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
Original
No Three in a Line at MathPickle

What it is

Build as many towers as the dots can hold, with no three on one straight line. That includes slanted lines, steep or shallow. On the Find Them All boards, the goal is every different biggest city.

The rules

A small field of dots with three towers on it, and above it the count 3/6: three towers standing of the six to build. Taps build a fourth, a fifth and a sixth tower, and the count goes up with each, to 6/6.3/64/65/66/6
Build the indicated number of minarets.
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
Never put three on a line.
Under a small field of dots, a shelf of three columns of boxes. A sixth tower makes a biggest city shaped like two arrows pointing in, its sight lines flash green, and it shrinks into the first column's one box. Then a second biggest city is built, its sight lines flash green, and it shrinks into the top box of the second column; the towers on the field flip across a dashed mirror into its mirror image, which shrinks into the box below. The last box stays empty: another biggest city is still to find.
Find all possible configurations.

What it gives you to practise

Spotting straight lines at every slant. Rows and columns are easy to check. The lines that go two across and one down, or three across and one down, are the ones players miss, and after a few boards they start to notice them. The puzzle also teaches symmetry. Each biggest city goes on the shelf with its quarter turns and mirror images, so two cities that look different can turn out to be the same city turned around. Cities that are not square often cannot fit two towers in every row, and working out why is part of the puzzle. Whether every square city can fit two in every row is still an open question in mathematics.

In a classroom

Henry Dudeney posed the puzzle in 1917. Gordon Hamilton of MathPickle teaches it from grade 1. The Quarter Turns and Two Mirrors packs come from his "No three minarets in a line" symmetry sheets: on these boards, each tower you build comes with its copies. The 4 × 4 city, Medina, is a good one to solve as a class. It has eleven biggest cities, but only four are really different; the rest are turns and mirror images of those four. Printed, the board is an empty field of dots, plus any towers the sheet starts with, ready for a pencil.

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