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Imagine having one ball and somehow turning it into two identical balls without adding any extra material. Sounds impossible, right? Surprisingly, mathematics has a theorem that describes exactly that! It's called the Banach–Tarski paradox, and it's one of the strangest results in mathematics. So, how does it work? In 1924, mathematicians Stefan Banach and Alfred Tarski proved that, under the usual mathematical assumptions, a solid ball in three-dimensional space can be split into a finite number of sets and rearranged to form two identical copies of the original ball. But there's a catch: these aren't normal pieces you'd get by cutting a ball with a knife. They're extremely unusual collections of mathematical points, so you couldn't physically separate them and put them back together like puzzle pieces. Why doesn't the ball's volume stay the same? Normally, if you cut an object into pieces and rearrange them, the total volume stays the same. But the sets used in the Banach–Tarski paradox are non-measurable. That means they can't be assigned ordinary volumes in the usual mathematical way, so the familiar rules of adding volumes don't apply. The proof also relies on a mathematical idea called the Axiom of Choice. This principle allows mathematicians to select elements from collections of sets, even in situations involving infinitely many sets where no explicit selection rule is given. Does this mean we can duplicate real objects? No! You can't use the Banach–Tarski paradox to make two basketballs from one, duplicate food, or create extra matter. It's a result about abstract mathematics, not a method for physically copying objects. Want to explore the paradox further? I've created a lesson about the Banach–Tarski paradox in Flaura Learn. You can explore the ideas in more detail, discover why the theorem works, and take a quiz with explanations to test your understanding.

Modular Arithmetic is where numbers loop back. It can be very complicated, but you can learn it very quickly. Take these two equations (down below), which one is right? 8 + 6 = 14 or 8 + 6 ≡ 2 You might be wondering what the "≡" means or what the difference is between "≡" and "=" is, well, the = means that the numbers left of the = are equal to the following. The ≡ means that the the answer won't be above the loop number, it will go back to 1 and count on from there. The loop number is basically the maximum number it will go to. So for our example from earlier, say it's a clock, so the loop number would be 12. 8 + 6 ≡ 2 For this example, 8 could be the time we went to bed. 6 could be the time we slept for. And 2 would be the outcome, remember in this example our loop number is 12 since its a clock example. You may be wondering how do we write that we loop back at that number. Well you could write that as: ( mod 12)* *That's for the clock example You can find Modular Arithmetic everywhere, In cars, on the odometer, On calendars, the months loop, In real life, time on a clock.