MEI Conference 2009
Proof by colouring:
a masterclass
Presenter: Bernard Murphy
bernard.murphy@mei.org.uk
© MEI 2009
Workshop F9
8 by 8
10 by 10
© MEI 2009
11 rows
8 rows
7 rows
© MEI 2009
Problems
These tiles are called from left to right:
straight tetromino,
L-tetromino,
T-tetromino,
skew tetromino
and square tetromino.
1. Show that an 8 x 8 grid cannot be covered by 15 T-tetrominoes and a square tetromino.
2. Show that an 8 x 8 grid cannot be covered by 15 L-tetrominoes and a square tetromino.
3. Show that a 10 x 10 grid cannot be covered by 25 T-tetrominoes.
4. Show that a 10 x 10 grid cannot be covered by 25 L-tetrominoes.
5. Show that an 8 x 8 grid cannot be covered by a square tetromino and 15 other
tetrominoes chosen from the straight tetrominoes and the Z-tetrominoes.
6. A rectangular floor is covered by 2 x 2 and 1 x 4 tiles. One tile smashed. There is a tile
of the other kind available. Show that the floor cannot be covered by rearranging the tiles.
7. Show that an 8 x 9 rectangle cannot be covered by 1 x 6 rectangles.
8. A beetle sits on each square of a 9 x 9 chessboard. At a signal each beetle crawls
diagonally onto a neighbouring square. Then it may happen that several beetles will sit on
some squares and none on others. Find the minimal possible number of free squares.
9. A 23 x 23 square is completely tiled by 1 x 1, 2 x 2 and 3x3 tiles. What is the
minimum number of 1 x 1 tiles needed?
Call this shape a hexagonal triangle of side 3.
10. Show that a hexagonal triangle of side 8 cannot be covered using one copy of
along with any combination of
and
11. Show that a hexagonal triangle of side 11 cannot be covered with 22 copies of
12. Show that a hexagonal triangle of side 7 cannot be covered with any combination of
the shapes
© MEI 2009
Blocking Tetrominoes
►
►
►
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Two tetrominoes require four blocking squares, one needs five and the other two need six.
Blocking Pentominoes
On the first chessboard below it is impossible to place the X pentomino without it overlapping one of the ten squares.
The minimum numbers of squares for all the pentominoes are given below. Can you find any other solutions?
I
F
F
F
I
L
N
P
P
F
I
L
N
P
P
F
I
L
N
P
I
L
F:14
I:12
N
L
L:16
T
T
T
T
U
U
U
U
V
W
U
V
W W
T
V
V
V
W W
X
X
X
X
N
N:16
Y
X
Y
Z
Z
Y
Z
Y
Z
Z
Y
P:16
T:14
U:16
X
© MEI 2009
V:16
W:15
X:10
Y:16
Z:14