superprismatic

I think you have to determine what the coins are to determine which way the head is facing. If the rotating coin were a US cent, then the head faces to the viewer's right. Nearly all other US coins (except for very recent mintings) have heads that face to the viewer's left. I suppose that you have to use the circumferences of the coins to find out which pair would satisfy the other conditions of the OP.

In sequencing DNA, technicians break up long molecules into smaller pieces because it is easier to sequence small segments than long ones. They must then reconstruct the target molecule by fitting the small segments together using overlaps as a guide. The small segments are sequenced beginning at either end in a random fashion. So, for example, the segment ATACAG may also be sequenced as GACATA. In this puzzle, I have chopped up many identical 50-long sequences into pieces of lengths 5, 6, 7, and 8. 30 of these pieces are: ATACAG TGACAT GTCTTA GTCGAGA AACGA CAAGG CAGTGTGA GTGGTGT CCGATGAC AGACAA TGTGA ATACAGTG CTGTG ACATA GTGTCG TACAGT GTCTTAG CATAA GGTGG GACTCCAG TGTGA CCAGTG TGTGACA GTGACATA TGTGGT ATTCTGA TGAATACA TAGCCG TGTGACCT CCTCA I have insured that these 30 pieces completely cover the 50-long sequence with which I started. Can you find that sequence?

It seems rather difficult to pull the wool over the Denizens!

I solved it with by exhaustion over all possible paths of moves less than 38. I saw that Martin Gardner's column of 1975 had a 38-move solution, so I started there.

I found a novice programmer mistake in my code which computed the values in my post #14. The value I get now is the same that bonanova and BMAD get. I'm relieved that my approach agrees with the concise generating function one.

Whoops! I overlooked the word "consecutive".