Tuesday, September 9, 2014

Eleusis: Problem Solving

The Problem

 Given the following Eleusis card set-up, find a rule to describe it. Then, list three more not yet played cards that can follow the set-up.

The Process

   My first initial thought when viewing the Eleusis card set-up was that color was not the sole factor of the rule. There were two blacks, two reds, black, red, four black. This did not appear to be any sort of pattern. Next I noticed how color affected cards of the same number being right and wrong. I found these four instances:
  • A black 10 could not follow a black 8 but a red 10 could.
  • A black 2 could follow a red 3 but a red 2 could not.
  • A black 6 could follow a black king (or 13) but a red 6 could not.
  • A black 8 could follow a black 7 but a red 8 could not.
   After discovering this, I was able to eliminate adding/subtracting of the cards to be part of the rule. Color would not affect the resulting sum or difference of the two numbers. Getting on the thought of addition and subtracting, I decided to look at the differences between the cards. At the time, I was thinking anything I discover will be helpful - whether I find a pattern or not. As a result I ended up with the following:

   In hindsight, I discovered what I already knew: that the difference between the numbers is irrelevant. Otherwise, if this was untrue any 10 should have been able to follow the 8 because the difference would still be the same. However, I stumbled upon something important. What is the value of the ace? In this rule, was the ace being viewed as 1 or 14, odd or even? To figure this out, I started to propose guesses as to what the rule may be.
  • If you go from even to even, you must change suite.
This instance only occurred once. So I ruled it out and reformulated the hypothesis.
  • If you go from even to even or odd to odd, you must change suite.
There were however cases not covered by this rule, so this was not it either. For example it could not explain why the 2 of hearts could not follow the 3 of diamonds. While neither of these were the rule, thinking and testing my guesses was helpful. I saw what did not work and began to notice my focus on odd/even and the suite/color of the card. So I made the following chart (assuming the ace is 1):

   This simplification of the information given, brought me to a pattern and a rule to describe the Eleusis set-up

The Solution

   The rule for the pictured Eleusis is: An even card must be followed by a red card, and an odd card must be followed by a black card. So based upon my rule I can play the following 3 cards:


    Describing my thought process was the easy part of solving this problem. Although, it took an effort to remind myself to record what all I had been thinking. Having stepped back and re-evaluated the problem several times over a couple days and still no answer, I thought to myself: perhaps it is unsolvable. Perhaps the diagram  is too limited to reveal the rule. Unwilling to give up and a strong desire to overcome the problem - it couldn't be that difficult could it? - I continued to search for the solution from where I left off each time. A lot of thinking, as well as various methods, went into solving this one problem. Even when I was not looking at the problem, I tried to think about it. It wasn't so much as arriving at a correct answer. After all, my rule only holds if the ace is 1 and there was no example in the diagram to confirm an even red card can follow an even red card. I kept trying because I enjoyed the complexity of the problem and simply did not want to give up. I wanted to make sense of the problem at hand and I did.

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