Maple calculations - Revision history

Teorth at 04:52, 10 July 2009

2009-07-10T04:52:14Z

← Older revision		Revision as of 21:52, 9 July 2009
Line 73:		Line 73:
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);
	[337, [a = 26, b = 89, c = 120, d = 80, e = 20, f = 1, g = 1]]		[337, [a = 26, b = 89, c = 120, d = 80, e = 20, f = 1, g = 1]]
	''This implies that <math>a+b+c+d+e+f+g+~~19g~~ \leq ~~356~~</math>''. Also shows that g=0 for 338+ point solutions.''		''This, together with the bound <math>c'_6=353</math>, implies that <math>a+b+c+d+e+f+g+16g \leq 353</math>''. Also shows that g=0 for 338+ point solutions.''
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),f>=1},assume=nonnegint,depthlimit=500,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),f>=1},assume=nonnegint,depthlimit=500,maximize);
	[353, [a = 24, b = 77, c = 179, d = 72, e = 0, f = 1, g = 0]]		[353, [a = 24, b = 77, c = 179, d = 72, e = 0, f = 1, g = 0]]
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	[80, [a = 24, b = 72, c = 180, d = 80, e = 0, f = 0, g = 0]]		[80, [a = 24, b = 72, c = 180, d = 80, e = 0, f = 0, g = 0]]
	''This shows that the only 356-point statistic is (24,72,180,80,0,0,0).''		''This shows that the only 356-point statistic is (24,72,180,80,0,0,0).''
	:X6 := {op(X6), A6+19*g <= ~~356~~};		:X6 := {op(X6), A6+16*g <= 353};
	:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};		:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};
	''xyzwuv1, xyzwuv2, xxxxxxx inequalities''		''xyzwuv1, xyzwuv2, xxxxxxx inequalities''

Teorth at 18:25, 13 June 2009

2009-06-13T18:25:43Z

@@ Line 67: / Line 67: @@
 ''These come from xxxxyz cubes, see Peake.991''
 :X6 := {op(X6),4*a+2*b+0*c+3*d+1*e+1*f+1*g <= 6,4*a+0*b+2*c+3*d+1*e+1*f+1*g <= 6,8*a+2*b+0*c+3*d+1*e+1*f+1*g <= 8,8*a+0*b+2*c+3*d+1*e+1*f+1*g <= 8,0*a+4*b+0*c+0*d+2*e+0*f+1*g <= 4,0*a+0*b+4*c+0*d+0*e+2*f+1*g <= 4,8*a+4*b+0*c+0*d+2*e+0*f+1*g <= 8,8*a+0*b+4*c+0*d+0*e+2*f+1*g <= 8};
-''These come from xxxyyz cubes, see Peake.991, Peake.993''</del>
+''These come from xxxyyz cubes, see Peake.991, Peake.993''
 :Y6 := subs([a=a/64,b=b/192,c=c/240,d=d/160,e=e/60,f=f/12],X6);
 :LPSolve(a+b+c+d+e+f+g,Y6,assume=nonnegint,depthlimit=500,maximize);
@@ Line 73: / Line 73: @@
 :LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);
   [337, [a = 26, b = 89, c = 120, d = 80, e = 20, f = 1, g = 1]]
-''This implies that <math>a+b+c+d+e+f+g+19g \leq 356</math>''.
+''This implies that <math>a+b+c+d+e+f+g+19g \leq 356</math>''.  Also shows that g=0 for 338+ point solutions.''
 :X6 := {op(X6), A6+19*g <= 356};
 :X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};

