A cafe’s temperamental jukebox contains 5 singles, numbered 1 to 5. On payment of £1, the jukebox always begins by playing single 3. The singles that follow are determined by a sequence of 11 moves entered by the payee, each move either being up (U) or down (D) one single, e.g.

UDUDUDUDUDU.

Unfortunately, the rickety old jukebox has a life of its own, and will only make every 1st, 2nd, 3rd, 4th or 5th move, where this number is randomly selected at payment. For instance, if the jukebox decides to play only every second move of the sequence above, then the singles played will be

3 (D—>) 2 (D—>) 1 (D—>) 5 (D—>) 4 (D—>) 3

The singles 1 and 5 are two of the cheesiest most irritating songs in existence, anathema to the cafe’s hipster clientele. Your task is to find the sequences of moves that avoid playing these cheesy songs, regardless of the number of moves the jukebox skips.

Please enter the sequence that would appear in last place in an alphabetical list of all possible sequences that will not antagonise the cafe's clientele.

Mathsbombe Competition 2017 is organised by the The Department of Mathematics at The University of Manchester.
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