The "handshake" thing can actually be solved by an equation!

Mathematics is a language, and equations are its basic expression. We all know that the universe follows certain laws. We call it science, and we describe these laws in the language of mathematics. When these laws are expressed in mathematical language, they are equations. Everything in the world, from the formation of galaxies to the pattern of freckles on a child's nose, is related to these equations. Whether you like it or not, whether you are a person who jumps on the feeling or a person who cares about order and details, every aspect of your life is dominated by equations. Equations don't care whether you understand them or not, they still control everything around you. So it's time to become more familiar with the world of mathematics. "Handshake" can also be solved by equations.

You are the official photographer for a summit of world leaders. You are an expert at capturing your subjects at their best. You have been asked to take as many handshakes as possible. But there is a problem. The last two heads of state will arrive in an hour. They are a pair of self-important, grandstanding clowns. Their hairstyles are confusing. Their political stance is even more questionable. None of the other 100 heads of state are willing to shake their hands. You know that once they arrive, all heads of state will stop shaking hands. Do you have time to take photos of every possible handshake before then?

The handshake problem is a widely studied area of ​​mathematics, and is well known for the various and interesting ways to solve it. To tackle the challenge, it is useful to know how many pictures you need to take in total. Let's look at the following groups of people and see how many handshakes they need to take.

Anna greets Bob, so there is only one handshake so far.

Kara comes in. She needs to shake hands with Anna and Bob, so the number of handshakes increases by 2.

Diego comes in. He needs to shake hands with Ana, Bob, and Carla, so the number of handshakes increases by 3.

When Edith arrives, she needs to shake hands 4 times. And so on. For example, when the 25th person arrives, he needs to shake hands with the 24 people who are already there. The nth person must shake hands with the n-1th person who is already there. So, to calculate how many handshakes are needed for 100 people, you need to calculate 1 + 2 + 3 +…+ 98 + 99.

Here you could simply spend some time with a calculator, but adding consecutive integers is another age-old math problem. One way to solve this problem is to use a triangle of dots. If you have five people and you need to shake hands 1 + 2 + 3 + 4 times, you can represent this with a triangle of dots:

You want a formula for the number of points inside a triangle.

The number of points in a rectangle is relatively easy to count, so if you add a triangle with the same number of points, you get a rectangle that is 4 points wide and 5 points high.

There are 4 × 5 = 20 points in the rectangle, which means that there must be 20 ÷ 2 = 10 points in each triangle. Generalizing, you can see that if there are r columns of triangles, the rectangle made up of two such triangles will have r + 1 rows. This means there are r × (r + 1) points in the rectangle. To work out the number of points in the original triangle, you have to halve this, yielding the formula:

Mathematicians are familiar with the formula and the story behind it. In the late 1700s, when German math wizard Carl Gauss was a schoolboy, his teacher gave him the task of adding all the numbers from 1 to 100. Legend has it that Gauss discovered this shortcut and solved the problem on the spot, much to the indignation and embarrassment of his lazy teacher. Gauss went on to become one of the greatest mathematicians of all time.

For n people shaking hands, you need r = n–1 rounds, because you want the number of rounds to be 1 less than the total number of people. Substituting r in the formula with n–1, we get:

That equals 4,950 handshakes. Each photo is expected to take only 10 seconds, but it will take 49,500 seconds, or 13 hours and 45 minutes. So there's no way to take all the photos in an hour. But the convention center didn't invite you to come because they don't want you to say there's nothing you can do. Let's see what you can do.

The President Who Set a World Record

On New Year's Day in 1907, US President Theodore Roosevelt held an open house at the White House so the public could come and meet their leader. By the time the doors closed, Roosevelt had shaken hands with 8,513 people. This set the world record for the most handshakes in a single day, a record that stood for nearly 60 years. The record for the longest handshake was set in 2011 by two pairs of people known as the "Handshake Clan," who shook hands after 33 hours and 3 minutes.

Suppose you start by taking a photo of the first round of handshakes of the heads of state. For 100 people, that's 50 handshakes, which will take 500 seconds to shoot. So let's see how many people you can take complete photos of in the remaining time. The time to shoot is the number of handshakes multiplied by 10 seconds, so:

You have 1 hour, which is 60 × 60 = 3,600 seconds. We have already used 500 seconds to make sure everyone has at least one photo, which leaves 3,100 seconds. The equation you need to solve is: 3,100 = 5 (n–1)n

First, divide both sides of the equation by 5:

620 = (n–1)n

Then expand the brackets:

This type of equation is called a quadratic equation because the unknown n is in the form of a square. This type of equation is not as easy to solve as a linear equation, but if you can get the equation to a point where one side equals zero, you can use the quadratic root formula (see the introduction). To get our equation to a point where one side equals zero, we subtract 620 from both sides of the equation:

The answer should be positive and rounded to one decimal place, giving n = 25.4. This means that you can take photos of 25 people shaking hands, which will take 5 × 24 × 25 = 3,000 seconds. This also leaves us with 100 seconds, which you can use to take 10 more photos or wipe the sweat off your forehead.

You can use a graph to represent all the handshakes of heads of state. If you treat 100 heads of state as points on a circle and use lines between the points to represent handshakes, you get a complicated graph. This graph is called the "Mystic Rose," perhaps because it looks a bit like the circular rose stained glass windows in old churches. What's really amazing is that, despite the fact that the graph is made entirely of straight lines, it appears to have concentric circles and curves. The following is a rose graph of 25 handshakes, created by Edward L. Platt using the Mystic Rose Generator on his personal website.

The Mystic Rose doesn't really help with your problem, but it sure looks great.

The organizers are very satisfied with your solution. When a large and eye-catching helicopter and a fleet of luxury cars park outside the venue, your mission is just completed.

Further reading: The Beauty of Equations: The Mathematical Formulas Behind Everything