Chico and Dico were a famous pair of magicians, and the following
trick used to be part of their repertoire:
From an untampered standard deck of 52 playing cards, a volunteer from the audience would pick five cards at random. He would give these to Chico. Chico would reorder them and give them back to the volunteer. This would all happen entirely without Dico seeing it. The volunteer would then show Dico the first four cards, according to the order decided by Chico, and Dico would magically be able to divine the identity of the fifth card. It is not difficult to show that no ESP and no sleight of hand is required to pull this trick off. An agreed strategy between Chico and Dico regarding how the cards should be ordered is all that is really required.
This month's riddle has to do with generalizations over this trick.
We will consider decks with an arbitrary number of cards, which we will
denote
The riddle this month will be composed of two parts. Answer
## Part 1:For what combinations ofn, k and j is the trick
possible? Prove your answer.
Note that in this part we ask for an exact characterization, and not mere bounds. ## Part 2:Consider the special casej=1. Describe a strategy for
Chico and Dico to follow, that will allow them to perform the trick with
an arbitrary k and with the largest possible associated n
(as determined in part 1, above).
Note that in this part of the riddle it is not enough to show that such a
strategy exists, nor even to give a recipe that will show how to come up
with such a strategy for any |
## List of solvers:## Both parts:Oded Margalit (7 May 5:00)## Part 1 only:## Part 2 only: |

Elegant solutions can be submitted to the puzzlemaster at __riddlesbrand.scso.com__.
Names of solvers will be posted on this page. Notify if you don't want
your name to be mentioned.

The solution will be published at the end of the month.

Enjoy!