Teorth at 18:08, 13 June 2009

2009-06-13T18:08:02Z

Show changes

Teorth at 01:20, 13 June 2009

2009-06-13T01:20:31Z

← Older revision		Revision as of 18:20, 12 June 2009
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	:LPSolve(a+b+c+d+e,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);
	[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]		[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]
	''~~This~~, ~~together with~~ the ~~bound <math>c'_4=43</math>, implies that~~ <math>a+b+c+d+5e \leq 43</math>''.		''off by three, thanks to the [[4D Moser brute force search]]. So we have <math>a+b+c+d+e+7e \leq 43</math>''.
	: LPSolve(c,{op(Y4),a+b+c+d+e=43},assume=nonnegint,depthlimit=100);		: LPSolve(c,{op(Y4),a+b+c+d+e=43},assume=nonnegint,depthlimit=100);
	[18, [a = 5, b = 20, c = 18, d = 0, e = 0]]		[18, [a = 5, b = 20, c = 18, d = 0, e = 0]]
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	:LPSolve(a+b+c+d+e+d/2,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+d/2,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);
	[40, [a = 8, b = 16, c = 12, d = 2, e = 1]]		[40, [a = 8, b = 16, c = 12, d = 2, e = 1]]
	''~~This~~, ~~together with~~ the 4D ~~data for 41+ point~~ Moser ~~sets, implies that~~ <math>a+b+c+d+e+d/2+3e \leq 43</math>''.		''off by 2.5, thanks to the [[4D Moser brute force search]]. So we have <math>a+b+c+d+e+d/2+11e/2 \leq 43</math>''.
	:X4 := {op(X4), A4+4e <= 43, A4 + 4d + 3*e <= 43};		:X4 := {op(X4), A4+7e <= 43, A4 + d/2 + 11e/2 <= 43};
	:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2*a+f <= 2};		:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2*a+f <= 2};
	''xyzw1, xyzw2, and xxxx cube inequalities''		''xyzw1, xyzw2, and xxxx cube inequalities''
Line 54:		Line 54:
	:Y5 := subs([a=a/32,b=b/80,c=c/80,d=d/40,e=e/10],X5);		:Y5 := subs([a=a/32,b=b/80,c=c/80,d=d/40,e=e/10],X5);
	:LPSolve(a+b+c+d+e+f,Y5,assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f,Y5,assume=nonnegint,depthlimit=100,maximize);
	[126, [a = 6, b = 40, c = 80, d = 0, e = 0, f = 0]]		[126, [a = 10, b = 38, c = 70, d = 8, e = 0, f = 0]]
	''Off by two! The truth is <math>c'_5=124</math>. But the 126 case can again be eliminated by hand		''Off by two! The truth is <math>c'_5=124</math>. But the 126 case can again be eliminated by hand
	:LPSolve(a+b+c+d+e+f,{op(Y5),f=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f,{op(Y5),f=1},assume=nonnegint,depthlimit=100,maximize);
	[~~117~~, [a = 15, b = 40, c = 40, d = 20, e = 1, f = 1]]		[115, [a = 16, b = 40, c = 40, d = 17, e = 1, f = 1]]
	''This, together with the bound <math>c'_5=124</math>, implies that <math>a+b+c+d+e+f+7f \leq 124</math>''.		''This, together with the bound <math>c'_5=124</math>, implies that <math>a+b+c+d+e+f+9f \leq 124</math>''.
	:X5 := {op(X5), A5+7*f <= 124};		:X5 := {op(X5), A5+9*f <= 124};
	:LPSolve(a+b+c+d+e+f,{op(Y5),e>=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f,{op(Y5),e>=1},assume=nonnegint,depthlimit=100,maximize);
	[~~124~~, [a = 6, b = 43, c = 74, d = 0, e = 1, f = 0]]		[125, [a = 11, b = 36, c = 68, d = 9, e = 1, f = 0]]
	~~''This gives an alternate proof that e=0 for 125-point Moser sets''~~
