Portion the Fair-Divider

The method, she explains, has three clean steps, and Advanced students should be able to name each. "One: count the total shares the ratio calls for. Two parts and three parts is five shares in all — that sum is the whole thing most people skip. Two: find what a single share is worth, by dividing the whole quantity by that total. Twenty counters over five shares is four counters per share. Three: hand out each portion — two shares, so eight; three shares, so twelve." She lays the two little piles on the bench. "Eight and twelve. They add back to twenty, and they stand in the ratio two to three, and no one can argue."

The load-bearing move, she stresses, is the middle step. "The whole trick is never the shares themselves. It is finding the value of one share first — the single unit that makes every other portion fall out on its own. Miss it, and you are guessing. Find it, and the division is not just correct but visibly fair, so plain that anyone can check your working."

She pushes them into the harder cases the academy loves to set. "Three-way splits work the same. Share ninety coins in the ratio two to three to four. Total shares?" Nine. "One share?" Ten coins. "So the portions are twenty, thirty, and forty — and they sum to ninety." She writes each portion as a fraction of the whole, too, for the ones who like the other view: two-ninths, three-ninths, four-ninths of ninety. "Both routes land in the same place, because a share is a ninth of the whole here. Counting shares and taking fractions of the whole are two doors into the same room."

And she teaches the check, because a fair split must be checkable. "Always add the portions back. If they do not sum to the whole you started with, you have made an error, and better you find it than the person you shorted. Fairness you cannot verify is only a promise. Fairness you can show is a proof."

A girl named Wren stayed after, stacking the counters, and said the part that stayed with her was not the arithmetic but the feeling — that a way of splitting things could be so plainly fair it took all the fear out of the sharing.

Portion sat with her at the long bench. "At my mother's table, a bad split could break a friendship that had lasted a hundred years," she said. "A good split — a fair one, done so everyone could see the working — kept the whole valley at peace. Fairness you can show is a kind of gift you give the whole room." Wren looked at the two honest piles, eight and twelve, settled on the bench, and felt the quiet relief of a thing shared without anyone being wronged.