A Domino Chain of Sums 
       
|1 3|2 0|

Place the dominos 1-3, and 2-0 together in a line. By taking all the pips in unbroken conjunction I can get all the numbers from 1 to 6 inclusive. Thus 1, 2, and 3 can be taken alone; then 1 and 3 make 4; 3 and 2 make 5; and the entire set makes 6. It would not have been allowed to take the 1 and the 2 to make 3. The numbers would not have been in conjunction.
  1. Take any two dominos from a double-6 set and place them in a chain so that you can make the pips in this way sum to any number from 1 to N, where N is as large as possible. As above, the dominos need not be placed 1 against 1, 2 against 2, and so on, as in play.
  2. Same question for three dominos.
  3. Same question for four dominos.

Source: Henry Ernest Dudeny, 536 Curious Problems & Puzzles, #481.

Solutions were received from Igor Volkov, Sandy Thompson, Kirk Bresniker, Larry Baum, and Joseph DiVincentis. Kirk's solution is included below and shows a few extensions: 
In addition to the solutions for 2,3 and 4 here is what I've found for
5 and 6 dominos:

1 1 4 3 found sums 1 .. 9
1 3 3 2 found sums 1 .. 9
2 5 1 3 found sums 1 .. 9
4 1 2 6 found sums 1 .. 9

1 1 1 5 5 4 found sums 1 .. 17
1 1 4 4 4 3 found sums 1 .. 17
1 1 6 4 2 3 found sums 1 .. 17
1 1 6 4 3 2 found sums 1 .. 17
1 3 6 2 3 2 found sums 1 .. 17
3 2 1 6 4 4 found sums 1 .. 17

1 3 6 6 6 2 3 2 found sums 1 .. 29

1 1 1 5 6 6 6 4 5 4 found sums 1 .. 39
1 3 6 6 4 6 6 2 3 2 found sums 1 .. 39
2 5 2 6 6 6 6 4 1 3 found sums 1 .. 39
5 1 2 6 6 6 6 4 3 4 found sums 1 .. 39

1 1 1 5 6 6 5 6 6 4 5 4 found sums 1 .. 50
2 5 2 6 6 5 6 6 6 4 1 3 found sums 1 .. 50

Interestingly, there is a relation between some of the patterns:

1 1     4 3 found sums 1 .. 9
1 1 4 4 4 3 found sums 1 .. 17

1 1 1 5             5 4 found sums 1 .. 17
1 1 1 5 6 6 6 4     5 4 found sums 1 .. 39
1 1 1 5 6 6 5 6 6 4 5 4 found sums 1 .. 50

1 3             3 2 found sums 1 .. 9
1 3         6 2 3 2 found sums 1 .. 17
1 3 6 6     6 2 3 2 found sums 1 .. 29
1 3 6 6 4 6 6 2 3 2 found sums 1 .. 39

The sum of the dots on the domino added is also the increase between
the maximum of the two patterns.
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