S/O 2. Victor Barocas presents a non-transitive dice game with the dice below and the expected win-loss-tie counts out of 36 games. When two players each select a die and roll it, the second player can always choose a die giving an advantage over the first. To set up such a game with N dice, Barocas asks how many faces a die must have, and how to number the faces.

B: 3, 3, 3, 3, 4, 4B against C 18W-16L-2T

A: 1, 2, 3, 4, 5, 6A against B 16W-14L-6T

C: 2, 2, 2, 4, 6, 6C against A 16W-14L-6T

The dice have unequal sums (C=22, A=21, B=20) favoring C over A and A over B. Non-transitively B beats C because all B faces exceed half the C faces, giving 18 wins, and matched faces give ties, thus fewer than 18 losses. To expand the notion to N dice, we seek the largest set {Di, i = 1, N} of k-sided dice, having all k face values in [1, k], for which:

(1)the sums increase according to (kD)i+1 = (kD)i + 1 ;

(2)the lowest-sum die D1 is favored over the highest DN ;

(3)one of {Di, i = 2, (N – 1)} is numbered 1, 2, 3, … k .

If k is even, condition (2) is met if k faces of D1 exceed (k/2) faces of DNand any other face is matched across D1 and DN. Dice satisfying (2) that span the widest range of sums are:

D1: (k – 1) faces = (x + 1); 1 face = (x + 1 + )

DN: (k/2) faces = x; 1 face = (x + 1 + ); (k/2 – 1) faces = k

Here  is an increment used to create the single matched face.

The sums of D1 and DN must bracket the sum of the die numbered 1 to k to allow condition (3), that is (kD)1 < {1, 2, 3, … k} < (kD)N. Inserting the explicit sum expressions yields:

k(x + 1) +  < k(k + 1)/2 < x(k/2) + (x + 1 + ) + k(k/2 – 1)

which restricts x to [2, (k/2 – 1)]. To maximize N for a given k we take x = 2. The number of unique sums in {Di, i = 1, N} is N = (kD)N – (kD)1 + 1 which reduces to N = (k2 – 6k + 8)/2. If k is odd, a similar analysis leads to N = (k2 – 5k + 6)/2. Hence the face count needed for N dice is the smaller of:

after each result, if fractional, is rounded up to the next integer of like type. Below are the required numbers of faces for dice sets up to N = 40. Odd face counts are more often optimal.

N / 3 / 4 / 5-10 / 11-12 / 13-21 / 22-24 / 25-36 / 37-40
k / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12

Face numbering for D2 through DN – 1 must satisfy condition (1), starting at level i = 1 with (kD)2 = (kD)1 + 1 where (kD)1 is known. At i 2, Di+1 is generable from Di by adding 1 to each of l faces and deducting 1 from (l – 1) others. Some choices of increments lead to a Di+1 lacking advantage over Di, but a feasible Di+1 emerges if l is minimized (often to one face) and the (k/2) faces having the smallest values are fixed across most Di. Dice thus generated for N = 12 appear below.

Die / Face Values / Sum / Advantage / W-L-T
D1 / 3 3 3 3 3 3 3 5 / 26 / D1 > D12 / 32-31-1
D2 / 2 2 3 3 4 4 4 5 / 27 / D2 > D1 / 28-21-15
D3 / 2 2 3 3 4 4 5 5 / 28 / D3 > D2 / 26-22-16
D4 / 2 2 3 3 4 5 5 5 / 29 / D4 > D3 / 26-22-16
D5 / 2 2 3 3 5 5 5 5 / 30 / D5 > D4 / 24-20-20
D6 / 2 2 3 3 5 5 5 6 / 31 / D6 > D5 / 24-20-20
D7 / 2 2 3 3 5 5 6 6 / 32 / D7 > D6 / 26-22-16
D8 / 2 2 3 3 5 6 6 6 / 33 / D8 > D7 / 26-22-16
D9 / 2 2 3 3 6 6 6 6 / 34 / D9 > D8 / 24-20-20
D10 / 2 2 3 3 6 6 6 7 / 35 / D10 > D9 / 24-20-20
D11 / 1 2 3 4 5 6 7 8 / 36 / D11 > D10 / 29-27-8
D12 / 2 2 2 2 5 8 8 8 / 37 / D12 > D11 / 29-27-8

S/O 2.

If a die numbered 1 through k (condition 3) is not required, the same general form of D1 and DN admits x = 1, leading to:

thus a fixed face count enables a larger dice set. Eight faces, for example, allow N = 15 dice as below (versus 12) and give enough latitude in face numbering to ensure wins exceed losses by at least four in every case.

Die / Face Values / Sum / Advantage / W-L-T
D1 / 2 2 2 2 2 2 2 8 / 22 / D1 > D15 / 32-28-4
D2 / 2 2 2 3 3 3 3 5 / 23 / D2 > D1 / 35-8-21
D3 / 2 2 2 3 3 3 4 5 / 24 / D3 > D2 / 23-19-22
D4 / 2 2 2 3 3 4 4 5 / 25 / D4 > D3 / 25-21-18
D5 / 1 1 2 3 4 5 5 5 / 26 / D5 > D4 / 29-25-10
D6 / 1 1 2 2 5 5 5 6 / 27 / D6 > D5 / 27-22-15
D7 / 1 1 2 2 5 5 6 6 / 28 / D7 > D6 / 26-22-16
D8 / 1 1 2 2 5 6 6 6 / 29 / D8 > D7 / 26-22-16
D9 / 1 1 2 2 6 6 6 6 / 30 / D9 > D8 / 24-20-20
D10 / 1 1 2 2 6 6 6 7 / 31 / D10 > D9 / 24-20-20
D11 / 1 1 2 2 6 6 7 7 / 32 / D11 > D10 / 26-22-16
D12 / 1 1 2 2 6 7 7 7 / 33 / D12 > D11 / 26-22-16
D13 / 1 1 2 2 7 7 7 7 / 34 / D13 > D12 / 24-20-20
D14 / 1 1 2 2 7 7 7 8 / 35 / D14 > D13 / 24-20-20
D15 / 1 1 1 1 8 8 8 8 / 36 / D15 > D14 / 28-24-12