There is one final possibility remaining.  If a card is placed next
  to another card, and they both have arrows pointing at each other,
  a battle ensues.  
                   card 1 | card 2
                    _____________
                    |\    |     |
                    |    -|-   -|
                    |/    |  | \|
                    -------------
  The above placement would result in a card battle.  Each battle has
  three phases where different numbers are displayed.
  Phase 1:
    * Each card has a power as discussed previously.  This value falls
      between the min and max listed in the table.  Each card also has
      a defense fitting the the above chart.
                 A            B
      Example : 4P22 attacks 1M01
      The first number that appears on card A is its attack power, say
      70 (4 = between 64 and 79).  Card A is a physical card, and card
      B has 0 physical defense, so the first number to appear on card
      B is its defense, say 7 (0 = between 0 and 16).
  Phase 2:
    # Next, the computer rolls a number between 0 and the number shown
      in phase 1.  This will be the *actual* attack or defense.  Let's
      say it rolls a 66 for card A, and 1 for card B.
  Phase 3:
    # The number rolled in phase 2 is subtracted from the number in
      phase 1.  This guarantees that the number will be positive, and
      the highest number wins.  So:
      Card A: 70 - 66 : 4
      Card B: 7  - 1  : 6
  There are a few things this should tell you :
  1.)  Higher rolls are BAD.  You want low rolls so less is subtracted
       from the total number.
  2.)  A weak card can defeat a strong card if the roll is in its favor,
       look at how the 1M01 defended itself against a 4P22.
  If you want to know how likely a card is to win a battle, here's the
  basic formula:
                 1 + Power of Weak
   100 * (1 -  ----------------------)
               2*(1+ Power of Strong)
  So in our example, you have:
                 1 + 7             142 - 8         136
   100 * ( 1 - --------- ) = 100 * ---  --- = 100 * --- = 94.4%
               2(1 + 70)           142 142         142
  So in our example, card A will win the battle 94% of the time.  But
  in the example of the battle, it lost because of a bad roll; that's
  the 6% it loses.
  But, since you may not know attack/defense values right away, you
  can get a basic idea.  Using our example again, take the maximum
  attack card A could have (79) and the lowest defense card B can have
  (0) and use our equation.  The result is 99.4%.  Now, take the minimum
  attack card A can have (64) and the maximum defense card B can have
  (15) and use our equation.  The result is 87.7%.
  So, in our example, if you don't know the values of the cards fighting
  it out, card A will have a 88-100% chance of beating card B if it
  is attacking.
  If you played card A, good job.  You would have won the card battle,
  but you had a 6% chance of losing, and lost.  Your opponent now
  controls both cards.  If you had won, you'd control both cards.