	:LPSolve(d,{op(Y5),a+b+c+d+e=125},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(d,{op(Y5),a+b+c+d+e=125},assume=nonnegint,depthlimit=100,maximize);
	[10, [a = 10, b = 35, c = 70, d = 10, e = 0, f = 0]]		[12, [a = 11, b = 35, c = 67, d = 12, e = 0, f = 0]]
	''Not good at all... we were able to get d=0 for 125-sets.''		''Not good at all... we were able to get d=0 for 125-sets.''
	:LPSolve(d,{op(Y5),a+b+c+d+e=126},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(d,{op(Y5),a+b+c+d+e=126},assume=nonnegint,depthlimit=100,maximize);
	[0, [a = 6, b = 40, c = 80, d = 0, e = 0, f = 0]]		[8, [a = 10, b = 37, c = 71, d = 8, e = 0, f = 0]]
	~~''Well, at least 126-point sets are OK (this is probably the unique possible statistics consistent with the linear inequalities)''~~
	:X6 := {op(X5), op(subs([a=b,b=c,c=d,d=e,e=f,f=g],X5)), op(subs([b=d,c=g],X2)), op(subs([b=c,c=e,d=g],X3))};		:X6 := {op(X5), op(subs([a=b,b=c,c=d,d=e,e=f,f=g],X5)), op(subs([b=d,c=g],X2)), op(subs([b=c,c=e,d=g],X3))};
	''xyzwu1, xyzwu2, xxxyyy, xxyyzz cube inequalities''		''xyzwu1, xyzwu2, xxxyyy, xxyyzz cube inequalities''
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	:Y6 := subs([a=a/64,b=b/192,c=c/240,d=d/160,e=e/60,f=f/12],X6);		:Y6 := subs([a=a/64,b=b/192,c=c/240,d=d/160,e=e/60,f=f/12],X6);
	:LPSolve(a+b+c+d+e+f+g,Y6,assume=nonnegint,depthlimit=500,maximize);		:LPSolve(a+b+c+d+e+f+g,Y6,assume=nonnegint,depthlimit=500,maximize);
	[~~361~~, [a = 28, b = 72, c = ~~166~~, d = 88, e = 5, f = 2, g = 0]]		[362, [a = 31, b = 84, c = 137, d = 86, e = 21, f = 3, g = 0]]
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);
	[352, [a = 28, b = 96, c = 120, d = 80, e = 25, f = 2, g = 1]]		[352, [a = 30, b = 96, c = 120, d = 80, e = 22, f = 3, g = 1]]
	''This implies that <math>a+b+c+d+e+f+g+9g \leq ~~361~~</math>''.		''This implies that <math>a+b+c+d+e+f+g+10g \leq 362</math>''.
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),e=0,f=0,g=0},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),e=0,f=0,g=0},assume=nonnegint,depthlimit=100,maximize);
	[360, [a = 32, b = 64, c = ~~136~~, d = ~~128~~, e = 0, f = 0, g = 0]]		[360, [a = 32, b = 60, c = 132, d = 136, e = 0, f = 0, g = 0]]
	:X6 := {op(X6), A6+9*g <= 361};		:X6 := {op(X6), A6+9*g <= 361};
	:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};		:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};
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	:Y7 := subs([a=a/128,b=b/448,c=c/672,d=d/560,e=e/280,f=f/84,g=g/14],X7);		:Y7 := subs([a=a/128,b=b/448,c=c/672,d=d/560,e=e/280,f=f/84,g=g/14],X7);
	:LPSolve(a+b+c+d+e+f+g+h,Y7,assume=nonnegint,depthlimit=500,maximize);		:LPSolve(a+b+c+d+e+f+g+h,Y7,assume=nonnegint,depthlimit=500,maximize);
	[~~1073~~, [a = 65, b = ~~213~~, c = ~~293~~, d = ~~339~~, e = ~~131~~, f = 27, g = 5, h = 0]]		[1079, [a = 64, b = 214, c = 306, d = 321, e = 141, f = 31, g = 2, h = 0]]
	:LPSolve(a+b+c+d+e+f+g+h,{op(Y7),h=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g+h,{op(Y7),h=1},assume=nonnegint,depthlimit=100,maximize);
	[~~1065~~, [a = 64, b = ~~210~~, c = 336, d = 280, e = 140, f = 31, g = 3, h = 1]]		[1067, [a = 56, b = 224, c = 336, d = 280, e = 140, f = 28, g = 2, h = 1]]

