Dice problem

Discussion in 'Math' started by Mark44, Apr 8, 2008.

  1. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Roll a pair of ordinary dice.
    1. Multiply the values of the two top faces.
    2. Multiply the values of the two bottom faces.
    3. Multiply the value on the top face of the first die by the value on the bottom face of the second die.
    4. Multiply the value on the top face of the second die by the value on the bottom face of the first die.
    5. Add the four numbers from the previous four steps.
    If you multiplied and added correctly you will always get the same number.

    Some questions:
    1. What's the number?
    2. Why do you always get that number? (Again, assuming that your arithmetic is correct.)
    Mark
     
  2. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Hint: The spots (pips?) on a die follow certain rules. If I can see three faces of a die, I can tell you how many spots are on the remaining faces.
     
  3. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    I haven't thrashed out a set of answers, but the first thing that springs to mind is the fact that opposite faces of a die always sum to 7. Is this a factor?

    Dave
     
  4. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    It most certainly is.
     
  5. hgmjr

    Moderator

    Jan 28, 2005
    9,030
    214
    Greetings Mark44,

    Before I blurt out my answer, I should ask if this is a homework assignment. I don't want to deprive you of the pleasure of deriving the answer yourself.

    In either case, what do you believe the answers to be.

    hgmjr

    PS. Dave, I think you are on to the key.
     
  6. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Nope, hgmjr, it's not a homework assignment, but I did give out a lot of homework assignments in the 21 years I was a teacher. It's actually a fairly simple problem once you understand how the spots are arranged on a die.
     
  7. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    It's half-past midnight here so I won't crack open my "Games-set". Pen and paper tomorrow lunchtime I think!

    I've come across many-a-maths problem with dice, and the root is always the fact that the opposite sides sum to 7 - something I discovered by accident when a curious youngster!

    Dave
     
  8. hgmjr

    Moderator

    Jan 28, 2005
    9,030
    214
    Since that is the case then I attach my answer below.

    hgmjr
     
  9. MusicTech

    Active Member

    Apr 4, 2008
    144
    0
    Yep, a simple case in point example of factor by grouping...

    Good example for an algebra 2 class.

    Mark22, do you teach Algebra 2 or Pre-Calc, by any chance?
     
  10. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    I taught all levels of high-school math for two years, and I taught all levels that we offered in the community college, near Seattle, where I worked for 18 years. There was also some student teaching before I got my job in the high school, plus a couple of quarters as a teaching assistant (I actually taught the class) in the university I went to.

    While I was working at the community college I taught everything from just plain arithmetic to algebra and precalculus with trig, engineering calculus, linear algebra, and differential equations. I also taught classes in BASIC, Pascal, one class in Modula-2, many, many C classes, a couple of classes in C++, and several classes in FORTRAN. During that time I taught myself Java and x86 assembler.
    Mark
     
  11. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Yep, it's 49. You can determine this with only two variables, though.

    Let x = no. of spots on the top face of die 1
    Let y = no. of spots on the top face of die 2

    Carrying out the multiplications described in the problem gives:
    xy + x (7 - y) + y (7 -x) + (7 - x)(7 - y)
    = xy + 7x - xy + 7y -xy + 49 - 7x - 7y + xy
    = 49

    Mark
     
  12. MusicTech

    Active Member

    Apr 4, 2008
    144
    0
    Wow, very impressive, that's all there is to say (to the reply two posts back, of course)
     
  13. uPC

    New Member

    Apr 8, 2008
    1
    0
    Variable designation is incorrect in solution paper...although the logic is clear.

    Better to correct it....
     
  14. drewlas

    New Member

    May 12, 2008
    7
    0
    Well,
    The numbers on the first die are x and (7-x) and,
    The numbers on the second die are y and (7-y).
    Just go from there.....

    drewlas
     
  15. recca02

    Senior Member

    Apr 2, 2007
    1,211
    0
    any idea why that is the case?
    Of course the dice won't become biased if we change it a little:confused:.
     
    Last edited: May 14, 2008
  16. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    It is either a) convention (i.e. the way it is always done with a reason lost in annals of time), or b) there is some statistical reason which ensures that by having opposite faces summing to 7 statistically we have a "fair dice".

    I honestly can't say which of the above it is, but a quick Google search doesn't shed any light on it.

    Dave
     
Loading...