Use mathematical induction to show that a rectangu- lar checkerboard with an even number of cell

Use mathematical induction to show that a rectangu- lar...

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish a statement for all natural numbers. It involves proving a base case and then showing that if the statement holds for an arbitrary case, it also holds for the next case. In tiling problems, induction is used to demonstrate that a property holds for increasingly larger boards by relying on the property holding for smaller boards.

Domino Tiling

Domino tiling refers to covering a given board completely with domino pieces, where each domino covers exactly two adjacent cells. The challenge in these problems often lies in ensuring that the pieces cover the board without gaps or overlaps, taking into account the board's structure and any modifications such as missing squares.

Checkerboard Coloring and Parity

A standard checkerboard is colored in an alternating pattern, meaning that adjacent squares are of opposite colors. This coloring helps in analyzing tiling problems, since each domino placed on the board will always cover one white and one black square. The parity (balance in the number of white and black squares) is crucial to determine whether a tiling is feasible, especially when squares are removed.

Invariant Principle

The invariant principle involves identifying a condition that remains unchanged under certain operations. In the context of domino tilings on a checkerboard, the invariant is that every domino covers one white and one black square. This invariance is utilized to prove that if one square of each color is removed from an even-sized board, the remaining squares can still be entirely covered by dominoes.

Transcript

00:01 We're asked to use mathematical induction to show that rectangular checkerboard with an even number of cells, one white and one black, by the statement.

00:20 We have that let pnk be the statement, a 2n by k.

00:40 So here we're assuming that there are an even number of rows.

00:45 And the thing is, he says an even number of cells, you have either the rows or the columns have to have an even number.

00:57 That loss of generality, we'll assume as any number of rows, and if not, they can just root get the checkerboards so that the columns become the rose.

01:06 Pnk is the statements, 2n times k checkerboard with one missing white and missing black square can be covered by dominoes and for the basis, say n equals 1, we're trying to prove p1k for all k in the positive integers.

02:00 So if you do this, like k, be a positive integer, consider a 2x4x4x4 with one missing white, one missing black square.

02:45 Let the two missing squares be the eighth and jth column for checkerboard.

03:17 If i is equal to j, then we just tally the checkerboard replacing dominoes vertically.

03:40 But if i is not equal to j, how the checkerboard replacing all dominoes vertically in the columns up to j minus 1 in the column.

03:55 From j plus 1 to n.

04:00 Sorry, there should be i minus 1 and the column from j plus 1 to n.

04:35 So up to i minus 1 and from j plus 1 to n, and then tile the remaining part by placing all dominoes horizontally, like bricks.

05:25 And so we have shown that statement p1 is true...

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