#Here (without proof) are the five hundred 'formulas' for 5-card Krypto.
#For method of generation, see the companion file  'fournums', which
#sketches the smaller case of 4-card Krypto.
#                                                rusin@math.niu.edu 2002/01/30

#If you perform the following steps, this file will also automatically create
#the 27,142 values obtainable from five numbers x1,x2,x3,x4,x5, using Maple.
#
#Save this file as   file500 , fire up Maple, and issue this command:
#    read file500:
#(strip the leading pound-sign!) After a few minutes' delay, there will be
#created a set  L  of the values. You can print them with one of these
#commands, but I don't advise any of them! (Too long)
#   for i to nops(L) do print(L[i]):od:   #longest, most legible
#   for i to nops(L) do lprint(L[i]):od:  #one formula per line
#   lprint(L):                            #compact, not very human-legible
#   save(L, MYFILE):                      #writes list to a file

#You can easily use them to search for ways to solve a Krypto hand.
#Try  play(a,b,c,d,e,f);   to search for ways to combine {a,b,c,d,e} to
#yield  f.

#What follows are the 500 formulas, then some Maple routines to
#play with them, and some miscellaneous commentary.



G:=[
#The 500 five-variable formulas for Krypto, sorted by symmetry:
#
#Two with symmetry group of order 120:
x1+x2+x3+x4+x5,
x1*x2*x3*x4*x5,
#
#Ten with group Sym(4), of order 24:
(x1+x2+x3+x4)-x5,
x5-(x1+x2+x3+x4),
(x1+x2+x3+x4)*x5,
(x1+x2+x3+x4)/x5,
x5/(x1+x2+x3+x4),
(x1*x2*x3*x4)+x5,
(x1*x2*x3*x4)-x5,
x5-(x1*x2*x3*x4),
(x1*x2*x3*x4)/x5,
x5/(x1*x2*x3*x4),
#
#22 with symmetry group  Sym(3) x <(4,5)>  of order 12:
(x4+x5)-(x1+x2+x3),
(x1+x2+x3)-(x4+x5),
(x1+x2+x3)*(x4+x5),
(x4+x5)/(x1+x2+x3),
(x1+x2+x3)/(x4+x5),
(x1+x2+x3)+(x4*x5),
(x4*x5)-(x1+x2+x3),
(x1+x2+x3)-(x4*x5),
(x1+x2+x3)*(x4*x5),
(x4*x5)/(x1+x2+x3),
(x1+x2+x3)/(x4*x5),
(x4+x5)+(x1*x2*x3),
(x4+x5)-(x1*x2*x3),
(x1*x2*x3)-(x4+x5),
(x4+x5)*(x1*x2*x3),
(x1*x2*x3)/(x4+x5),
(x4+x5)/(x1*x2*x3),
(x4*x5)+(x1*x2*x3),
(x1*x2*x3)-(x4*x5),
(x4*x5)-(x1*x2*x3),
(x4*x5)/(x1*x2*x3),
(x1*x2*x3)/(x4*x5), 
#
#12 with dihedral symmetry group of order 8 (<(1 2),(3 4), (1 3)(2 4)>):
x5+(x1+x2)*(x3+x4),
x5+(x1*x2)+(x3*x4),
x5-(x1+x2)*(x3+x4),
x5-(x1*x2)-(x3*x4),
(x1+x2)*(x3+x4)-x5,
(x1*x2)+(x3*x4)-x5,
((x1*x2)+(x3*x4))*x5,
((x1+x2)*(x3+x4))*x5,
((x1*x2)+(x3*x4))/x5,
((x1+x2)*(x3+x4))/x5,
x5/((x1*x2)+(x3*x4)),
x5/((x1+x2)*(x3+x4)),
#
#46 with symmetry group Sym(3) of order 6:
((x1+x2+x3)-x4)*x5,
((x1+x2+x3)-x4)/x5,
x5/((x1+x2+x3)-x4),
(x4-(x1+x2+x3))*x5,
(x4-(x1+x2+x3))/x5,
x5/(x4-(x1+x2+x3)),
(x1+x2+x3)*x4+x5,
(x1+x2+x3)*x4-x5,
x5-(x1+x2+x3)*x4,
((x1+x2+x3)*x4)/x5,
