Math PicnicEvery sheet is a promise

Color it first, step by step

A board with two corners cut off, and a box of dominoes.

Practice Practice sheetNew questions
The question

Can the rest be covered exactly by double lockers, each covering two squares that share an edge, with none left over and none overlapping? Say YES or NO, and say how you know.

Color the board, then count the two colors

6 × 4board− 2 corners
each domino:1 dark + 1 light
11dark=11light
YES
  1. A 6 by 4 board with two corners cut out, and the question is whether dominoes can cover the rest exactly — each domino over two squares that touch. Trying to lay them settles nothing; a coloring does.
  2. Color it like a chessboard. A domino always sits on two squares that share an edge, and those are one dark and one light — never two of a color. So a full covering needs exactly as many dark squares as light ones.
  3. The two removed corners are at the same end of an edge, so on this even board one is dark and one is light: 11 dark and 11 light are left, equal, which passes the test. And it can be done — that edge's row keeps an even run and every other row is full, so lay them along each row.
  4. So the answer is YES.

How it works.

  1. 01

    A 6 by 4 board with two corners cut out, and the question is whether dominoes can cover the rest exactly — each domino over two squares that touch. Trying to lay them settles nothing; a coloring does.

  2. 02

    Color it like a chessboard. A domino always sits on two squares that share an edge, and those are one dark and one light — never two of a color. So a full covering needs exactly as many dark squares as light ones.

  3. 03

    The two removed corners are at the same end of an edge, so on this even board one is dark and one is light: 11 dark and 11 light are left, equal, which passes the test. And it can be done — that edge's row keeps an even run and every other row is full, so lay them along each row.

  4. 04

    So the answer is YES.

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