Here is a single decker EIGHT-SANDWICH. There are 150 different 8-sandwiches, plus their mirror images.

Sandwiches are made under the conditions that, between any pair of ones in the list there is one digit, between any pair of twos there are two digits, between any pair of threes there are three digits and so on.

An extension to the sandwich number problem is to construct lists where the same sandwich condition is met, but each digit occurs three times, rather than twice. . Apart from the trivial 000, there are solutions for the 24-digit sandwiches containing three each of the digits 1 to 8.

Finding triple-8-sandwiches is already a somewhat daunting task for the empiricists, but you may like to try writing a program to find them.

The rationalists may like to think about when it is impossible to find triple-n-sandwiches. For which values of n can you prove that no triple-n-sandwiches exist?

This problem first appeared on NRICH in September 1997 and the solution there gives a program for printing out all the solutions for a given single decker n-sandwich.

There is a simple proof of the non-existence of single decker n-sandwiches for certain values of n given in a Further Inspiration article written by Paul Cockayne.

Thank you Paul Cockayne for suggesting this problem.