Monday, July 7, 2014

A card-deck challenge

Hello Mathemathinking readers,

today's post is a bit special in two ways: First, I'm not Cory. My name is Bernhard, and I was Cory's Math office mate at UBC in Vancouver.

Janelle, Cory and me at a hike last summer

Several times Cory and I chatted about writing a blog together, and I'm happy to finally contribute to his page. In fact, it will be a double header: I want to challenge you to a fun quiz, and will give a week's time to find a solution, before I reveal my answer. Email full solutions to me, or discuss the problem in the comments below (hints are welcome, but please don't post full solutions in the comments). Ready? Here we go:

You need a partner, and a standard 52-card deck with four suits and 13 cards per suit (no jokers). Five random cards are drawn from the deck. While you see none of the cards, your partner is allowed to view all five. Your partner then decides which four of those cards she wants to reveal to you and in which order. Is there an ordering scheme that you and your partner can agree on beforehand that allows you to determine the fifth (hidden) card from the choice and order of the four cards that your partner shows?

If you think that's always possible, explain your scheme.
Otherwise, explain why such a general algorithm can not exist.

Can you find a scheme that uniquely determines the fifth hidden card?
To clarify: Since your partner can choose which four cards to show, he also chooses the card you have to guess. Also, all the information you get is the rank and the suit of the four cards, and the order in which they are revealed. There is no additional information in the way the cards are revealed (eg your partner can not flip some of the cards to signal additional information).

Good luck, I look forward to your answers!