Subject: Dominoes

Name: Mark

Who is asking: Teacher Level: Secondary

Question: A standard dominoe set consists of 28 pieces, from double-zero to double-six.

1. Is it possible to arrange all those pieces in a straight line in such a way that the dots of any pair of adjacent pieces match? Please include picture
   
   
2. Is it possible to arrange them in a circle and still meet the conditions in 1?

Both are possible. We won't include a picture, but show you how to build the circle starting from smaller circles:

Start with a list of all 28 dominoes:

	[0,0], [0,1], [0,2], [0,3], [0,4], [0,5], [0,6],
	[1,1], [1,2], [1,3], [1,4], [1,5], [1,6],
	[2,2], [2,3], [2,4], [2,5], [2,6],
	[3,3], [3,4], [3,5], [3,6],
	[4,4], [4,5], [4,6],
	[5,5], [5,6],
	[6,6]

Then, starting as 0, start to make a chain and keep at it as long as you can (crossing off from the list the dominoes that you use):

[0,1][1,2][2,3][3,4][4,5][5,6][6,0][0,0][0,2][2,3][3,0][0,4][4,5][5,0].

When you get stuck, the last number on the last dominoe is the same as the one you started with (in this case 0). Can you figure out why?

Now notice that on the remaining dominoes, there are an even number of 1, an even number of 2, and so on. Start with a 1 and make a chain as long as you can with the dominoes that are left:

[1,3][3,4][4,1][1,1][1,5][5,6][6,1].

You get stuck again, but you can insert the chain that you just completed between the first and second dominoes of your first chain.

And you keep on going in the same way: make chains as long as you can with the dominoes that are left, and when you get stuck insert the new chain in the original chain.

Note: If you try to make a chain in this way with the 21 dominoes that do not contain a 6, you will find that you cannot. But if you try to make a chain with the 15 dominoes that contain only numbers among 0, 1, 2, 3, 4, you will succeed using the method above. What is the pattern here?