Diversity City

Every zone counts the different sizes next door, and the city scores their average — so the fewer, busier zones win.

The Diversity City board

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Grades
3–6
Subjects
Multiplying & Dividing, Shapes & Space
Kind
Puzzle
Cost
Free, no account, no ads
Original
Diversity City at MathPickle

What it is

The city is a square grid, and it is cut into square zones: drag out a square and it is built, tap it and it falls back into single lots. Every zone wears a number, which is how many different sizes of zone share an edge with it — a zone touching two 1 × 1s and a 3 × 3 scores 2, not 3. The city scores the average of all those numbers, and each city has three averages to reach, one after another. Done works the score out in front of you: the points climb, the zones climb, and then one is divided by the other.

The rules

Cut the city into square zones so that every zone has neighbours of many different sizes.

A grid of small squares, each showing a 1. A finger drags out a three by three square, which is built when it lets go, and the numbers around it change. Then two two by two squares are built.11111111111111111111111112×23×3111111112221212122211211312121222211211323
Cover the big square in zones.
One small square in the city is ringed and the squares it does not touch go dark. Its four neighbours flash. One at a time each sends a swatch of its colour to a strip below: a one by one, a two by two, a three by three, and a second one by one that lands on the first and disappears. The three different swatches pulse, and the ringed square shows 3.3
Neighbouring zones touch along an edge. A zone's number counts how many different sizes of neighbours it has.
A city of 11 squares, each showing its number. The squares light up one after another; underneath, one counter adds up their numbers and another counts the squares, together, reaching 20 and 11. A division sign appears between them, then about 1.82.222112113231111222432641751863117213831692181022011÷≈ 1.82
The city scores the average of all its zones.

What it gives you to practise

Averages, and what moves one. Every zone counts once on each side of the division, so a zone scoring below the city’s average pulls it down unless it lifts its neighbours by more, and a city of fewer, busier zones beats one cut into many small ones. The colours do the counting of sizes, since every size has its own, and holding a zone down lights just the zones it touches. The score stays hidden until Done is pressed, so each city is a guess about an average checked by working it out.

In a classroom

Gordon Hamilton, MathPickle, 2020. His 7 × 7 city in the video scores 30 ÷ 12 = 2.5, which he called pretty good; here every 7 × 7 goal is above it. The cities run from 4 × 4 to 12 × 12. Up to 8 × 8 the best possible city has been found by checking every one of them by computer; from 9 × 9 up, the goals sit under the best city a computer search has found, and a class may beat it. Every city that is scored stays on a wall beside the board, so the climb is there to talk about afterwards. The board prints as an empty grid, for graph paper and a pencil.

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The whole puzzle at one address, with every board in it. Whoever opens it stays inside it, and their progress stays in their own browser.

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