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We used Maple™ to study exotic sacks containing one three sided die and one \(k\) sided die.
Here is the new function we wrote for this calculation:
The function m3 takes an integer \(k\) and checks each integer \( \lceil \frac{k}{4} \rceil \leq m \leq \lceil \frac{k}{2} \rceil \) to see if swapping \( \chi_m\) from a fair \(k\)- sided die with the factor \( (x^2+x+1) \) from a fair 3-sided die results in a strict pair of dice. Once it finds a nonstrict pair, it checks that all larger values of \(m\) also give nonstrict pairs. In this way, the function computes the quantity \(M_3(k)\) defined in the paper and checks Conjecture 31 part 1. (The lack of output from the second for loop below is a lack of counterexamples---that is, this part of the conjecture is true in the range we compute.)
We send the output both to the screen (the printf commands) and to a log file (the fprintf commands).
m3 := proc(k)::integer;
local m, mk, mm, pm, maxm;
global tc, vaguec;
vaguec := 0;
maxm := iquo(k,2);
mk := maxm;
for m from iquo(k,4) to maxm do
pm := expand(quo(1.0 * x^(k)-1.0, (1.0*x-1.0)*f(m, k),x)*f(1,3));
if (signtype(pm,k) = -1) then
mk := m-1;
printf("%4d %4d %+-14.7f %+-14.7f ", k, mk, evalf(mk/k), evalf((mk/k) - (5/12)));
fprintf(tc,"%4d %4d %+-14.7f %+-14.7f ", k, mk, evalf(mk/k), evalf((mk/k) - (5/12)));
break;
end if;
end do;
for mm from (m+1) to maxm do
pm := expand(quo(1.0 * x^(k)-1.0, (1.0*x-1.0)*f(mm, k),x)*f(1,3));
if (signtype(pm,k) >= 0) then printf("**** %4d ", mm); fprintf(tc, "**** %4d ", mm); end if;
end do;
if (vaguec > 0) then printf(" [%d]", vaguec);fprintf(tc, " [%d]", vaguec); end if;
printf("\n");fprintf(tc,"\n");
return mk;
end proc;
Here we show the output of m3 for \( 6 \leq k \leq 200\). We also check part 4 of Conjecture 31. (Here is the full output file.)
kmin := 6: kmax := 200: mindiff := kmin+143:
printlevel := 0:
with(LinearAlgebra):
mthree := Vector(kmax):
tcname := sprintf("triconelog%dto%d.txt",kmin,kmax);
diffname := sprintf("triconedifflog%dto%d.txt",kmin,kmax);
tc := fopen(tcname, WRITE):
tdiff := fopen(diffname, WRITE):
printf("\n k M(k) M(k)/k M(k)/k-5/12\n\n"):
fprintf(tc, " k M(k) M(k)/k M(k)/k-5/12\n\n"):
fclose(tc):
for k from 6 to kmax do
tc := fopen(tcname, APPEND):
mthree[k] := m3(k):
if (k >= mindiff) then
diffmk := mthree[k]-mthree[k-143] -60:
if (diffmk <>0) then
fprintf(tdiff, "%4d %4d\n", i, diffmk):
end if:
end if:
fclose(tc):
end do;
fclose(tdiff):
k M(k) M(k)/k M(k)/k-5/12
6 2 +0.3333333 -0.0833333
7 2 +0.2857143 -0.1309524
8 3 +0.3750000 -0.0416667
9 3 +0.3333333 -0.0833333
10 4 +0.4000000 -0.0166667
11 4 +0.3636364 -0.0530303
