Unit 2 · Topic 08 · Number Play
Ms. Rao lined up seven children of different heights and asked each to say 0, 1, or 2 based on the children standing right beside them.
Some shouted 2 because both neighbors were taller. Others said 0 because no neighbor beat them — and those children turned out to be supercells.
The same local rule works for numbers in a row: a supercell beats every cell touching it directly.
"A supercell is a cell whose number is greatest among the cells directly touching it," said Ms. Rao, writing a row on the board: 645, 700, 233, 215, 1809. Kabir checked each cell against only its immediate neighbors. 645 loses to 700 on its right. 700 beats both 645 and 233 — that makes 700 a supercell. Because 700 qualifies, its neighbor 233 cannot: a supercell and its neighbor can never both be supercells. Kabir continued: 215 loses to 233; 1809 loses to 215. Only one supercell in the whole row: 700.
The next day Ms. Rao lined up seven children of different heights in random order. "Each child will say a number — 0, 1, or 2 — based on the children standing right beside you," she explained. Say 2 if both neighbors are taller than you. Say 1 if exactly one neighbor is taller and one is shorter. Say 0 if no neighbor is taller — you are taller than everyone touching you. Anaya noticed: the children who say 0 are taller than their neighbor children. They are the supercells of the line! Saying 0 does not mean everyone has the same height.
Kabir tested a pattern: what if every child stands in strictly ascending order, shortest to tallest? Each middle child has one taller neighbor on the right and one shorter on the left — all say 1. The tallest child at the end has only a shorter neighbor on the left — says 0. Nobody has two taller neighbors, so nobody says 2. In a line of five, four say 1 and exactly one says 0. For N children in ascending order, N minus 1 say 1.
Ms. Rao gave one more number row. Arrange 51, 29, 24, 20, and 10 from largest to smallest. Only 51 beats its single neighbor 29. Every middle number is beaten by the cell on its left, and 10 loses to 20. Just one supercell again. "Numbers tell us things about their neighbors," said Ms. Rao, "when you compare locally, patterns appear."
A supercell is a cell whose number is greatest among the cells directly touching it — its immediate left and right neighbors only.
In the height line game, each child says 0, 1, or 2: say 2 if both neighbors are taller, say 1 if exactly one neighbor is taller, say 0 if no neighbor is taller. Children who say 0 are the supercells.
The first and last cells (or corner children) have only one neighbor to compare with.
Two supercells can never be immediate neighbors — if both were supercells, each would need to be bigger than the other.
In strictly ascending order: exactly one child says 0, N minus 1 say 1, and nobody says 2.
Maximum supercells in a row of n cells: (n+1)/2 when n is odd, and n/2 when n is even.
Line up five friends by height in ascending order. Predict who says 0, 1, or 2, then check.
Find all supercells in the row 51, 29, 24, 20, 10 arranged from largest to smallest.
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