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In the heart of a bustling warehouse, a lone worker named Alex faced a daunting task: unloading a truck brimming with oranges into 25 kg tray boxes. This job usually took hours and left workers exhausted. But Alex had a brilliant idea — a strategy inspired by the mathematical principle of induction and the domino effect, a perfect example of bottom-up reasoning.
What is Bottom-Up Reasoning?
Bottom-up reasoning starts with specific observations or cases and builds up to a general conclusion. In this case, Alex observed the efficiency of the domino effect and applied it to the task at hand, building a general strategy from a specific example.
The Base Case (n=1)
Alex started by positioning the first empty box under the truck’s chute. With a determined look, Alex released the oranges, filling the box to the brim. As soon as the box was full, a fellow worker named Jamie, who had been waiting eagerly, swooped in. With a swift motion, Jamie replaced the full box with an empty one, ready for the next round.
The Inductive Step
Assume that for n boxes, Alex successfully fills each box to 25 kg using the domino effect, and Jamie replaces each filled box with an empty one. We need to show that the process works for n+1 boxes.
- After the nth Box: Jamie removes the nth filled box and places the (n+1)th empty box under the chute.