Teorth at 20:33, 1 June 2009

2009-06-01T20:33:46Z

← Older revision		Revision as of 13:33, 1 June 2009
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	:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};		:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};
	:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2)), 2*a+d <= 2};		:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2)), 2*a+d <= 2};
	''These are the inequalities inherited from ~~xyz1~~, ~~xyz2~~, ~~xxxx~~ cubes (averaged over symmetries)''		''These are the inequalities inherited from xy1, xy2, xxx cubes (averaged over symmetries)''
	:Y3 := subs([a=a/8,b=b/12,c=c/6,d=d],X3);		:Y3 := subs([a=a/8,b=b/12,c=c/6,d=d],X3);
	:LPSolve(a+b+c+d,Y3,assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d,Y3,assume=nonnegint,depthlimit=100,maximize);

Teorth at 05:50, 24 May 2009

2009-05-24T05:50:05Z

← Older revision		Revision as of 22:50, 23 May 2009
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	''This, together with the 4D data for 41+ point Moser sets, implies that <math>a+b+c+d+e+d/2+3e \leq 43</math>''.		''This, together with the 4D data for 41+ point Moser sets, implies that <math>a+b+c+d+e+d/2+3e \leq 43</math>''.
	:X4 := {op(X4), A4+4e <= 43, A4 + 4d + 3*e <= 43};		:X4 := {op(X4), A4+4e <= 43, A4 + 4d + 3*e <= 43};
	:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2*a+f <= 2}		:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2*a+f <= 2};
	''xyzw1, xyzw2, and xxxx cube inequalities''		''xyzw1, xyzw2, and xxxx cube inequalities''
	:X5 := {op(X5), 4a+b+2c+2d+e+f <= 11/2, 2e+f <= 2, 4a+4c+2d+e+f <= 6, 4a+2b+2c+2d+e+f <= 6, 7a+b+2c+2d+e+f <= 7, 4a+2e+f <= 4, 8a+4c+2d+e+f <= 8, 8a+2b+2c+2d+e+f <= 8, 4a+2b+2d+2e+f <= 6, 8a+2b+2d+2*e+f <= 8};		:X5 := {op(X5), 4a+b+2c+2d+e+f <= 11/2, 2e+f <= 2, 4a+4c+2d+e+f <= 6, 4a+2b+2c+2d+e+f <= 6, 7a+b+2c+2d+e+f <= 7, 4a+2e+f <= 4, 8a+4c+2d+e+f <= 8, 8a+2b+2c+2d+e+f <= 8, 4a+2b+2d+2e+f <= 6, 8a+2b+2d+2*e+f <= 8};

Teorth at 04:55, 24 May 2009

2009-05-24T04:55:01Z

← Older revision		Revision as of 21:55, 23 May 2009
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	: LPSolve(c,{op(Y4),a+b+c+d+e>=42},assume=nonnegint,depthlimit=100);		: LPSolve(c,{op(Y4),a+b+c+d+e>=42},assume=nonnegint,depthlimit=100);
	[12, [a = 6, b = 24, c = 12, d = 0, e = 0]]		[12, [a = 6, b = 24, c = 12, d = 0, e = 0]]
	''This shows that 42+ point sets have c at least 18		''This shows that 42+ point sets have c at least 12''
	:LPSolve(d,{op(Y4),a+b+c+d+e >= 41},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(d,{op(Y4),a+b+c+d+e >= 41},assume=nonnegint,depthlimit=100,maximize);
	[4, [a = 7, b = 17, c = 13, d = 4, e = 0]]		[4, [a = 7, b = 17, c = 13, d = 4, e = 0]]

Teorth at 01:44, 24 May 2009

2009-05-24T01:44:49Z

← Older revision		Revision as of 18:44, 23 May 2009
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	[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]		[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]
	''This, together with the bound <math>c'_4=43</math>, implies that <math>a+b+c+d+5e \leq 43</math>''.		''This, together with the bound <math>c'_4=43</math>, implies that <math>a+b+c+d+5e \leq 43</math>''.
			: LPSolve(c,{op(Y4),a+b+c+d+e=43},assume=nonnegint,depthlimit=100);
			[18, [a = 5, b = 20, c = 18, d = 0, e = 0]]
			''This shows that 43-point sets have c at least 18.''
			: LPSolve(c,{op(Y4),a+b+c+d+e>=42},assume=nonnegint,depthlimit=100);
			[12, [a = 6, b = 24, c = 12, d = 0, e = 0]]
			''This shows that 42+ point sets have c at least 18
	:LPSolve(d,{op(Y4),a+b+c+d+e >= 41},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(d,{op(Y4),a+b+c+d+e >= 41},assume=nonnegint,depthlimit=100,maximize);
	[4, [a = 7, b = 17, c = 13, d = 4, e = 0]]		[4, [a = 7, b = 17, c = 13, d = 4, e = 0]]
	''Off by 1 again: the truth here is d=3''		''This shows that 41+ point Moser sets have d at most 4. Off by 1 again: the truth here is d=3''
			:LPSolve(d,{op(Y4),a+b+c+d+e >= 40},assume=nonnegint,depthlimit=100,maximize);
			[6, [a = 8, b = 16, c = 10, d = 6, e = 0]]
			''This shows that 40+ point Moser sets have d at most 6.''
	:LPSolve(a+b+c+d+e+d/2,{op(Y4),a+b+c+d+e<=40},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+d/2,{op(Y4),a+b+c+d+e<=40},assume=nonnegint,depthlimit=100,maximize);
	[43, [a = 8, b = 16, c = 10, d = 6, e = 0]]		[43, [a = 8, b = 16, c = 10, d = 6, e = 0]]

