A Square and New Dice

Two simpler problems this week:
  1. In a 3x3 grid, place the numbers 1 to 8 in the outside squares such that on each side of the grid, the two corners sum to the number between them.
  2. One of my daughter's new games is called "Sum Swamp". It comes with two regular dice and a third (6-sided) "operation" die. The operation die has three "+" symbols and three "-" symbols. To determine the distance to move, all three dice are rolled. If a "+" appears, the numbers on the regular dice are added. If a "-" appears, the smaller (or equal) number is subtracted from the larger (or equal) number. What are the probabilities of each possible roll, and what is the average roll for these dice?

Source: Gabe, citing Hard to solve brain teasers. Original.


Solutions were received from Rick Meyer, Stephane Higueret, Colin Bown, Joseph DeVincentis, Samantha Levin, Peter Sim, Mete Kart, John Hewson, Sudipta Das, Richard Karis, Francesc Sunol, Vinko Marinkovic, Charles Tatum, Mike Zinniel, Stan Neevel.
  1. From Colin Bown:
       1 7 6 
       4   8 
       3 5 2 
    
    This is made simpler by a couple of realizations:

  2. From Joseph DeVincentis:
    When a + is rolled (half the time), the probabilities are the same as for a normal pair of dice. When a - is rolled, an unfamiliar set of probabilities results:

    Probabilities for various rolls:

    Sum (with +)   Prob.    Difference (with -)   Prob.
       2           1/72               0           6/72
       3           2/72               1          10/72
       4           3/72               2           8/72
       5           4/72               3           6/72
       6           5/72               4           4/72
       7           6/72               5           2/72
       8           5/72
       9           4/72
      10           3/72
      11           2/72
      12           1/72
    
    Overall probabilities for each roll:
    Roll   Prob.
      0    6/72
      1   10/72
      2    9/72
      3    8/72
      4    7/72
      5    6/72
      6    5/72
      7    6/72
      8    5/72
      9    4/72
     10    3/72
     11    2/72
     12    1/72
    Average roll is easily computed from the above.
    0 * 6/72 + 1 * 10/72 + 2 * 9/72 + ... = 322/72 =~ 4.472

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