Yooklid

C. Dudley Langford, a Scottish mathematician, was watching his little boy play with colored blocks. There were two blocks of each color, and the child had piled six of them in a column in such a way that one block was between the red pair, two blocks were between the blue pair, and three were between the yellow pair. Substitute digits 1, 2, 3 for the colors and the sequence can be represented as 312132.

This is the unique answer (not counting its reversal as being different) to the problem of arranging the six digits so that there is one digit between the 1′s and there are two digits between the 2′s and three digits between the 3′s.

Langford tried the same task with four pairs of differently colored blocks and found that it too had a unique solution. Can you discover it? A convenient way to work on this easy problem is with eight playing cards: two aces, two deuces, two threes, and two fours. The object then is to place them in a row so that one card separates the aces, two cards separate the deuces, and so on.

There are no solutions to “Langford’s problem,” as it is now called, with five or six pairs of cards. There are 26 distinct solutions with seven pairs. The reader will find it a pleasant exercise to work out some of its solutions. No one knows how to determine the number of distinct solutions for a given number of pairs except by exhaustive trial-and-error methods, but perhaps you can discover a simple method of determining whether there is a solution.

I will post the solution to this problem on Christmas day. For those of you who might not have arrived at the solutions by then, you can count that as my Christmas present to you.