Teorth at 00:43, 16 March 2009

2009-03-16T00:43:00Z

← Older revision		Revision as of 17:43, 15 March 2009
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	:X1 := {2*a+b <= 2, b <= 1};		:X1 := {2*a+b <= 2, b <= 1};
	:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};		:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};
	:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2))};		:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2)), 2*a+d <= 2};
	''These are the inequalities inherited from xyz1, xyz2 cubes (averaged over symmetries)''		''These are the inequalities inherited from xyz1, xyz2, xxxx cubes (averaged over symmetries)''
	:X3 := {op(X3), 8a+6b+6c+2d<=11, 4a+4b+3c+d<=6, 7a+3b+3c+d <= 7, 4a+6c+2d<=7, 5a+3c+d<=5~~, 2a+d <= 2~~};		:Y3 := subs([a=a/8,b=b/12,c=c/6,d=d],X3);
			:LPSolve(a+b+c+d,Y3,assume=nonnegint,depthlimit=100,maximize);
			[16, [a = 4, b = 6, c = 6, d = 0]]
			''This gives a proof of <math>c'_3 \leq 16</math>.
			:X3 := {op(X3), 8a+6b+6c+2d<=11, 4a+4b+3c+d<=6, 7a+3b+3c+d <= 7, 4a+6c+2d<=7, 5a+3*c+d<=5};
	''This comes from the linear inequalities obeyed by the 3D extremals.''		''This comes from the linear inequalities obeyed by the 3D extremals.''
	:X4 := {op(X3), op(subs([a=b,b=c,c=d,d=e],X3)), op(subs([b=c,c=e],X2))};		:X4 := {op(X3), op(subs([a=b,b=c,c=d,d=e],X3)), op(subs([b=c,c=e],X2))};

