Imagine a crisp, snowy morning. The world is hushed, and the only sounds are the crunch of boots and the gentle whisper of the wind. Now, picture two people, a child and their father, setting out to measure the perimeter of a circular garden. They start at the same spot, walk in the same direction, but their strides are different. The child’s steps are 54 centimeters long, while the father’s are a more substantial 72 centimeters.
As they complete their circuit, they notice something interesting: the snowy ground isn't covered in a continuous line of footprints. Instead, there are distinct marks, and remarkably, only 60 unique footprints remain. This isn't magic; it's a fascinating interplay of mathematics and their differing paces.
The core of this puzzle lies in understanding when their footprints will align. This happens at points that are common multiples of their step lengths. To find the first point where their steps perfectly coincide after the start, we need to find the Least Common Multiple (LCM) of 54 and 72.
Let's break down the numbers: 54 can be expressed as 2 × 3³, and 72 as 2³ × 3². To find the LCM, we take the highest power of each prime factor present in either number. So, we have 2³ (from 72) and 3³ (from 54). Multiplying these together, 2³ × 3³ = 8 × 27 = 216. This means that every 216 centimeters, their footprints will land on the exact same spot.
Now, how does this help us find the garden's perimeter? In each 216-centimeter segment, the child takes 216 / 54 = 4 steps, and the father takes 216 / 72 = 3 steps. If we simply added these up (4 + 3 = 7), we'd be double-counting the spot where their footprints overlap at the end of that 216 cm interval. Since they start at the same point and end at the same point (assuming they complete full circles), the overlap at the end of each 216 cm segment is the crucial factor. So, within each 216 cm cycle, there are 4 (child's steps) + 3 (father's steps) - 1 (the overlapping step) = 6 unique footprints.
We know there are a total of 60 unique footprints. If each 216 cm segment contributes 6 unique footprints, then the total number of these 216 cm segments must be 60 footprints / 6 footprints per segment = 10 segments.
Therefore, the total perimeter of the garden is simply the number of segments multiplied by the length of each segment: 10 segments × 216 centimeters per segment = 2160 centimeters.
To put that into a more familiar unit, 2160 centimeters is equal to 21.6 meters. It’s a beautiful illustration of how seemingly simple observations can lead us to deeper mathematical principles, and how numbers can help us measure and understand the world around us, even on a snowy morning.
