Pica and Serra

Pica studied the run-length numbers written beside each row and column.

Serra studied the visibility numbers perched at the ends of the lines.

"Beginners see a blank grid with just some numbers around the edge and freeze," said Pica. "'How can I fill the middle when the clues are all on the outside?' But that's the beauty of it. The border tells you the inside — if you know how to read it. Watch."

Pica went first, reading the run-length numbers beside a line — because those little border-numbers describe exactly which cells get filled, if you reason them out.

"Here's a row, ten cells wide, blank," said Pica. "And beside it, the clue: 7. That means somewhere in this row is a single unbroken run of seven filled cells." Pica studied it. "Now — seven filled cells in a row of ten. Where can't they NOT be? Watch this." Pica reasoned it out. "If the run is squeezed all the way left, it fills cells one through seven. All the way right, cells four through ten. And any way you slide it, cells four through seven get filled in every single case. So without knowing exactly where the run sits, I know for certain those four middle cells are filled." Pica shaded them in, certain. "That's run-length reasoning — the numbers on the edge, plus the size of the line, force the overlap in the middle. Bigger clues in tighter spaces force even more. Little by little, edge-number by edge-number, a hidden picture emerges from the border — and often, when you've filled enough, you realize the numbers were secretly drawing a picture the whole time."

Then Serra tilted her head to read a visibility number at the end of a line — because a single number, counting what can be seen, pins down the heights inside.

"My clues are even sneakier," said Serra. "Imagine each cell in this line holds a tower, of some height one through five, all different. And my clue, sitting at the end of the line, is a number: how many towers you can see looking down the line from that end — where a taller tower hides all the shorter ones behind it." Serra pointed to a clue reading 1. "A one? That means from this end, you can see only one tower. And that's only possible if the very first tower is the tallest — the five — because it hides everything behind it. So that cell is forced: it must be the five." Serra placed it, certain. Then a clue reading 5 at another end. "And a five means you can see all of them — which only happens if they climb up in perfect order, shortest to tallest: one, two, three, four, five. Forced, the entire line, from a single number." Serra smiled. "One little visibility clue on the edge, and I've deduced heights I never directly saw. The border counts what's visible; I reason back to what must be standing there."

And between the two of them — Pica reading the run-lengths, Serra reading the sightlines — the whole grid filled in from its edges inward, a hidden interior conjured entirely out of the numbers written around its border. Not one guess. Just careful reading of the clues that had been sitting in plain sight the whole time.