No Three in a Line

Fit in as many towers as the dots can hold with never three on one straight line — and start to see the slanted lines nobody draws.

The No Three in a Line board

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Grades
1–6
Subjects
Shapes & Space, Logic
Kind
Puzzle
Cost
Free, no account, no ads
Original
No Three in a Line at MathPickle

What it is

The city is a field of dots, and a tap builds a tower on one. The only rule is that no three towers may stand on one straight line: not across, not down, not on any slant, however long or shallow. The number over the field says how many towers the biggest city holds. Done checks the rule and nothing else, so every city that keeps it is accepted, and when Done finds three in a line it draws that line from edge to edge.

The rules

Build the biggest city of towers the dots can hold — with never three towers on one straight line.

A small field of dots. Taps build four towers on four dots, one at a time; then a tap on one of the towers takes it down again. Last, a finger presses on an empty dot and draws down over the next one, and a tower stands on each.
Tap a dot to build a tower on it, or hold and draw across the dots. Tap a tower to take it down.
Five towers stand on a field of dots. A line is swept across the field through three of them on a slant, two across for every one down, counting one, two, three — and it turns red: three towers on one straight line is not allowed.123
No three towers may stand on one straight line: not across, not down, not on any slant. Done shows the line it finds.
A sixth tower makes the biggest city the dots hold; its sight lines flash green and it shrinks into the top box of a column on the shelf. The towers left on the field flip across a dashed mirror into a second city of the same kind, which shrinks into the box below. An empty box beside the column stands for another kind of city just as big.
Done puts each biggest city on the shelf and fills in its turns and mirror images below it. An empty box is a city still to find.
A city with a quarter-turn arrow at its centre: one tap builds a tower, and three copies swing round the centre to their places. Then a city with two dashed mirrors on its diagonals: one tap, and the copies slide across the mirrors.
When a turn or two mirrors are drawn on the city, every tower you build comes with its copies.

What it gives you to practise

Seeing the lines nobody draws. Rows and columns are easy to watch; the slants that go two across for one down, or three across for one down, are the ones that catch a city out, and after a few boards they start to show up without being looked for. Then symmetry: every biggest city is kept on a shelf with its quarter turns and mirror images, so two cities that look different can turn out to be the same one turned round. The odd-shaped cities often cannot hold two towers in every row, and finding out why is the puzzle. Whether every square city can is a question mathematicians have not settled.

In a classroom

Henry Dudeney set the puzzle in 1917; Gordon Hamilton of MathPickle teaches it from grade 1, and his "No three minarets in a line" symmetry sheets are the Quarter Turns and Two Mirrors packs, where every tower is built together with its copies. The 4 × 4 city, Medina, is a good one to do together: it has eleven biggest cities, and only four really different ones. The board prints as an empty city of dots, with any minarets the sheet gives, for 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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Near it on the shelf