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.

Play No Three in a Line

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.

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.

The rules

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

Tap a dot to build a tower on it, or hold and draw across the dots. Tap a tower to take it down.

No three towers may stand on one straight line: not across, not down, not on any slant. Done shows the line it finds.

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.

When a turn or two mirrors are drawn on the city, every tower you build comes with its copies.