Broken TV

Fill a board square by square with the smallest number the rule allows, and see the picture the order paints.

The Broken TV board

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Grades
2–6
Subjects
Counting & Estimating, Logic
Kind
Puzzle

What it is

A square board is filled one square at a time, and the rule decides every number: in every row and every column there can be only one 1, two 2s, three 3s, four 4s and so on, and each new square gets the smallest number that keeps it so. The order the squares are filled in paints a picture. The first square is always a 1. In Fill the Screen the line is walked square by square, like a king in chess, and has to fill every square so that the few numbers given come out where they are given. In Find the Line the finished picture is given, and the line that painted it is the answer. Fill from Anywhere and Start Anywhere take away even the first square, and not every 1 in a picture can begin a line. Static ends on whole screens painted by lines chosen at random.

The rules

A four by four board with a line through it, nearly finished. Its last five squares are filled one at a time, along the bottom and up the left side, and every one of them is a 3: each row and column already holds its 1 and its two 2s.1111222222333333
Each row and each column in your TV can have at most one one, two twos, three threes etc.
The next square is empty. Its row already has a 1, and its column already has two 2s, so neither 1 nor 2 can go there. It gets 3, the smallest number it can.111223
When you visit a new square it automatically gets the smallest number possible.
A line has reached the middle of a board, with a dashed 2 to its right and a dashed 3 to its left. Stepping right paints a 1 where the board asks for a 2: the 1 pushes the 2 aside and is ringed red. Stepping left paints a 2 where the board asks for a 3: the 2 pushes the 3 aside and is ringed red.223211221232
Some squares can only be filled with a specific number.

What it gives you to practise

Counting along a row and down a column at the same time, and checking each count against a rule that never changes. Every square asks the same question: is there a 1 here already, are there two 2s, three 3s? The first time a number appears is the interesting moment, because a square only gets a bigger number once every smaller one is used up in its row or its column. Fill the Screen turns the rule round: for a 3 to come out where one is given, its row or its column must already hold a 1, and its row or its column two 2s, and the line must still leave itself a way through every other square, so the walk is a plan. Find the Line turns it round again, from the picture back to the order that made it.

In a classroom

A good first question is to draw a short line on a blank board and ask what number each square will get before it is written in. The 81 boards run from a picture of eight squares to a screen of forty-three. On most of the boards that give a few numbers, every one of them matters: without it, some other line would do.

Share or embed

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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