12 5 +0.4166667 +0.0000000
13 5 +0.3846154 -0.0320513
14 5 +0.3571429 -0.0595238
15 6 +0.4000000 -0.0166667
16 6 +0.3750000 -0.0416667
17 7 +0.4117647 -0.0049020
18 7 +0.3888889 -0.0277778
19 7 +0.3684211 -0.0482456
20 8 +0.4000000 -0.0166667
21 8 +0.3809524 -0.0357143
22 9 +0.4090909 -0.0075758
23 9 +0.3913043 -0.0253623
24 10 +0.4166667 +0.0000000
25 10 +0.4000000 -0.0166667
26 10 +0.3846154 -0.0320513
27 11 +0.4074074 -0.0092593
28 11 +0.3928571 -0.0238095
29 12 +0.4137931 -0.0028736
30 12 +0.4000000 -0.0166667
31 13 +0.4193548 +0.0026882
32 13 +0.4062500 -0.0104167
33 13 +0.3939394 -0.0227273
34 14 +0.4117647 -0.0049020
35 14 +0.4000000 -0.0166667
36 15 +0.4166667 +0.0000000
37 15 +0.4054054 -0.0112613
38 15 +0.3947368 -0.0219298
39 16 +0.4102564 -0.0064103
40 16 +0.4000000 -0.0166667
41 17 +0.4146341 -0.0020325
42 17 +0.4047619 -0.0119048
43 18 +0.4186047 +0.0019380
44 18 +0.4090909 -0.0075758
45 18 +0.4000000 -0.0166667
46 19 +0.4130435 -0.0036232
47 19 +0.4042553 -0.0124113
48 20 +0.4166667 +0.0000000
49 20 +0.4081633 -0.0085034
50 20 +0.4000000 -0.0166667
51 21 +0.4117647 -0.0049020
52 21 +0.4038462 -0.0128205
53 22 +0.4150943 -0.0015723
54 22 +0.4074074 -0.0092593
55 23 +0.4181818 +0.0015152
56 23 +0.4107143 -0.0059524
57 23 +0.4035088 -0.0131579
58 24 +0.4137931 -0.0028736
59 24 +0.4067797 -0.0098870
60 25 +0.4166667 +0.0000000
61 25 +0.4098361 -0.0068306
62 26 +0.4193548 +0.0026882
63 26 +0.4126984 -0.0039683
64 26 +0.4062500 -0.0104167
65 27 +0.4153846 -0.0012821
66 27 +0.4090909 -0.0075758
67 28 +0.4179104 +0.0012438
68 28 +0.4117647 -0.0049020
69 28 +0.4057971 -0.0108696
70 29 +0.4142857 -0.0023810
71 29 +0.4084507 -0.0082160
72 30 +0.4166667 +0.0000000
73 30 +0.4109589 -0.0057078
74 31 +0.4189189 +0.0022523
75 31 +0.4133333 -0.0033333
76 31 +0.4078947 -0.0087719
77 32 +0.4155844 -0.0010823
78 32 +0.4102564 -0.0064103
79 33 +0.4177215 +0.0010549
80 33 +0.4125000 -0.0041667
81 33 +0.4074074 -0.0092593
82 34 +0.4146341 -0.0020325
83 34 +0.4096386 -0.0070281
84 35 +0.4166667 +0.0000000
85 35 +0.4117647 -0.0049020
86 36 +0.4186047 +0.0019380
87 36 +0.4137931 -0.0028736
88 36 +0.4090909 -0.0075758
89 37 +0.4157303 -0.0009363
90 37 +0.4111111 -0.0055556
91 38 +0.4175824 +0.0009158
92 38 +0.4130435 -0.0036232
93 39 +0.4193548 +0.0026882
94 39 +0.4148936 -0.0017730
95 39 +0.4105263 -0.0061404
96 40 +0.4166667 +0.0000000
97 40 +0.4123711 -0.0042955
98 41 +0.4183673 +0.0017007
99 41 +0.4141414 -0.0025253
100 41 +0.4100000 -0.0066667
101 42 +0.4158416 -0.0008251
102 42 +0.4117647 -0.0049020
103 43 +0.4174757 +0.0008091
104 43 +0.4134615 -0.0032051
105 44 +0.4190476 +0.0023810
106 44 +0.4150943 -0.0015723
107 44 +0.4112150 -0.0054517
