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## After 90 Years, Mathematicians Finally Solved the Most Notorious Ramsey Problem

“Everyone’s tried to figure it out,” says the scientist who actually did. • For 90 years, mathematicians have been stumped by a problem that attempts to explain how many people would need to be invited to a party so at least 4 people always know one another.
• To finally crack the case, researchers from UC San Diego delved into the world of finite geometry and found a pseudograph to estimate an answer to the question .
• The researchers hope that this approach could help estimate other mathematical functions that are still left unsolved.

Okay ... let’s back up a little.

Named after mathematician Frank Ramsey, who first proved his eponymous theorem in the 1920s, Ramsey problems seek to find order in disorder. Ramsey’s original theory says that if a graph is large enough—in mathematics , a graph is a collection of points and lines between those points—you can find order within all that chaos. Chiefly, the theorem states that a set of points will have no lines between them, or a set of points with all possible lines between them (aka cliques). This is expressed as r(s, t ), where s means “points with lines” and t means “points without lines.”

## More From Popular Mechanics While this is enough to make a non-mathematician dizzy, the concept is usually described as “the theorem on friends and strangers,” and is often allegorically explained as the idea that a party of six people will inevitably produce three people who know each other and three people who don’t. This is expressed as r(3,3), with the answer being six. (While I’m sure you have five close friends you can invite to a party, we’re talking about points and lines on a graph here—not actually people.)

“It’s a fact of nature , an absolute truth,” Jacques Verstraete, whose groundbreaking finding is currently under review with the journal Annals of Mathematics , says in a press release . “It doesn’t matter what the situation is or which six people you pick—you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

Expanding this idea, mathematicians Paul Erdös and George Szekeres discovered that r(4,4) equals 18 in 1935, and the solution to r(5,5) is still unknown. However, Verstraete and UC San Diego researcher Sam Mattheus have found an estimated solution for r(4, t ), where t means that “points without lines” is variable.

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations ,” Verstraete said in the press statement. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

Verstraete first saw the problem stated in a book about Erdös written by two UC San Diego researchers. These problems are usually solved using random graphs, but this Ramsey problem required more out-of-the-box thinking.

Using a concept known as pseudorandom graphs, Verstaraete discovered a solution for the r(3, t ) in 2019 . But the technique hit a brick wall when applied to the notoriously tricky r(4, t ) problem, so Verstraete ventured into other areas of mathematics—specifically, finite geometry —to figure out the answer.

“Many people have thought about r(4, t )—it’s been an open problem for over 90 years ,” Verstraete said in a press statement. “But it wasn’t something that was at the forefront of my research. Everybody knows it’s hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Drawing on Mattheus’s expertise in the field, the researchers discovered a pseudograph in finite geometry called an O’Nan configuration that worked for their problem. After a year of fine-tuning the math, Verstraete found that the answer was very close to a cubic function of t . This means that if you want a party where there will always be four people who know one another then, you need to invite t 3 people who don’t know each other. Reminder: this is just a close estimate, not an exact answer.

Now that it has produced a close estimate, Verstraete hopes that the technique could help open up new mathematical frontiers that are still waiting to be explored. Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough. Solution to Riddle of the Week #42 Solution to Riddle of the Week #16 Solution to Riddle of the Week #29 Why Some People Think 2+2=5 Get a Die to Roll Any Way You Want…With Math! Scientists Create Quantum 'Alice Rings' Scientists Find Boat Load of Three-Body Solutions Solution to Riddle of the Week #6 AI Is Helping Build the 'Periodic Table of Shapes" Solution to Riddle of the Week #53 Forget 50/50, Coin Tosses Have a Bias

• Math Tables

## Table of 29

Tables are the basics of Maths we have learned in schools. Table of 29 gives the repeated addition of number 29, repeated for a certain number of times. For example, 5 buckets of 29 bottles each, gives the sum of 29 + 29 + 29 + 29 + 29 = 145. Therefore, the total number of bottles in all the buckets are 145. Or we can also write it as, 5 x 29 = 145, where 5 is the number of buckets and 29 is the number of bottles in each bucket.

