tmua.fyi Paper 2 Challenge 2 · Question 10

Which of the following establishes that a $10\times10$ board cannot be tiled by $1\times4$ pieces, each of which covers four squares in a row or in a column?

100

is not divisible by

4

, so no tiling exists.

BColour the board like a chessboard. Each piece covers two squares of each colour, and the two colours occur unequally often.

CNo piece can be placed in a corner of the board.

DThe board has an even number of squares, so any tiling would use an odd number of pieces.

ENothing establishes it: a tiling does exist.

FColour the square in row

i

and column

j

with

i+j

modulo

4

. Each piece covers all four colours once, so a tiling would need

25

squares of each colour; in fact they occur

25,26,25

and

24

times.

GColour the columns in the repeating pattern

1,2,3,4,1,2,\ldots

. Each piece covers all four colours, and the colours occur

30,30,20

and

20

times.

HA tiling would use

25

pieces, and

25

is odd.

examday

tmua.fyi Paper 2 Challenge 2 · Q10