User:Whiteknight1066

OK, this is my test page.

I've got this account to help my Dad's mate get his article on nontransitive dice published.

The basic text is:

How to Make Nontransitive Dice, Dual Nontransitive Dice and Dual Nontransitive Dice Magic Squares

Also called Efron’s Dice, Nontransitive dice (NTD) are like paper-rock-scissors.

The first die has a better than 50% chance of beating the second, the second has a better than 50% chance of beating the third and so on with the last die having a better than 50% chance of beating the first die.

Though NTD can be real dice, in general a DIE is a collection of quantities, FACES are the quantities and a SET is a working whole. Thus three boxes each holding three numbered balls is a SET of three three-FACED DICE. For example: Box A’s balls are numbered 4, 4 and 1 Box B’s balls are numbered 3, 3 and 3 Box C’s balls are numbered 5, 2 and 2

Play high number wins and randomly pick balls from boxes A and B. Picking either of A’s 4’s beats all three of B’s balls. Picking A’s I loses to all three of B’s balls. So, on average, A beats B 6 out of 9 picks [66.7%]. Likewise B beats C 6 out of 9 picks [66.7%] and C beats A 5 out of 9 picks [55.6%]. Tally the wins [17] and the picks [27] to get a win ratio of 63.0% for the set.

To make NTD sets use Trepal’s method based on work published in Scientific American [July 1998, page 111] its three steps are best shown by example. Start with numbers or letters arranged from big to small, such as the alphabet with A bigger than B, B bigger than C etc. To make four three-faced dice choose any three four-letter groups. The groups can share some letters such as ABCD and DEFG sharing D, but no groups can be identical. For example: ABCD, FGHI and KLMN.

Stack the groups and mark each end of row letter with parenthesis, A		B	C	[D] F		G	H	[I] K		L	M	[N]

The stacking arranges the quantities from big to small with the biggest being to the left in rows and at the top in columns. In rows, A beats B, B beats C etc but D wraps around and loses to A. Parenthesis mark the losers.

To rotate a row move its ending letter to its start e.g.

A B C [D] rotates to [D] A B C, C [D] A B, B C [D] A.

Rotate rows so each column has more winners than losers thus making an NTD set.

A B C [D]		rotates to		A B C [D] F G H [I]					G H [I] F K L M [N]					M [N] K L

Many other arrangements are possible and work just as well, but putting all the losers in a diagonal line helps in making duals and magic squares explained in the next sections. The columns are the dice and the rows are the faces. The NTD set is:

1st die:	faces A, G, M: beats the 2nd die 6 out of 9 picks [66.7%] 2nd die:	faces B, H, N: beats the 3rd die 5 out of 9 picks [55.6%] 3rd die:	faces C, I, K: beats the 4th die 5 out of 9 picks [55.6%] 4th die:	faces D, F, L beats the 1st die 5 out of 9 picks [55.6%]

The set’s win ratio is 58.3%

Special rule: When groups share letters such as A B C and CDE sharing C, rotate to put the shared letters in the same columns. For example:

A B C [D]		rotates to		A B C [D] B C D [E]					[E] B C D C D E [F]			 		E [F] C D

The more shared letters the higher the set’s win ratio, this set has 72.2% while the previous example’s set, which has no shared letters, is 58.37. Shared letters and having more dice per set increases the win ratio. Having more faces per die decreases it. Thus six three faced dice have a higher win ratio than three six-faced dice.

With DUAL NTD sets one set’s dice and faces are the other set’s faces and dice. To make duals start with stacked groups, mark the row ends then rotate the rows to get the first set. To get the second set, mark the column bottoms then rotate the columns.

For example:

9  8   [7]       then     9   8   [7]      then   9   8    7     then    9    4   [2] 6  5   [4]       	   5   [4]   6             5   4    6             5   [3]   7 3   2   [1]               [1]   3   2             [1]  [3] [2]            [1]   8    6

The first NTD set has columns for dice, rows for face numbers and goes left to right. 1st die is 9,5,1. 2nd die is 4,3,8. 3rd die is 2,7,6.

The second NTD set has rows for dice, columns for face numbers and goes bottom to top. 1st die is 1,8,6. 2nd die is 5,3,7. 3rd die is 9,4,2.

In both sets each die beats its neighbour 5 out of 9 picks. [55.6%]

In MAGIC SQUARES all columns, rows and diagonals add to the same amount. If dual NTD sets are square, have an odd number of numbers per side and use consecutive numbers starting with one [as in the above example] they can become magic squares. Find the middle number: in the above example’s 1 to 9 range, it is 5. Get it into the square’s center by moving columns from one side to the other, moving rows top and bottom or both.

9  4   2			moves the third column to become	2   9   4 5  3   7								7   5   3 1   8   6								6   1   8

Now, with 5 in the center all columns, rows and diagonals add to 15: a magic square.

Note: Made by any method most odd number sided magic squares are Dual NTD sets.

References: George Trepal’s work published in Scientific American July 1998, page 111