Imagine you’ve invented the coolest pets ever—nano-rabbits! These tiny, fluffy creatures multiply super fast. In your lab, you have 36 habitat cells set up in an upside-down pyramid shape, with 8 cells in the top row. The first cell has one rabbit, the second has two, and so on, until the eighth cell, which has eight rabbits. The rest of the cells are empty… for now.
These rabbits are special because each one can breed with every rabbit in the cells next to it, creating one baby rabbit each time. The baby rabbits fall into the cell directly below their parents’ cells, and they grow up and start reproducing almost immediately. Each cell can hold an enormous number of rabbits—(10^{80}) to be exact—before they escape.
You’ve calculated that the bottom cell will have a 46-digit number of rabbits, which seems safe. But just as you start the experiment, your assistant rushes in with bad news. A rival lab has messed with your code, cutting off all the zeros at the end of your results. Now, you don’t know if the bottom cell can handle all the rabbits, and the experiment is already running!
With your calculators broken, you need to figure out how many trailing zeros the final number of rabbits will have. Trailing zeros are important because they tell you how big the number is. But how can you find out the number of trailing zeros without calculating the entire number?
Here’s the trick: trailing zeros in a number come from multiplying numbers that end in zero or from multiplying a number ending in 5 with an even number. So, let’s look at the second row of cells to find any patterns. In this row, two cells have numbers with trailing zeros—20 rabbits in the fourth cell and 30 in the fifth cell. But there are no numbers ending in 5, so we won’t see any more numbers ending in 5 in the rows below.
To find out how many trailing zeros the final number will have, we count the zeros in each factor. For example, (10 times 100 = 1,000) has three trailing zeros. So, let’s multiply the numbers in the fourth and fifth cells down the pyramid. Both 20 and 30 have one zero each, so their product will have two trailing zeros. If you multiply one of these with a number that doesn’t end in zero, the product will have only one trailing zero.
As you continue this process down to the bottom cell, you end up with 35 trailing zeros. This means the final number of rabbits is a 46-digit number with 35 zeros added, making it an 81-digit number. That’s way too many rabbits for the habitat to hold!
Realizing the danger, you quickly pull the emergency switch just before the seventh generation of rabbits matures, narrowly avoiding a rabbit overload disaster.
This riddle teaches us about multiplication and how to find trailing zeros in large numbers. It’s a fun way to explore math concepts and see how they apply to real-world scenarios, even if those scenarios involve futuristic nano-rabbits!
Imagine you’re setting up your own nano-rabbit habitat. Create a physical model of the pyramid using paper or blocks. Label each cell with the number of rabbits it starts with, just like in the article. This will help you visualize how the rabbits multiply and move through the pyramid. Share your model with the class and explain how the rabbits reproduce in your setup.
Use a computer or tablet to simulate the reproduction process of the nano-rabbits. Create a simple program or use a spreadsheet to track the number of rabbits in each cell over several generations. Observe how the numbers grow and identify any patterns. Discuss with your classmates how this simulation helps you understand the concept of exponential growth.
Work in pairs to explore the concept of trailing zeros. Use different numbers to practice finding trailing zeros by multiplying numbers that end in zero or a combination of numbers ending in 5 and even numbers. Create a chart showing your results and present your findings to the class, explaining why trailing zeros are important in large numbers.
Create your own math riddle involving multiplication and trailing zeros, inspired by the nano-rabbits scenario. Swap riddles with a classmate and try to solve each other’s challenges. This activity will help you think creatively and apply your understanding of the concepts in a fun way.
Participate in a class discussion about how the concepts of multiplication and trailing zeros apply to real-world scenarios. Think about situations where understanding large numbers and their properties might be important, such as in computing or science. Share your thoughts and listen to your classmates’ ideas to gain a broader perspective.
Here’s a sanitized version of the provided YouTube transcript:
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After years of experiments, you’ve finally created the pets of the future—nano-rabbits! They’re tiny, they’re fuzzy, and they multiply rapidly. In your lab, there are 36 habitat cells arranged in an inverted pyramid, with 8 cells in the top row. The first cell has one rabbit, the second has two, and so on, up to eight rabbits in the last one. The other rows of cells are empty… for now.
