Can you solve it? The club sandwich problem

Here are two puzzles I ordered on room service.

1) A number sandwich is a line of digits such that there is one digit sandwiched between the 1s, two digits sandwiched between the 2’s, three digits sandwiched between the 3’s, and so on. For example, 312132 is a number sandwich with the digits 1,2 and 3

Construct a number sandwich with the digits 1,2,3 and 4. (Each digit will appear exactly twice.)

2) A number club sandwich is a number sandwich in which each digit appears exactly three times. The same rules as above apply: one digit is sandwiched between any two consecutive 1s, two digits are sandwiched between any two consecutive 2s, and so on.

Construct a number club sandwich with the digits 1 to 9. To help you out, I’ve placed five digits in their correct positions.

Number sandwiches are more commonly known in mathematics as Langford pairings, after the Scottish mathematician C. Dudley Langford who once noticed that his child had stacked three pairs of coloured blocks such that there was one block between the red pair, two blocks between the blue pair, and three blocks between the green pair. Langford (senior!) found a number sandwich for 3, 4, 7, 8, 11 and 15 pairs, and submitted his results to the Mathematical Gazette in 1958. (Homework for the reader: find them!)

I’ll be back at 5pm UK today with the answers. NO SPOILERS PLEASE! Instead please discuss sandwiches (mathematical or otherwise), or other interesting mathematical games derived from toddlers’ playing with their toys.

As it’s the last column before Christmas, it would be remiss of me not to mention my latest book, So You Think You’ve Got Problems?, a compendium of 150 puzzles and the stories behind them. There are logic puzzles, word puzzles, lateraling-thinking puzzles and much more. If you enjoy this column, I promise you will enjoy my book. I hope so, anyway, since I wrote it with you people in mind!

You can buy it at the Guardian Bookshop or in any other good bookshop!

Further reading: John E. Miller has a great page on Langford pairings here. Data Genetics also has an excellent article here.

Source: The number club sandwich puzzle was devised by Bernado Recamán Santos, a maths educator in Colombia, who is also the inventor of the mesmerising Recamán sequence.