1 2 3 4 5 5 6 Understanding MSAs Form

This is my recommendation, which is an update of the old "machete order", which is already known among fans:Rogue One: If you've never seen a Star Wars film it's a good idea to start with a modern movie so it will catch you. In addition, it does not require to know almost anything about the other films. It introduces us to the rebel alliance from its origins and also has a much better introduction to Darth Vader than other films. If this movie catches you, the ending will leave you waiting to see what's next, which is ...Episode 4 - A New Hope: This film was the first one to be produced, and it introduces us to Luke, the hero, learning what a Jedi is and meeting Princess Leia and the Rebel Alliance she fights for. Then we continue with ...Episode 5 - The Empire Strikes Back: Where we see how Luke Skywalker continues to learn about the Force and manages to fight with Darth Vader himself. The film ends with an incredible plot twist that will leave those who have not seen the saga thinking. Therefore, it is time to make a break. Let's not see the next episode but let's learn how it was that the whole galaxy signNowed this point...Episode 1 - The Phantom Menace (Optional): Only if you want to see all the movies or if you have children at home, watch this movie. It includes an action filled “car” race and we see the planet Naboo and Senator Palpatine. It also includes a child and an alien clown who do not contribute anything to the saga.Episode 2 - Attack of the Clones: We still do not know how the evil Galactic Empire was born but in this film we see that the Republic did not live in peace and needed serious changes. We also meet the members of the Jedi order, with their successes and mistakes. And we obviously meet Anakin Skywalker, a rebellious Jedi who wants to leave everything out of love and revenge.Episode 3 - Revenge of the Sith: The war peaks and the Galactic Empire is born while the Jedi Order ends. The film has one of the best endings of the entire saga. Now that we know the whole history of the Empire, we can go back to know how this story will end...Episode 6 - Return of the Jedi: Luke already knows the truth and now the Rebels must end the Empire at once. But the Empire has a new, bigger, Death Star. In the end, we will have an unexpected good ending. But it's not like they're going to live happily ever after because...Episode 7 - The Force Awakens: The survivors of the Empire return after 30 years as The First Order. General Leia forms the Resistance fight and to understand what this First Order is. A young stranger named Rey ends up involved in this war without knowing why, but she discovers that the Force is also strong with her, so she wants to train herself.Solo - A Star Wars Story: Probably ‘The Force Awakens’ left you thinking about Han Solo. So let’s take a break to understand where he comes from and how his friendship with Chewie was formed.Episode 8 - The Last Jedi: Although Rey finds Luke Skywalker, he refuses to train her because he has had terrible experiences training people. He has changed, and now Luke thinks that the Jedi order must cease to exist in its traditional form.If we add all the movies, including Episode 8, watching the whole saga takes almost 22 hours. So find time in your agenda and prepare a lot of pop corn for a saga that is totally worth the time.Edit: Added ‘Solo: A Star Wars Story’ between episodes 7 and 8 to keep this list updated.

