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By starting with the double six, a full set of dominoes are used by a person to lay a line of dominoes in one direction only until the supply is is finished or you cannot carry on. What will the number at the end of the last domino and why? I know the answer: it is 6 but I don't know how to prove it (since I actually tried to lay dominoes in one direction). Can anyone help me? The second part of the question says that the same method is used but on both directions of the double six. The person discovers that he can use all the dominoes that he has available, but someone says that he had hid one before he started. Can the person identify immediately (without checking the hole set) which domino has been hidden?

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After the double six, you need to lay down a piece with a six and some other number. And if you now focus on the sixes, you will find that there are five other poeces with a six on it, and they can only be played as pairs. So, the last six will have to be played as the last piece with the six as the last number.

For the second question: if you take the first domino chain that went only one way from the double sixes to the single six, and you cut that chain somewhere and past that chunk on the other end of the double sixes (with the original end piece with the six against the double sixes now), then you can remove a piece from one of the ends ... But since that was the conecting piece in the original chain, that piece will have the numbers that are now at the end of both ends of the resulting domino chain.

Of course, this only shows that if you create a new domino chain from an old one in this particular way and then remove one from the end, the removed piece will have the two numbers at the end ... But this does not mean that if you create the chain some other way, the missing piece still has to have the two numbers at the ends. In fact, as we will see, this is not always true.

What is true is that any chain that uses all dominoes will have to have the same number on either end: the double of each number will have to be between two pieces with that number (or be at the very end), and otherwise you can only place that number in pairs. So, if you have one number on one end, you have to get that same number on the other end.

This means that if the missing piece is not a double piece, you will get an odd number of pieces with that that number as a single number, and so a piece with that number will have to go at the ends. In other words: if you see two different numbers at the ends of the chain, then you know the missing piece is the piece with those two numbers.

However: if the missing piece is a double piece (e.g. Double 4), then this is no longer true, and indeed you can make a single domino-chain where both ends end in, say, a 3. So, if you see the same number at the ends of the chain, then you know that some double piece is missing, but you can't tell which one!