The rabbits are hermaphroditic, and each rabbit in a given cell will breed once with every rabbit in the horizontally adjacent cells, producing exactly one offspring each time. The newborn rabbits will drop into the cell directly below the two cells of their parents, and within minutes will mature and reproduce in turn. Each cell can hold (10^{80}) nano-rabbits—an incredibly large number—before they break free.
Your calculations have given you a 46-digit number for the count of rabbits in the bottom cell—plenty of room to spare. But just as you pull the lever to start the experiment, your assistant runs in with terrible news. A rival lab has sabotaged your code, cutting off all the zeros at the end of your results. This means you don’t actually know if the bottom cell will be able to hold all the rabbits—and the reproduction is already underway!
To make matters worse, your devices and calculators are all malfunctioning, so you only have a few minutes to work it out by hand. How many trailing zeros should there be at the end of the count of rabbits in the bottom habitat? And do you need to pull the emergency shut-down lever? Pause the video now if you want to figure it out for yourself.
There isn’t enough time to calculate the exact number of rabbits in the final cell. The good news is we don’t need to. All we need to figure out is how many trailing zeros it has. But how can we know how many trailing zeros a number has without calculating the number itself?
What we do know is that we arrive at the number of rabbits in the bottom cell through a process of multiplication. The number of rabbits in each cell is the product of the number of rabbits in each of the two cells above it. There are only two ways to get numbers with trailing zeros through multiplication: either multiplying a number ending in 5 by any even number, or by multiplying numbers that have trailing zeros themselves.
Let’s calculate the number of rabbits in the second row and see what patterns emerge. Two of the numbers have trailing zeros—20 rabbits in the fourth cell and 30 in the fifth cell. But there are no numbers ending in 5. Since the only way to get a number ending in 5 through multiplication is by starting with a number ending in 5, there won’t be any more down the line either.
That means we only need to worry about the numbers that have trailing zeros themselves. A neat trick to figure out the amount of trailing zeros in a product is to count and add the trailing zeros in each of the factors—for example, (10 times 100 = 1,000).
So let’s take the numbers in the fourth and fifth cells and multiply down from there. 20 and 30 each have one zero, so the product of both cells will have two trailing zeros, while the product of either cell and an adjacent non-zero-ending cell will have only one. When we continue all the way down, we end up with 35 zeros in the bottom cell.
If you’re not too stressed about the potential nano-rabbit situation, you might notice that counting the zeros this way forms part of Pascal’s triangle. Adding those 35 zeros to the 46-digit number we had before yields an 81-digit number—too big for the habitat to contain! You rush over and pull the emergency switch just as the seventh generation of rabbits was about to mature—very close to disaster.
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This version maintains the original content’s essence while removing any potentially sensitive or inappropriate language.
Multiplying – The mathematical operation of scaling one number by another. – Example sentence: When multiplying 3 by 4, the result is 12.
Rabbits – In mathematics, often used in word problems to represent quantities or groups. – Example sentence: If there are 5 rabbits and each rabbit has 3 carrots, how many carrots are there in total?
Trailing – Refers to the sequence of zeros at the end of a whole number. – Example sentence: The number 1000 has three trailing zeros.
Zeros – The digits that represent the absence of a value in a number, often used to denote place value. – Example sentence: In the number 2050, there are two zeros.
Multiplication – The process of combining equal groups to find the total number of items. – Example sentence: Multiplication is a faster way to add the same number several times.
Number – A mathematical object used to count, measure, and label. – Example sentence: The number 7 is a prime number because it has no divisors other than 1 and itself.
Cells – In mathematics, often refers to individual units in a grid or table, such as in a spreadsheet. – Example sentence: Each cell in the table contains a different number.
Product – The result of multiplying two or more numbers together. – Example sentence: The product of 6 and 7 is 42.
Factors – Numbers that can be multiplied together to get another number. – Example sentence: The factors of 12 are 1, 2, 3, 4, 6, and 12.
Patterns – Repeated or recurring sequences in numbers or shapes. – Example sentence: Recognizing patterns can help solve complex algebra problems.
