The Game of Primes

I just read a post called "Math Notes" by Greg Ross at Futility Closet, and as always, it was very interesting, read it all.
But one part in particular got my mind turning. He wrote:
1
19
197
1979
19793
197933
1979339
19793393 and
197933933 are all prime.
It struck me that this could be a great student game/project.

Player one picks a number, a single prime digit.
Player two must pick a second digit so that the two form a two digit prime.
Play continues until one can not make a prime.

An interesting alternative rule cold be to allow the subsequent numbers to be added at either the front or the end of the string.

It might also be interesting to create a "map" of the strings possible..


For example the one above would be on a mapping that starts with one and then branches to 11, 13, 17, and 19 The 11 could go to 113 but then would be a dead end as 1131, 1133, 1135, and 1137 (and 1139) are all factorable. The 13 node can be extended to 131, 137, 139.
Students might amaze themselves with the long strings they could create.

For the string that started all this above, one might add 1979339333 is also prime, but if you wished, you could use 1979339339 which is prime as well.  That's a quick ten digit string of primes.  It is not easy for students to test if either of those is the end of the string, but along the way they have learned something about testing primes.

A good solitare version might be to start with the digits 0,1,2,3,4,5,6,7,8,9 and try to build a tree that leads to a ten digit prime with all the numerals by adding one digit at a time.