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Arrangements of pips on a die

Posted By: Timothy Chow
Date: Monday, 8 March 2010, at 9:55 p.m.

In Response To: Do high dice win more often? (Marv Porten)

Marv Porten wrote:

dice that had these numbers: 1, 3, 4, 5, 5, 6 (no 2 and two 5s)

This reminds me of a remark that Danny Kleinman makes in his book Vision Laughs at Counting with Advice to the Dicelorn, which I recently purchased. He observes that even after you enforce opposite sides of a die to sum to seven, there is still a choice of "handedness"; on a standard die, the faces with 1, 2, and 3 on them are arranged counterclockwise, but you could construct a mirror-image die on which they are arranged clockwise. He remarks that he can see no practical difference between the two, but would not be surprised if some player were to insist on one form or the other.

Kleinman was of course mocking the superstitions of certain backgammon players. However, there is in fact a legitimate reason to insist on the standard orientation. Namely, if you train yourself to recognize standard dice instantly, then you can spot illegal dice of the form above (with no 2 and two 5's for example) much more readily. Specifically, if the cheater swaps in a die which has a 5 where it should have a 2, and you see the illicit 5 along with 2 adjacent faces, then you will notice right away that something is amiss because the three numbers will be arranged in the opposite orientation from what they should be. For example, on a standard die, 1-3-5 are arranged counterclockwise, but on the illicit die, 1-3-5* (where 5* denotes the illicit 5) are arranged clockwise.

One can go even further. The 2, 3, and 6 can each be printed on the die in two different orientations, which are rotated 90 degrees from each other. On a standard die, if you draw a diagonal line connecting the dots on the 2, another diagonal line connecting the dots on the 3, and a third diagonal line that makes the 6 into an uppercase "N", then these three diagonal lines will all meet at the 2-3-6 corner (which the mathematician John Conway calls the "home" vertex). In principle the 2, 3, or 6 could be printed so that these diagonal lines point in a "wayward" rather than a "homeward" direction (or some could be homeward and others wayward). Again, if you train yourself to recognize the standard dice instantly, then you will be better equipped to spot illicit dice.

Messages In This Thread

• Do high dice win more often?
  Max Urban -- Monday, 8 March 2010, at 1:43 p.m.
  • Do high dice win more often?
    Matt Cohn-Geier -- Monday, 8 March 2010, at 3:39 p.m.
  • Do high dice win more often?
    W Womack -- Monday, 8 March 2010, at 5:41 p.m.
  • Do high dice win more often?
    Randy -- Monday, 8 March 2010, at 6:22 p.m.
  • Do high dice win more often?
    Marv Porten -- Monday, 8 March 2010, at 6:41 p.m.
    • Do high dice win more often?
      Matt Cohn-Geier -- Monday, 8 March 2010, at 6:52 p.m.
      • Do high dice win more often?
        Jason Lee -- Monday, 8 March 2010, at 7:06 p.m.
      • Do high dice win more often?
        joe freedman -- Monday, 8 March 2010, at 11:04 p.m.
    • Do high dice win more often?
      Chuck Bower -- Monday, 8 March 2010, at 7:36 p.m.
    • Arrangements of pips on a die
      Timothy Chow -- Monday, 8 March 2010, at 9:55 p.m.
      • Arrangements of pips on a die
        Jason Lee -- Monday, 8 March 2010, at 11:42 p.m.
        • Arrangements of pips on a die
          Timothy Chow -- Tuesday, 9 March 2010, at 1:00 a.m.
          • Arrangements of pips on a die
            Bob Koca -- Tuesday, 9 March 2010, at 4:04 a.m.
            • Arrangements of pips on a die
              Timothy Chow -- Tuesday, 9 March 2010, at 4:10 a.m.
              • Arrangements of pips on a die
                Tom Keith -- Tuesday, 9 March 2010, at 11:33 a.m.
            • Arrangements of pips on a die
              Jason Lee -- Tuesday, 9 March 2010, at 6:41 a.m.
              • Arrangements of pips on a die
                Bob Koca -- Tuesday, 9 March 2010, at 7:13 a.m.