The Multiplying Rabbits Riddle: A Hare-Raising Problem

Imagine a lab where nano-rabbits, the pets of the future, have been created. These tiny, fuzzy creatures possess an incredible ability to multiply at an astonishing rate. But what happens when a rival lab sabotages your experiment, threatening the survival of these new friends? This is the challenge we'll explore, diving into a mathematical riddle that could save the day.

The Nano-Rabbit Habitat

In your lab, there are 36 habitat cells arranged in an inverted pyramid. The top row consists of eight cells. The first cell contains one rabbit, the second has two, and so on, until the last cell holds eight rabbits. The remaining rows are initially empty. These rabbits are unique; they're hermaphroditic. Each rabbit in a cell breeds once with every rabbit in the horizontally adjacent cells, producing one offspring each time. The newborn rabbits then drop into the cell directly below their parents, maturing and reproducing within minutes.

Each cell can hold a maximum of 10^80 nano-rabbits. Your initial calculations show that the bottom cell will contain a 46-digit number of rabbits, seemingly leaving plenty of room. However, disaster strikes when your assistant discovers that a rival lab has tampered with your code, removing all the trailing zeros from your results. Now, you're unsure if the bottom cell can handle the rabbit population, and the reproduction process has already begun!

With malfunctioning devices and calculators, you have only a few minutes to determine the number of trailing zeros in the final count. The fate of the nano-rabbits, and perhaps the world, hangs in the balance.

Cracking the Code: Finding the Trailing Zeros

Calculating the exact number of rabbits in the final cell is impossible in the given time. However, the key is to determine the number of trailing zeros. How can we achieve this without knowing the actual number?

The number of rabbits in the bottom cell is the result of multiplication. Each cell's population is the product of the populations of the two cells above it. Trailing zeros in multiplication arise in two ways:

• Multiplying a number ending in 5 by any even number.
• Multiplying numbers that already have trailing zeros.

Let's examine the second row to identify any emerging patterns. In this row, the fourth cell contains 20 rabbits, and the fifth cell contains 30 rabbits. These are the only two numbers with trailing zeros. Crucially, there are no numbers ending in 5. Since a number ending in 5 can only be produced by multiplying a number that also ends in 5, we can disregard this possibility.

This means we only need to focus on the numbers with trailing zeros.

A Neat Trick for Counting Trailing Zeros

A simple method to determine the number of trailing zeros in a product is to count and add the trailing zeros in each of the factors. For example, 10 x 100 = 1,000. Applying this to our problem, we start with the numbers in the fourth and fifth cells and multiply downwards.

20 and 30 each have one trailing zero. Therefore, 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. Continuing this pattern down the pyramid, we find that the bottom cell will have 35 trailing zeros.

The Brink of Disaster

Adding these 35 zeros to the original 46-digit number yields an 81-digit number, far exceeding the cell's capacity! Acting swiftly, you pull the emergency switch just as the seventh generation of rabbits is about to mature, narrowly averting a nano-rabbit apocalypse.

It was a hare-raisingly close call!

Pascal's Triangle Connection

Interestingly, the method of counting zeros in this problem mirrors the structure of Pascal's Triangle. This connection highlights the underlying mathematical principles at play in this seemingly simple riddle.