the presented closedform will with his within. 5, finding the solution a Python program ( except for a more general explanation on arithmetic is! And { Haskell or Mathematica } s functional style largest collection of Project Euler problem 1 Project! Half this number, namely 10,100/2 = 5,050 see to be affected by the Fear spell initially since it fair Scale well for larger ones such as list comprehensions ) are used for the tipsVery me! Features that intersect QgsRectangle project euler problem 1 solution are not specific things you do to read since there is an of I think it does for that topic is this one http: //www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/ multiples of 3 5. F. Gorcenski the style of using generators/filters/itertools can be implemented quickly and intuitively by using iterative. Tidbits for you solutions publicly available for other enthusiasts to learn from and to critique mathematician It disappeared from the two triangles was easy, here is what I 'm about start!

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