108 45 +0.4166667 +0.0000000
109 45 +0.4128440 -0.0038226
110 46 +0.4181818 +0.0015152
111 46 +0.4144144 -0.0022523
112 46 +0.4107143 -0.0059524
113 47 +0.4159292 -0.0007375
114 47 +0.4122807 -0.0043860
115 48 +0.4173913 +0.0007246
116 48 +0.4137931 -0.0028736
117 49 +0.4188034 +0.0021368
118 49 +0.4152542 -0.0014124
119 49 +0.4117647 -0.0049020
120 50 +0.4166667 +0.0000000
121 50 +0.4132231 -0.0034435
122 51 +0.4180328 +0.0013661
123 51 +0.4146341 -0.0020325
124 52 +0.4193548 +0.0026882
125 52 +0.4160000 -0.0006667
126 52 +0.4126984 -0.0039683
127 53 +0.4173228 +0.0006562
128 53 +0.4140625 -0.0026042
129 54 +0.4186047 +0.0019380
130 54 +0.4153846 -0.0012821
131 54 +0.4122137 -0.0044529
132 55 +0.4166667 +0.0000000
133 55 +0.4135338 -0.0031328
134 56 +0.4179104 +0.0012438
135 56 +0.4148148 -0.0018519
136 57 +0.4191176 +0.0024510
137 57 +0.4160584 -0.0006083
138 57 +0.4130435 -0.0036232
139 58 +0.4172662 +0.0005995
140 58 +0.4142857 -0.0023810
141 59 +0.4184397 +0.0017730
142 59 +0.4154930 -0.0011737
143 60 +0.4195804 +0.0029138
144 60 +0.4166667 +0.0000000
145 60 +0.4137931 -0.0028736
146 61 +0.4178082 +0.0011416
147 61 +0.4149660 -0.0017007
148 62 +0.4189189 +0.0022523
149 62 +0.4161074 -0.0005593
150 62 +0.4133333 -0.0033333
151 63 +0.4172185 +0.0005519
152 63 +0.4144737 -0.0021930
153 64 +0.4183007 +0.0016340
154 64 +0.4155844 -0.0010823
155 65 +0.4193548 +0.0026882
156 65 +0.4166667 +0.0000000
157 65 +0.4140127 -0.0026539
158 66 +0.4177215 +0.0010549
159 66 +0.4150943 -0.0015723
160 67 +0.4187500 +0.0020833
161 67 +0.4161491 -0.0005176
162 67 +0.4135802 -0.0030864
163 68 +0.4171779 +0.0005112
164 68 +0.4146341 -0.0020325
165 69 +0.4181818 +0.0015152
166 69 +0.4156627 -0.0010040
167 70 +0.4191617 +0.0024950
168 70 +0.4166667 +0.0000000
169 70 +0.4142012 -0.0024655
170 71 +0.4176471 +0.0009804
171 71 +0.4152047 -0.0014620
172 72 +0.4186047 +0.0019380
173 72 +0.4161850 -0.0004817
174 73 +0.4195402 +0.0028736
175 73 +0.4171429 +0.0004762
176 73 +0.4147727 -0.0018939
177 74 +0.4180791 +0.0014124
178 74 +0.4157303 -0.0009363
179 75 +0.4189944 +0.0023277
180 75 +0.4166667 +0.0000000
181 75 +0.4143646 -0.0023020
182 76 +0.4175824 +0.0009158
183 76 +0.4153005 -0.0013661
184 77 +0.4184783 +0.0018116
185 77 +0.4162162 -0.0004505
186 78 +0.4193548 +0.0026882
187 78 +0.4171123 +0.0004456
188 78 +0.4148936 -0.0017730
189 79 +0.4179894 +0.0013228
190 79 +0.4157895 -0.0008772
191 80 +0.4188482 +0.0021815
192 80 +0.4166667 +0.0000000
193 80 +0.4145078 -0.0021589
194 81 +0.4175258 +0.0008591
195 81 +0.4153846 -0.0012821
196 82 +0.4183673 +0.0017007
197 82 +0.4162437 -0.0004230
198 83 +0.4191919 +0.0025253
199 83 +0.4170854 +0.0004188
200 83 +0.4150000 -0.0016667