Visual Patterns

Three figures made of tiles go up on the board: figure 1 has 3, figure 2 has 6, figure 3 has 12 — each figure holding two complete copies of the one before it. Two questions, always the same: how many tiles in figure 10? and how many in figure $n$? Last year’s patterns grew by adding; these grow by multiplying — and that is the point.

How we play

1. Study the figures in silence. See the structure, not the total.
2. Predict figure 10 from how you see the pattern growing.
3. Defend your count of figure $n$ by pointing at the picture.

Three ways to see figure 10

• “Every figure is two of the one before, so figure 10 is the seed of 3 doubled nine times: $3\times 2^9=1536$.”
• “Ten doublings of 3 is 3072, but that overshoots by one doubling — the seed itself uses no doubling: $\frac{3\times 2^{10}}{2}=1536$.”
• “Each figure is 3 clusters, and the clusters double: figure 10 is 3 clusters of $2^9$ tiles each.”

Three expressions — $3\times 2^{n-1}$, $\frac{3\times 2^n}{2}$, $3\cdot 2^{n-1}$ counted by clusters — one pattern. The Exponent Laws show they were always the same count.

One variation

Table the totals — 3, 6, 12, 24 — and difference them. The differences are 3, 6, 12: the pattern’s growth is the pattern itself, one step behind. A constant ratio is the geometric sequence’s fingerprint — the same one Patterns That Count dusts for, and Sequences and Their Rules names.

The general lives inside the specific

Nobody counts figure 10 tile by tile. The way you see figure 3 — two of yesterday, three doubling clusters — is already the formula for figure $n$. Say what you see, and the algebra writes itself.