Problem 1
The opposite corners of an 6 by 6 grid have been removed.
Can the remaining shape be covered using 17 dominoes?
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Opposite corners are the same colour.
Each domino covers one square of each colour…
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But if two squares of opposite colour are removed…
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How many tilings have six dominoes in these positions?
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Bernard Murphy, MEI
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Bernard Murphy, MEI
So the 6 red dominoes uniquely determined the tiling.
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Can we do better than 6 dominoes?
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Bernard Murphy, MEI
Problem 2 Can we do better still?
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Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Bernard Murphy, MEI
Number o rows
Number of columns
1
2
3
4
5
6
7
8
n
2
0
1
1
1
2
2
2
3
 n  1
 3 
4
0
1
6
0
2
8
0
3
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3?
?
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The rectangle on the left has area 56 square units and is fault free.
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Problem 3
What is the smallest rectangle (by area) that can be tiled fault free?
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Bernard Murphy, MEI
Problem 4
Valhalla
Val and Hal take it in turns to place a domino on the grid. Val
places dominoes vertically and Hal places dominoes
horizontally.
You can choose to be either Val or Hal. Your opponent then
chooses whether to go first or second. The first player
unable to place a domino loses.
Can you find a winning strategy?
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● ●
● ●
●
●
-----------------
Bernard Murphy, MEI
●
-----------------
●
●
-----------------
●
●
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
Clever!
Bernard Murphy, MEI
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
6
3
5
6
3
1
1
5
Bernard Murphy, MEI
2
2
3
2
0
2
0
1
3
5
4
6
6
1
6
0
4
2
3
1
0
4
2
2
3
0
2
5
4
4
1
5
3
0
0
0
6
6
3
5
6
4
5
4
1
4
1
5
Problem 5
How many ways are there of tiling a 2×n rectangle using dominoes?
n
No of
ways
1
2
3
1
2
3
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4
5
6
7
8
9
How many ways are there of tiling a 2×n rectangle using dominoes?
n
No. of
ways
1
2
3
4
5
6
7
8
9
1
2
3
5
8
13
21
34
55
Bernard Murphy, MEI
How many ways are there of tiling a 2×n rectangle using dominoes?
n
No. of
ways
1
2
3
4
5
6
7
8
9
1
2
3
5
8
13
21
34
55
Why Fibonacci?
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3
1
2
4
3
0
0
3
6
6
3
6
4
1
2
5
4
2
0
6
0
1
5
2
1
6
5
1
5
3
2
6
2
2
4
4
3
0
0
4
5
5
1
6
3
5
4
1
0
1
2
4
3
6
0
5
5
2
1
2
4
4
1
0
6
6
3
2
2
2
4
0
5
3
0
5
6
5
1
3
0
0
4
6
2
1
4
3
1
6
5
5
0
4
1
5
1
6
6
0
5
4
3
2
3
4
6
1
3
3
0
2
6
6
5
4
1
1
1
5
4
1
3
4
0
3
6
3
4
0
0
2
6
0
6
5
5
6
4
2
4
5
2
1
3
0
0
2
5
3
3
0
3
4
4
3
5
1
1
6
0
2
2
2
1
6
5
2
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
0
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
3
4
5
6
4
5
6
5
6
6
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
0
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
3
4
5
6
4
5
6
5
6
6
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
http://www.puzzlesmix.net/modules/domino/domino.php#clean
Bernard Murphy, MEI
0
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
3
4
5
6
4
5
6
5
6
6