Surge the Growth-Racer

Surge had learned to watch growth the hard way, back home. His family kept a little jam-kitchen, and young Surge's job was to stir. One summer two neighbours each lent the kitchen a way of pairing up jars for labelling — and for a small batch, both ways took about the same afternoon. So the family picked whichever they liked.

Then came the big harvest, and the batch was ten times the size. One neighbour's method took ten times the afternoon — long, but bearable. The other neighbour's method took a hundred times the afternoon, and the jam spoiled on the shelf before the labelling was half done. "Same two methods," Surge remembered thinking, stirring in dismay. "They looked like twins in spring. But one grew gently and one grew monstrously — and the difference only showed up when the batch got big." He never forgot it. From then on he stopped asking "how long does this take?" and started asking the only question that had saved the jam: "how does it grow?"

When he was grown, Surge raced the long road to DiscreteQuest, because he'd heard it cared about what happened when things got large. The head of the academy was an old owl with patient eyes. "What is order of growth?"

Surge didn't slow down even to answer. "It's the shape of how the work grows when the job gets bigger. I don't time one job — one job hides everything. I double the job, and I watch what the time does. Double the work? That's gentle; it'll scale. Quadruple the work for a doubled job? That surges, and it'll drown the moment things get big." He bounced on his long legs. "A method that's fast for ten and hopeless for ten thousand isn't a fast method. It's a small-batch method wearing a disguise." "And two methods that tie on a small job?" the owl asked. "Then the small job told you nothing," Surge said. "Make it bigger. The growth-shapes always separate in the end." The owl nodded, slow and sure. "You are appointed."

Surge's favourite thing was rescuing a student from a method that only looked fine. A careful young tortoise named Bram once brought him a plan he was proud of. "I found a way to check every pair of students for a matching badge! It works perfectly — I tested it on my study group of four." "Four," Surge said kindly. "And how many pairs was that?" Bram counted. "Six." "Now — the whole school. Three hundred students. How many pairs?" Bram's face fell as he worked it out. "...tens of thousands." "Right. Your method checks every pair, so when the group doubles, the pairs nearly quadruple. Four students, six checks. Three hundred students, forty-odd thousand. It surged." Bram looked crestfallen, and Surge nudged him gently. "This isn't a scolding. It's the most useful thing you'll ever learn about a method — before you trust it on the whole school, ask how it grows. You just saved yourself a very long afternoon."