This riddle was posted to WhatsApp with the claim that only few will figure it out.1 + 4 = 52 + 5 = 123 + 6 = 215 + 8 = ?What do you think the answer is?The problem went viral and generated over 3 million comments with people arguing about the correct answer. What do you think the correct answer is?Answer To 1 + 4 = 5 PuzzleA mathematician might take a literal approach.1 + 4 = 52 + 5 = 123 + 6 = 215 + 8 = ?The first equation is true, the second and third are false, and the answer to the equation should be 13.But riddles like this are not about literally interpreting mathematical symbols. They are about identifying a pattern in the set of equations and applying it to the unknown.The answer that jumped into my mind is to add the first number to the product of the two numbers to get the answer. That is:a + b means a + abThis works for the known equations.1 + 4 means 1 + 1(4) = 52 + 5 means 2 + 2(5) = 123 + 6 means 3 + 3(6) = 21Applying the pattern to the last equation gives the answer of 45.5 + 8 means 5 + 5(8) = 45The answer of 45 is valid according to this interpretation. Furthermore, many other IQ tests have this sort of pattern where you take two numbers and find a hidden equation that involves simple operations like addition, subtraction, multiplication, division, and exponentiation.But some people interpreted the problem differently and arrived at a different answer.Another interpretation: running totalSome people felt the pattern was a running total: add the result in the previous line to the new numbers to get the new answer.The first line is valid mathematically.1 + 4 = 5For the next line, take this result of 5 and add it to the new numbers to get the new answer.5 + 2 + 5 = 12Do the same thing for the next line: add the previous line’s result of 12 to the new numbers to get the next result.12 + 3 + 6 = 21To solve the puzzle, do the same for the final line.21 + 5 + 8 = 34This pattern results in an answer of 34, and many people suggested this answer is more valid.So what is the answer? Is it 34 or 45? While we cannot know for sure what the puzzle maker had in mind, there is a way to reconcile these two approaches. It turns out the running total can also lead to the answer of 34, if you decide to fill in the pattern a bit more.Running total missing linesSuppose the answer in each line is the running sum total of the previous result and the new numbers.1 + 4 = 55 + 2 + 5 = 1212 + 3 + 6 = 21Notice that in each line the new numbers are incremented 1 more from the previous line. For example, 1 + 4 is turned into 2 + 5; so both the numbers 1 and 4 are increased by 1. Then 2 + 5 is increased to 3 + 6.We can continue this pattern, so the next lines would be 4 + 7, and then 5 + 8.What is the running total when we include these “missing” lines?1 + 4 = 55 + 2 + 5 = 1212 + 3 + 6 = 2121 + 4 + 7 = 3232 + 5 + 8 = 45We once again get to the answer of 45, which is somewhat surprising!In fact, the running total with the missing lines generates the same answer, line by line, as the algebraic result from the first approach:a + b means a + ab = a(1 + b)How can we see the two approaches are the same? On line 5, there are 4 previous lines. We can make 8 by pairing the first number of each line with the second number of another line: we can pair 1 with 7, 2 with 6, 3 with 5, and 4 with 4. These are 4 pairs of 8. The final line has another 8. This means we need to take 5 and add 8 to it 5 more times, which is 5 + 5(8) = 45.In general, line n has the equation n + (n + 3), which is equal to the result n + n(n + 3) = n(n + 4).Let’s prove this formula holds by induction. Assuming the formula is true up to line n, we then consider the next line. In the next line n + 1, we add the numbers (n + 1) + (n + 4). The result in line n is n(n + 4), so when we add (n + 1) + (n + 4) we get:n(n + 4) + (n + 1) + (n + 4)= n2+ 6n + 5= (n + 1)(n + 5)= (n + 1)[(n + 1)) + 4]And this completes the induction.Most people believe the answer is either 45–with the equation a + ab–or 34–with a running total. Since the running total can also get to the answer of 45 when extending the pattern to missing lines, many believe that 45 is the answer that makes the most sense.A Third Interpretation: The Answer is 41.There was an another obscure answer I found.The idea is to evaluate the sums in base 10, and then convert the answer into descending number bases 6, 5, 4, etc. line by line. So the pattern is:1 + 4 = 5 (base 10) = 5 (base 6)2 + 5 = 7 (base 10) = 5×1 + 2 = 12 (base 5)3 + 6 = 9 (base 10) = 4×2 + 1 = 21 (base 4)So if we do the same for the last line, we would need to use base 3 to get:5 + 8 = 13 (base 10) = 3×4 + 1 = 41 (base 3)So if you got 41, that’s another way people have seen the pattern. It’s a bit more involved to explain what is going on, and you can’t really reconcile the result with the “missing line” interpretation. But several people did see this pattern.

How many 6 digit numbers can be formed from 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6?Answer: 31,321.I've broken the problem into 11 cases, or sets. A set consists of groups of digits. For example, [math]\;\{411\}\;[/math] is a set containing 3 groups: 4 of a particular digit, 1 of a different digit, and 1 of still a different digit.For each set, I first calculate the total groups and then the total permutations per group. The two are multiplied together to get the count for the set.To avoid being overly long, I'll explain no further and hope that the pattern of computation is obvious. Remember that the count of each digit from 1 to 6 is the same as the value of the digit.I’m assuming that the reader is familiar with the choose function. For completeness and clarity, I’ve left in extraneous choose operations (that evaluate to 1).[math]\begin{array}{|l|lcl|r|}\hline \textrm{Set}&\textrm{Groups:}&&\textrm{Permutations}&\textrm{Count}\\\hline \{6\} &{6 \choose 6}&\cdot&{1 \choose 1}&1\\ \{51\} &{6 \choose 5}{1 \choose 1}&\cdot&{2 \choose 1}{5 \choose 1}&60\\ \{42\} &{6 \choose 4}{2 \choose 2}&\cdot&{3 \choose 1}{4 \choose 1}&180\\ \{411\} &{6 \choose 4}{2 \choose 1}{1 \choose 1}&\cdot&{3 \choose 1}{5 \choose 2}&900\\ \{33\} &{6 \choose 3}{3 \choose 3}&\cdot&{4 \choose 2}&120\\ \{321\} &{6 \choose 3}{3 \choose 2}{1 \choose 1}&\cdot&{4 \choose 1}{4 \choose 1}{4 \choose 1}&3840\\ \{3111\} &{6 \choose 3}{3 \choose 1}{2 \choose 1}{1 \choose 1}&\cdot&{4 \choose 1}{5 \choose 3}&4800\\ \{222\} &{6 \choose 2}{4 \choose 2}{2 \choose 2}&\cdot&{5 \choose 3}&900\\ \{2211\} &{6 \choose 2}{4 \choose 2}{2 \choose 1}{1 \choose 1}&\cdot&{5 \choose 2}{4 \choose 2}&10800\\ \{21111\}&{6 \choose 2}{4 \choose 1}{3 \choose 1}{2 \choose 1}{1 \choose 1}&\cdot&{5 \choose 1}{5 \choose 4}&9000\\ \{111111\}&{6 \choose 1}{5 \choose 1}{4 \choose 1}{3 \choose 1}{2 \choose 1}{1 \choose 1}&\cdot&{6 \choose 6}&720\\\hline &&&&31321\\\hline \end{array}[/math]I find it interesting that my answer is a prime number.Brute force:$ python