Generally, people have a good feeling for how much cake is in half a cake. But do you know exactly how much most of a cake is? Researchers have conducted experiments to find out how much of a cake ‘most’ is.

‘Most’ doesn’t have an exact definition like ‘half’ does. Dictionaries often define ‘most’ to be more than half, but less than the whole. However, when people say ‘most’, they might not mean the same thing as it says in the dictionary.

The researchers wanted to know what people understood when they heard ‘most’ in real life. They organised volunteers from all around the world and played them a video of a conversation. During the conversation, one of the people would use the word ‘most’ to describe something. For example, they might have said that most of the biscuits in a jar were chocolate. After watching the conversation, the volunteers were asked questions, and the researchers used those answers to work out what ‘most’ meant.

This survey found that when people were talking, ‘most’ meant something between 80 per cent and 95 per cent (see picture). This is a much smaller range than the dictionary definition of between 50 per cent and 100 per cent.

You might not find this research very surprising, and that’s normal. This was a survey to see what people think, so it’s natural for you to feel the same way about ‘most’ as other people. If you don’t agree with the results of the survey, that’s okay too. The dictionary says that ‘most’ can be anything more than half and less than all. Of course, it might be helpful to keep in mind what ‘most’ means to other people, or at least to most of them.

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More and more forms of life are becoming extinct, as people destroy habitats, hunt animals and introduce pests. Individual extinctions are often just the start of the problems facing an ecosystem under threat. The elimination of one species can cause disturbances in an ecosystem that could lead to further species being wiped out. Now, mathematicians have developed techniques to predict these flow-on effects, and suggest ways of saving these fragile ecosystems.

Scientists have investigated many cases where the removal of one species causes another species to fail. For example, in the north-western Atlantic Ocean, the overfishing of large sharks meant that they were no longer eating as many cownose rays. The population of these rays skyrocketed, and as a result, they ate more scallops, driving the scallops almost to extinction.

Eliminating one species can unbalance an entire ecosystem, which can cause another extinction. This new extinction can cause even more extinctions, leading to a cascade that can decimate an ecosystem, until it finds a new stable state.

Researchers from Northwestern University in the United States have now developed techniques that can help rangers to reduce the impacts of these cascades, and in some cases, prevent them entirely. The researchers begin with a food web, showing which species rely on each other for food. Food webs are an example of an important mathematical idea called a network. By analysing the links between the species, they can identify which species are the most critical to keep in an ecosystem, and which threaten other species. They can then use the network model to test different intervention strategies that could help reduce the effect of an extinction.

It can be a lot easier and faster to change an ecosystem by removing species, such as by hunting animals, than it is to replace individuals. And with the help of mathematicians and networks, removing some species can stabilise an ecosystem, and hopefully cause less destruction than simply letting the ecosystem sort itself out.

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Get a piece of chewing gum, chew it for a bit and then take it out of your mouth and stretch it. As it gets longer, the gum will get thinner and thinner. This happens with all kinds of materials from rubber to steel. But not all materials get thinner when you stretch them. There are strange substances, called auxetic materials, which actually get wider as you stretch them longer. Now mathematicians are helping chemists and engineers understand what makes these materials work.

Although auxetic materials seem really advanced, scientists have known about them for more than 100 years. A German scientist named Woldemar Voigt did some experiments and found that fool’s gold (iron pyrite) was probably auxetic. Other experiments have suggested that many other materials could be auxetic, including paper and even living bone. However, until recently it has been very hard to work out exactly how an auxetic material will respond to forces.

A team of scientists from the University of Malta decided to investigate. The researchers imagined an auxetic material as a collection of strong rectangles that are joined at the corners. When the material is stretched, the rectangles don’t bend, but they rotate against one another (see picture). This rotation causes the gaps between the rectangles to grow, which allows the material to expand in all directions at once.

The researchers used their idea to write a computer program that could predict how real auxetics would respond to forces. Their program (called a ‘model’) gave good results for a wide range of auxetic materials. Having a good model means that engineers and scientists can work out how auxetic materials are going to behave without having to build them. Making new products out of these strange materials should now be a lot easier.

What are auxetics used for?

Auxetic materials get wider when you stretch them, but the reverse is also true. If you compress an auxetic material, it gets thinner as well. This shrinking means that the material becomes more dense (tightly packed) in the part that is getting squeezed, which could help absorb impacts. Engineers think that auxetic materials might be good for protective equipment, from bike helmets to body armour.

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This is a geomagic square. If you take the shapes in any row, column or main diagonal, you can rearrange them to make a rectangle that is 3 squares tall and five squares wide.

Over 2500 years ago, Chinese mathematicians discovered a grid of numbers with special properties that we now call a magic square. If you add the numbers in any row across, or column down, or either of the two diagonals of the grid, you get the same total. Magic squares have been studied for centuries by mathematicians from around the world. Now, a recreational mathematician named Lee Sallows has come up with an entirely different type of magic square that adds shapes together, not numbers.

Lee was studying magic squares when he realised that the numbers could represent lengths. He took the magic square he had been using, and replaced each of the numbers with a line of a certain length. If you took all the lines in a column or a row and put the lines end to end then the total distance was the same, no matter which row or column you chose.

He then wondered if he could make a more interesting magic square out of shapes, not just lines. After some thinking, he managed to come up with a set of nine different shapes that he organised into a square. If he took all the pieces in the top row, he could fit them together to make a shape. If he took the pieces from any other row or column, or either of the two diagonals, he could make exactly the same shape. Lee called his new invention the ‘geomagic square’.

He soon discovered that there were lots of things that you can do with his geomagic squares. For example, it is impossible to make a 2×2 magic square with numbers unless all the numbers are exactly the same. Lee couldn’t find a 2×2 geomagic square either, although he did get quite close. After hearing about the problem, another recreational mathematician named Frank Tinkelenberg did his own investigation and managed to find a 2×2 geomagic square.

Lee is not a trained mathematician, but he continues to make new discoveries with his geomagic squares. Sometimes new and interesting maths is discovered simply by having fun.

The top universities for the study of mathematics vary greatly in rankings depending on which organization is doing the grading. The rankings also differ between United States rankings, and worldwide rankings. This will cover both of those sets of ratings, with the United States rankings first.

The number one university for the study of mathematics in the Unites States, according to the US News & World Report, is the Massachusetts Institute of Technology. MIT is located in Cambridge, near Harvard, and is generally accepted as one of the best US universities. Tied for second place are Harvard University, Princeton University, Stanford University and University of California, Berkeley. And last on their list is the University of Chicago.

In worldwide rankings, three American Universities came out on top. Rated number one worldwide by the Academic Ranking of World Universities is Princeton University in New Jersey. Ranked second is the University of California, Berkeley, with Harvard University coming in third.