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An example from the United States shows how cheating with counties can lead to unfair results.
Imagine you decide to have lunch and make a ham sandwich. Suddenly you slipped, and your lunch flew in all directions: ham under the plate, one piece of bread on the floor, another on the ceiling. It sounds ridiculous, but math says that there is a way to carefully cut this chaos into two equal parts in one movement – with a large knife or machete. This is possible thanks to a special theorem called the "ham sandwich theorem". It says that you can divide any three objects into equal parts in one straight cut, even if they are scattered everywhere.
This theorem isn't just about sandwiches. It can work with any objects and in any number of dimensions. For example, on a piece of paper (which is two dimensions), there will always be a line that divides any two shapes in half. If we add a third object, we will need three-dimensional space to find a plane (like a machete cutting a room in half) that will divide them all at the same time.
Now imagine that this principle applies not to lunch, but to something more serious, such as elections and the division of electoral districts. In the United States, for example, politicians divide the country into districts in such a way as to win elections, even if the majority of people vote for another party. Using math, you can" slice " these counties so that the results are fair, but you can also make sure that one side gets an unfair advantage.
As mentioned, the ham sandwich theorem has much less capricious implications for the perennial problem of electoral fraud in politics. In the United States, state governments divide their states into electoral districts, and each district elects a member to the House of Representatives. Electoral fraud is the practice of deliberately delineating the boundaries of electoral districts for political gain. For a simplified example, imagine a state with a population of 80 people. 75 percent of them (60 people) support the purple party, while 25 percent (20 people) prefer the yellow party. The state will be divided into four districts of 20 people each. It seems fair that three of these districts (75 percent) should be purple and one yellow, so that the state's congressional representation matches the population's preferences. However, a clever cartographer can bend the boundaries of districts in such a way that there will be 15 purple and five yellow voters in each district. That way, the purple party will hold a majority in every county, and 100 percent of the state's representation will come from the purple party, not 75 percent. In fact, with a sufficiently large number of voters, any advantage one party has over the other (say, 50.01 percent purple versus 49.99 percent yellow) can be used to win each district; just make sure that 50.01 percent of each district supports the majority.
Of course, such districts look extremely artificial. A seemingly obvious way to limit county scams is to impose restrictions on county forms and ban the tentacular monstrosities that we often see on American electoral maps. Indeed, many states introduce such rules. While it may seem that requiring counties to have "normal" shapes could significantly improve the situation, smart researchers have applied a certain geometric theorem to show how ridiculous this is. Let's go back to our example: 80 voters, 60 of whom support purple and 20 yellow. The ham sandwich theorem tells us that regardless of their distribution, we can draw a straight line with exactly half of the purple voters and half of the yellow voters on either side (30 purple and 10 yellow on both sides). Now treat each side of the cut as its own ham sandwich problem, dividing each half with its own straight line so that there are 15 purple and five yellow ones in each resulting region. Purple ones now have the same scam advantage as before (they win every county), but the resulting regions are all simple with straight-line borders!
Even if you force politicians to divide districts into equal and simple parts, they will still be able to manipulate the borders in such a way as to win. Mathematics and politics are closely linked, and sometimes just a ham sandwich is enough to understand this.
Imagine you decide to have lunch and make a ham sandwich. Suddenly you slipped, and your lunch flew in all directions: ham under the plate, one piece of bread on the floor, another on the ceiling. It sounds ridiculous, but math says that there is a way to carefully cut this chaos into two equal parts in one movement – with a large knife or machete. This is possible thanks to a special theorem called the "ham sandwich theorem". It says that you can divide any three objects into equal parts in one straight cut, even if they are scattered everywhere.
This theorem isn't just about sandwiches. It can work with any objects and in any number of dimensions. For example, on a piece of paper (which is two dimensions), there will always be a line that divides any two shapes in half. If we add a third object, we will need three-dimensional space to find a plane (like a machete cutting a room in half) that will divide them all at the same time.
Now imagine that this principle applies not to lunch, but to something more serious, such as elections and the division of electoral districts. In the United States, for example, politicians divide the country into districts in such a way as to win elections, even if the majority of people vote for another party. Using math, you can" slice " these counties so that the results are fair, but you can also make sure that one side gets an unfair advantage.
As mentioned, the ham sandwich theorem has much less capricious implications for the perennial problem of electoral fraud in politics. In the United States, state governments divide their states into electoral districts, and each district elects a member to the House of Representatives. Electoral fraud is the practice of deliberately delineating the boundaries of electoral districts for political gain. For a simplified example, imagine a state with a population of 80 people. 75 percent of them (60 people) support the purple party, while 25 percent (20 people) prefer the yellow party. The state will be divided into four districts of 20 people each. It seems fair that three of these districts (75 percent) should be purple and one yellow, so that the state's congressional representation matches the population's preferences. However, a clever cartographer can bend the boundaries of districts in such a way that there will be 15 purple and five yellow voters in each district. That way, the purple party will hold a majority in every county, and 100 percent of the state's representation will come from the purple party, not 75 percent. In fact, with a sufficiently large number of voters, any advantage one party has over the other (say, 50.01 percent purple versus 49.99 percent yellow) can be used to win each district; just make sure that 50.01 percent of each district supports the majority.
Of course, such districts look extremely artificial. A seemingly obvious way to limit county scams is to impose restrictions on county forms and ban the tentacular monstrosities that we often see on American electoral maps. Indeed, many states introduce such rules. While it may seem that requiring counties to have "normal" shapes could significantly improve the situation, smart researchers have applied a certain geometric theorem to show how ridiculous this is. Let's go back to our example: 80 voters, 60 of whom support purple and 20 yellow. The ham sandwich theorem tells us that regardless of their distribution, we can draw a straight line with exactly half of the purple voters and half of the yellow voters on either side (30 purple and 10 yellow on both sides). Now treat each side of the cut as its own ham sandwich problem, dividing each half with its own straight line so that there are 15 purple and five yellow ones in each resulting region. Purple ones now have the same scam advantage as before (they win every county), but the resulting regions are all simple with straight-line borders!
Even if you force politicians to divide districts into equal and simple parts, they will still be able to manipulate the borders in such a way as to win. Mathematics and politics are closely linked, and sometimes just a ham sandwich is enough to understand this.