## Table of 29 Chart The multiplication tables are of very useful from Classes 1st to 12th. The calculations we do are sometimes time-consuming as it consists of multiplication problems. Therefore, if we memorize this maths table , we can solve the related problems easily and quickly. Also, aspirants of competitive exams that include maths aptitude questions can memorize 29 for quick calculations.

The table of 29 is given below is little difficult but could be memorised.

## 29 Times Table

Below is the table of 29 from numbers 1 to 20 times.

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## Math Puzzles Questions

How many squares do you see? Number of squares in the given figure is

16 + 9 + 19 + 1 = 45.

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A cuboid shaped wooden block has 6 cm length, 4 cm breadth and 1 cm height. Two faces measuring 4 cm x 1 cm are coloured in black. Two faces measuring 6 cm x 1 cm are coloured in red. Two faces measuring 6 cm x 4 cm are coloured in green. The block is divided into 6 equal cubes of side 1 cm (from 6 cm side), 4 equal cubes of side 1 cm(from 4 cm side).

How many cubes will have green colour on two sides and rest of the four sides having no colour ? There are 16 small cubes attached to the outer walls of the cuboid. Therefore remaining inner small cubes will be the cubes having two sides green coloured. So the required number = 24 - 16 = 8

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How many times can you subtract 10 from 100?

How many times can you subtract 10 from 100, for this if we go through logically, it is only 1 time. For the first time if we subtract 10 from 100, then there will no 100 to subtract from.

Hence, it is only 1 time we can subtract 10 from 100.

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A number of cats got together and decided to kill between them 999919 rats. Every cat killed an equal number of rats. Each cat killed more rats than there were cats. How many rats do you think that each cat killed ?

Given total number of rats = 999919

999919 = 1000000 - 81 = 1000 2   -   9 2  = (1000+9)(1000-9) = 1009 x 991. Since there were more rats than there were cats, 991 cats killed 1009 rats each.

If A is substituted by 4, B by 3, C by 2, D by 4, E by 3, F by 2 and so on, then what will be total of the numerical values of the letters of the word SICK? Total value = 4 + 2 + 2 + 3 = 11.

Find the missing number in the following puzzle? The logic in the given puzzle:

2+3 = 8 3+7 = (7+2)x3 = 27 4+5 = (5+3)x4 = 32 5+8 = (8+4)x5 = 60 6+7 = (7+5)x6 = 72

7+8 = (8+6)x7 = 98.

Arrange four nines(9) and a one(1) and only one mathematical symbol to make it equal to 100 ?

199 - 99  = 100

Two Fathers and Two Sons Riddle

Two fathers and two sons sat down to eat eggs for breakfast. They ate exactly three eggs, each person had an egg. The riddle is for you to explain how

one of the 'fathers' is also a grandfather. Therefore the other father is both a son and a father to the grandson. In other words, the one father is both a son and a father.

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The numbers x , y , z and w have an average equal to 25. The average of x , y and z is equal to 27. Find w. Find x , y , z so that the numbers 41 , 46 , x , y , z have a mean of 50 and a mode of 45. A is a constant. Find A such that the equation 2x + 1 = 2A + 3 (x + A) has a solution at x = 2.

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22. Mathematicians Finally Solve the Most Notorious Ramsey Problem

Expanding this idea, mathematicians Paul Erdös and George Szekeres discovered that r (4,4) equals 18 in 1935, and the solution to r (5,5) is still unknown. However, Verstraete and UC San Diego ...

23. Table of 29

Tables are the basics of Maths we have learned in schools. Table of 29 gives the repeated addition of number 29, repeated for a certain number of times. For example, 5 buckets of 29 bottles each, gives the sum of 29 + 29 + 29 + 29 + 29 = 145. Therefore, the total number of bottles in all the buckets are 145. Or we can also write it as, 5 x 29 ...

24. 79+ Math Puzzles Questions With Answers And Explanation

697 114353. Q: A cuboid shaped wooden block has 6 cm length, 4 cm breadth and 1 cm height. Two faces measuring 4 cm x 1 cm are coloured in black. Two faces measuring 6 cm x 1 cm are coloured in red. Two faces measuring 6 cm x 4 cm are coloured in green. The block is divided into 6 equal cubes of side 1 cm (from 6 cm side), 4 equal cubes of side ...