Teorth at 16:16, 14 March 2009

2009-03-14T16:16:18Z

← Older revision		Revision as of 09:16, 14 March 2009
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	:X1 := {2*a+b <= 2, b <= 1};		:X1 := {2*a+b <= 2, b <= 1};
	:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};		:X2 := {4a+2b+c <= 4, 2a+c <= 2, 2b+c <= 2, c <= 1};
	:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2)), 8a+6b+6c+2d<=11, 4a+4b+3c+d<=6, 7a+3b+3c+d <= 7, 4a+6c+2d<=7, 5a+3c+d<=5, 2a+d <= 2};		:X3 := {op(X2), op(subs([a=b,b=c,c=d],X2))};
			''These are the inequalities inherited from xyz1, xyz2 cubes (averaged over symmetries)''
			:X3 := {op(X3), 8a+6b+6c+2d<=11, 4a+4b+3c+d<=6, 7a+3b+3c+d <= 7, 4a+6c+2d<=7, 5a+3c+d<=5, 2a+d <= 2};
	''This comes from the linear inequalities obeyed by the 3D extremals.''		''This comes from the linear inequalities obeyed by the 3D extremals.''
	:X4 := {op(X3), op(subs([a=b,b=c,c=d,d=e],X3)), op(subs([b=c,c=e],X2))~~, 416a+32b+224c+48d+16e <= 96, 816a+32b+224c+48d+16e <= 128~~};		:X4 := {op(X3), op(subs([a=b,b=c,c=d,d=e],X3)), op(subs([b=c,c=e],X2))};
	''The op(subs([b=c,c=e],X2) term here reflects the presence of diagonal 2D cubes such as xxyy in 4D with two appearances of each wildcard. ~~The last two inequalities~~ come from Peake.986		''The op(subs([b=c,c=e],X2) term here reflects the presence of diagonal 2D cubes such as xxyy in 4D with two appearances of each wildcard.''
			:X4 := {op(X4),4a+2b+3c+2d+e <= 6, 8a+2b+3c+2d+e <= 8};
			''These come from the xxyz slices, see Peake.986 or Peake.993''
	:Y4 := subs([a=a/16,b=b/32,c=c/24,d=d/8],X4);		:Y4 := subs([a=a/16,b=b/32,c=c/24,d=d/8],X4);
			:LPSolve(a+b+c+d+e,Y4,assume=nonnegint,depthlimit=100,maximize);
			[44, [a = 4, b = 16, c = 24, d = 0, e = 0]]
			''off by one! The true maximum is of course <math>c'_4=43</math>. But one can eliminate the above 44-point set by hand''
	:LPSolve(a+b+c+d+e,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);
	[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]		[39, [a = 8, b = 16, c = 12, d = 2, e = 1]]
	''This, together with the bound <math>c'_4=43</math>, implies that <math>a+b+c+d+5e \leq 43</math>''.		''This, together with the bound <math>c'_4=43</math>, implies that <math>a+b+c+d+5e \leq 43</math>''.
			:LPSolve(d,{op(Y4),a+b+c+d+e >= 41},assume=nonnegint,depthlimit=100,maximize);
			[4, [a = 7, b = 17, c = 13, d = 4, e = 0]]
			''Off by 1 again: the truth here is d=3''
	:LPSolve(a+b+c+d+e+d/2,{op(Y4),a+b+c+d+e<=40},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+d/2,{op(Y4),a+b+c+d+e<=40},assume=nonnegint,depthlimit=100,maximize);
	[43, [a = 8, b = 16, c = 10, d = 6, e = 0]]		[43, [a = 8, b = 16, c = 10, d = 6, e = 0]]
	:LPSolve(a+b+c+d+e+d/2,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+d/2,{op(Y4),e=1},assume=nonnegint,depthlimit=100,maximize);
	[40, [a = 8, b = 15, c = 12, d = 3, e = 1]]		[40, [a = 8, b = 16, c = 12, d = 2, e = 1]]
	''This, together with the 4D data for 41+ point Moser sets, implies that <math>a+b+c+d+e+d/2+3e \leq 43</math>''.		''This, together with the 4D data for 41+ point Moser sets, implies that <math>a+b+c+d+e+d/2+3e \leq 43</math>''.
	:X4 := {op(X4), A4+4e <= 43, A4 + 4d + 3*e <= 43};		:X4 := {op(X4), A4+4e <= 43, A4 + 4d + 3*e <= 43};
	:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2a+f <= 2, 4a+b+2c+2d+e+f <= 11/2, 2e+f <= 2, 4a+4c+2d+e+f <= 6, 4a+2b+2c+2d+e+f <= 6, 7a+b+2c+2d+e+f <= 7, 4a+2e+f <= 4, 8a+4c+2d+e+f <= 8, 8a+2b+2c+2d+e+f <= 8, 4a+2b+2d+2e+f <= 6, 8a+2b+2d+2e+f <= 8};		:X5 := {op(X4), op(subs([a=b,b=c,c=d,d=e,e=f],X4)), 2*a+f <= 2}
	''~~The last ten~~ inequalities ~~here~~ come from Peake.~~988~~''		''xyzw1, xyzw2, and xxxx cube inequalities''
			:X5 := {op(X5), 4a+b+2c+2d+e+f <= 11/2, 2e+f <= 2, 4a+4c+2d+e+f <= 6, 4a+2b+2c+2d+e+f <= 6, 7a+b+2c+2d+e+f <= 7, 4a+2e+f <= 4, 8a+4c+2d+e+f <= 8, 8a+2b+2c+2d+e+f <= 8, 4a+2b+2d+2e+f <= 6, 8a+2b+2d+2*e+f <= 8};
			''These inequalities come from xxyyz cubes, see Peake.989''
	:X5 := {op(X5),8a+4b+2c+2d+4e+2f <=11,4a+2b+1c+2d+2e+1f <= 6,0a+0b+2c+0d+0e+1f <= 2,4a+4b+1c+0d+2e+1f <= 6,7a+2b+1c+1d+2e+1f <= 7,8a+2b+1c+2d+2e+1f <= 8,4a+0b+2c+0d+0e+1f <= 4,4a+0b+2c+2d+2e+1f <= 6,8a+0b+2c+2d+2e+1f <= 8,8a+4b+1c+0d+2e+1f <= 8};		:X5 := {op(X5),8a+4b+2c+2d+4e+2f <=11,4a+2b+1c+2d+2e+1f <= 6,0a+0b+2c+0d+0e+1f <= 2,4a+4b+1c+0d+2e+1f <= 6,7a+2b+1c+1d+2e+1f <= 7,8a+2b+1c+2d+2e+1f <= 8,4a+0b+2c+0d+0e+1f <= 4,4a+0b+2c+2d+2e+1f <= 6,8a+0b+2c+2d+2e+1f <= 8,8a+4b+1c+0d+2e+1f <= 8};
	''These inequalities come from Peake.991''		''These inequalities come from xxxyz cubes, see Peake.991''
	:Y5 := subs([a=a/32,b=b/80,c=c/80,d=d/40,e=e/10],X5);		:Y5 := subs([a=a/32,b=b/80,c=c/80,d=d/40,e=e/10],X5);
			:LPSolve(a+b+c+d+e+f,Y5,assume=nonnegint,depthlimit=100,maximize);
			[126, [a = 6, b = 40, c = 80, d = 0, e = 0, f = 0]]
			''Off by two! The truth is <math>c'_5=124</math>. But the 126 case can again be eliminated by hand
	:LPSolve(a+b+c+d+e+f,{op(Y5),f=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f,{op(Y5),f=1},assume=nonnegint,depthlimit=100,maximize);
	[117, [a = 15, b = 40, c = 40, d = 20, e = 1, f = 1]]		[117, [a = 15, b = 40, c = 40, d = 20, e = 1, f = 1]]
	''This, together with the bound <math>c'_5=124</math>, implies that <math>a+b+c+d+e+f+7f \leq 124</math>''.		''This, together with the bound <math>c'_5=124</math>, implies that <math>a+b+c+d+e+f+7f \leq 124</math>''.
			:X5 := {op(X5), A5+7*f <= 124};
	:LPSolve(a+b+c+d+e+f,{op(Y5),e>=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f,{op(Y5),e>=1},assume=nonnegint,depthlimit=100,maximize);
	[124, [a = 6, b = 43, c = 74, d = 0, e = 1, f = 0]]		[124, [a = 6, b = 43, c = 74, d = 0, e = 1, f = 0]]
	''This gives an alternate proof that e=0 for 125-point Moser sets''		''This gives an alternate proof that e=0 for 125-point Moser sets''
	:X5 := {op(X5), A5+~~7*f <~~= ~~124~~};		:LPSolve(d,{op(Y5),a+b+c+d+e=125},assume=nonnegint,depthlimit=100,maximize);
			[10, [a = 10, b = 35, c = 70, d = 10, e = 0, f = 0]]
			''Not good at all... we were able to get d=0 for 125-sets.''
			:LPSolve(d,{op(Y5),a+b+c+d+e=126},assume=nonnegint,depthlimit=100,maximize);
			[0, [a = 6, b = 40, c = 80, d = 0, e = 0, f = 0]]
			''Well, at least 126-point sets are OK (this is probably the unique possible statistics consistent with the linear inequalities)''
	:X6 := {op(X5), op(subs([a=b,b=c,c=d,d=e,e=f,f=g],X5)), op(subs([b=d,c=g],X2)), op(subs([b=c,c=e,d=g],X3))};		:X6 := {op(X5), op(subs([a=b,b=c,c=d,d=e,e=f,f=g],X5)), op(subs([b=d,c=g],X2)), op(subs([b=c,c=e,d=g],X3))};
	''~~The op(subs([b=c~~,~~c=e~~,~~d=g]~~,~~X3) term here reflects the presence of diagonal 3D cubes such as~~ xxyyzz ~~in 6D with two appearances of each wildcard.~~''		''xyzwu1, xyzwu2, xxxyyy, xxyyzz cube inequalities''
	:X6 := {op(X6),8a+4b+2c+2e+4f+2g <= 11,0a+0b+2c+0e+0f+1g <= 2,4a+2b+1c+2e+2f+1g <= 6,7a+2b+1c+1e+2f+1g <= 7,4a+0b+2c+0e+0f+1g <= 4,4a+0b+2c+2e+2f+1g <= 6,8a+0b+2c+2e+2f+1g <= 8,8a+2b+1c+2e+2f+1g <= 8,4a+4b+1c+0e+2f+1g <= 6,8a+4b+1c+0e+2f+1g <= 8};		:X6 := {op(X6),8a+4b+2c+2e+4f+2g <= 11,0a+0b+2c+0e+0f+1g <= 2,4a+2b+1c+2e+2f+1g <= 6,7a+2b+1c+1e+2f+1g <= 7,4a+0b+2c+0e+0f+1g <= 4,4a+0b+2c+2e+2f+1g <= 6,8a+0b+2c+2e+2f+1g <= 8,8a+2b+1c+2e+2f+1g <= 8,4a+4b+1c+0e+2f+1g <= 6,8a+4b+1c+0e+2f+1g <= 8};
	''These come from xxxxyz ~~diagonals~~, see Peake.991''		''These come from xxxxyz cubes, see Peake.991''
	:~~<del>~~X6 := {op(X6),4a+2b+0c+3d+1e+1f+1g <= 6,4a+0b+2c+3d+1e+1f+1g <= 6,8a+2b+0c+3d+1e+1f+1g <= 8,8a+0b+2c+3d+1e+1f+1g <= 8,0a+4b+0c+0d+2e+0f+1g <= 4,0a+0b+4c+0d+0e+2f+1g <= 4,8a+4b+0c+0d+2e+0f+1g <= 1,8a+0b+4c+0d+0e+2f+1g <= 1};		:X6 := {op(X6),4a+2b+0c+3d+1e+1f+1g <= 6,4a+0b+2c+3d+1e+1f+1g <= 6,8a+2b+0c+3d+1e+1f+1g <= 8,8a+0b+2c+3d+1e+1f+1g <= 8,0a+4b+0c+0d+2e+0f+1g <= 4,0a+0b+4c+0d+0e+2f+1g <= 4,8a+4b+0c+0d+2e+0f+1g <= 8,8a+0b+4c+0d+0e+2f+1g <= 8};
	''These come from xxxyyz ~~diagonals~~, see Peake.991''</del>		''These come from xxxyyz cubes, see Peake.991, Peake.993''</del>
	:Y6 := subs([a=a/64,b=b/192,c=c/240,d=d/160,e=e/60,f=f/12],X6);		:Y6 := subs([a=a/64,b=b/192,c=c/240,d=d/160,e=e/60,f=f/12],X6);
	:LPSolve(a+b+c+d+e+f+g,Y6,assume=nonnegint,depthlimit=500,maximize);		:LPSolve(a+b+c+d+e+f+g,Y6,assume=nonnegint,depthlimit=500,maximize);
	[361, [a = 28, b = 72, c = 166, d = 88, e = 5, f = 2, g = 0]]		[361, [a = 28, b = 72, c = 166, d = 88, e = 5, f = 2, g = 0]]
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),g=1},assume=nonnegint,depthlimit=100,maximize);
	[352, [a = 27, b = 96, c = 120, d = 80, e = 26, f = 4, g = 1]]		[352, [a = 28, b = 96, c = 120, d = 80, e = 25, f = 2, g = 1]]
	''This implies that <math>a+b+c+d+e+f+g+9g \leq 361</math>''.		''This implies that <math>a+b+c+d+e+f+g+9g \leq 361</math>''.
	:LPSolve(a+b+c+d+e+f+g,{op(Y6),e=0,f=0,g=0},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g,{op(Y6),e=0,f=0,g=0},assume=nonnegint,depthlimit=100,maximize);
Line 51:		Line 72:
	:X6 := {op(X6), A6+9*g <= 361};		:X6 := {op(X6), A6+9*g <= 361};
	:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};		:X7 := {op(X6), op(subs([a=b,b=c,c=d,d=e,e=f,f=g,g=h],X6)), 2*a+h <= 2};
			''xyzwuv1, xyzwuv2, xxxxxxx inequalities''
	:Y7 := subs([a=a/128,b=b/448,c=c/672,d=d/560,e=e/280,f=f/84,g=g/14],X7);		:Y7 := subs([a=a/128,b=b/448,c=c/672,d=d/560,e=e/280,f=f/84,g=g/14],X7);
	:LPSolve(a+b+c+d+e+f+g+h,Y7,assume=nonnegint,depthlimit=500,maximize);		:LPSolve(a+b+c+d+e+f+g+h,Y7,assume=nonnegint,depthlimit=500,maximize);
	[1073, [a = 64, b = ~~215~~, c = ~~298~~, d = ~~308~~, e = ~~168~~, f = 15, g = 5, h = 0]]		[1073, [a = 65, b = 213, c = 293, d = 339, e = 131, f = 27, g = 5, h = 0]]
	:LPSolve(a+b+c+d+e+f+g+h,{op(Y7),h=1},assume=nonnegint,depthlimit=100,maximize);		:LPSolve(a+b+c+d+e+f+g+h,{op(Y7),h=1},assume=nonnegint,depthlimit=100,maximize);
	[1065, [a = 63, b = ~~212~~, c = ~~335~~, d = 280, e = 140, f = 30, g = 4, h = 1]]		[1065, [a = 64, b = 210, c = 336, d = 280, e = 140, f = 31, g = 3, h = 1]]