x5/((x1+x2+x3)*x4),
(x1+x2+x3)/x4+x5,
(x1+x2+x3)/x4-x5,
x5-(x1+x2+x3)/x4,
x4/(x1+x2+x3)+x5,
x4/(x1+x2+x3)-x5,
x5-x4/(x1+x2+x3),
(x1+x2+x3)*(x4-x5),
(x1+x2+x3)/(x4-x5),
(x4-x5)/(x1+x2+x3),
(x1+x2+x3)+(x4/x5),
(x1+x2+x3)-(x4/x5),
(x4/x5)-(x1+x2+x3),
(x4+(x1*x2*x3))*x5,
(x4+(x1*x2*x3))/x5,
x5/(x4+(x1*x2*x3)),
((x1*x2*x3)-x4)*x5,
((x1*x2*x3)-x4)/x5,
x5/((x1*x2*x3)-x4),
(x4-(x1*x2*x3))*x5,
(x4-(x1*x2*x3))/x5,
x5/(x4-(x1*x2*x3)),
(x1*x2*x3)/x4+x5,
(x1*x2*x3)/x4-x5,
x5-(x1*x2*x3)/x4,
x4/(x1*x2*x3)+x5,
x4/(x1*x2*x3)-x5,
x5-x4/(x1*x2*x3),
(x1*x2*x3)+(x4-x5),
(x4-x5)-(x1*x2*x3),
(x4-x5)*(x1*x2*x3),
(x4-x5)/(x1*x2*x3),
(x1*x2*x3)/(x4-x5),
(x1*x2*x3)+(x4/x5),
(x1*x2*x3)-(x4/x5),
(x4/x5)-(x1*x2*x3),
#
#102 with symmetry group <(1 2),(3 4)> of order 4:
#two mults:
(x1*x2)-(x3*x4)+x5,
(x1*x2)-(x3*x4)-x5,
((x1*x2)-(x3*x4))*x5,
((x1*x2)-(x3*x4))/x5,
x5/((x1*x2)-(x3*x4)),
(x1*x2)/(x3*x4)+x5,
(x1*x2)/(x3*x4)-x5,
x5-(x1*x2)/(x3*x4),
(x5+(x1*x2))*(x3*x4),
(x5+(x1*x2))/(x3*x4),
(x3*x4)/(x5+(x1*x2)),
((x1*x2)-x5)*(x3*x4),
((x1*x2)-x5)/(x3*x4),
(x3*x4)/((x1*x2)-x5),
(x5-(x1*x2))*(x3*x4),
(x5-(x1*x2))/(x3*x4),
(x3*x4)/(x5-(x1*x2)),
(x1*x2)/x5+(x3*x4),
(x1*x2)/x5-(x3*x4),
(x3*x4)-(x1*x2)/x5,
x5/(x1*x2)+(x3*x4),
x5/(x1*x2)-(x3*x4),
(x3*x4)-x5/(x1*x2),
#
#two adds:
((x1+x2)-(x3+x4))*x5,
((x1+x2)-(x3+x4))/x5,
x5/((x1+x2)-(x3+x4)),
(x1+x2)/(x3+x4)+x5,
(x1+x2)/(x3+x4)-x5,
x5-(x1+x2)/(x3+x4),
(x1+x2)*x5+(x3+x4),
(x1+x2)*x5-(x3+x4),
(x3+x4)-(x1+x2)*x5,
(x1+x2)*x5/(x3+x4),
(x3+x4)/((x1+x2)*x5),
((x1+x2)-x5)*(x3+x4),
((x1+x2)-x5)/(x3+x4),
(x3+x4)/((x1+x2)-x5),
(x5-(x1+x2))*(x3+x4),
(x5-(x1+x2))/(x3+x4),
(x3+x4)/(x5-(x1+x2)),
(x1+x2)/x5+(x3+x4),
(x1+x2)/x5-(x3+x4),
(x3+x4)-(x1+x2)/x5,
x5/(x1+x2)+(x3+x4),
x5/(x1+x2)-(x3+x4),
(x3+x4)-x5/(x1+x2),
#
#one mult, one add
((x1+x2)+(x3*x4))-x5,
((x1+x2)+(x3*x4))*x5,
x5/((x1+x2)+(x3*x4)),
((x1+x2)+(x3*x4))/x5,
(x1+x2)-(x3*x4)-x5,
x5-(x1+x2)-(x3*x4),
((x1+x2)-(x3*x4))*x5,
((x1+x2)-(x3*x4))/x5,
x5/((x1+x2)-(x3*x4)),
((x1+x2)*(x3*x4))+x5,
((x1+x2)*(x3*x4))-x5,
x5-((x1+x2)*(x3*x4)),
(x1+x2)*(x3*x4)/x5,
x5/((x1+x2)*(x3*x4)),
(x1+x2)/(x3*x4)+x5,
(x1+x2)/(x3*x4)-x5,
x5-(x1+x2)/(x3*x4),
#
(x1*x2)-(x3+x4)+x5,
((x1*x2)-(x3+x4))*x5,
((x1*x2)-(x3+x4))/x5,
x5/((x1*x2)-(x3+x4)),
(x1*x2)/(x3+x4)+x5,
(x1*x2)/(x3+x4)-x5,
x5-(x1*x2)/(x3+x4),
#
((x1+x2)-x5)*(x3*x4),
((x1+x2)-x5)/(x3*x4),
(x3*x4)/((x1+x2)-x5),
(x5-(x1+x2))*(x3*x4),
(x5-(x1+x2))/(x3*x4),
(x3*x4)/(x5-(x1+x2)),
(x1+x2)*x5+(x3*x4),
(x1+x2)*x5-(x3*x4),
(x3*x4)-(x1+x2)*x5,
(x1+x2)*x5/(x3*x4),
(x3*x4)/((x1+x2)*x5),
(x1+x2)/x5+(x3*x4),
(x1+x2)/x5-(x3*x4),
(x3*x4)-(x1+x2)/x5,
x5/(x1+x2)+(x3*x4),
x5/(x1+x2)-(x3*x4),
(x3*x4)-x5/(x1+x2),
#
(x5+(x1*x2))*(x3+x4),
(x5+(x1*x2))/(x3+x4),
(x3+x4)/(x5+(x1*x2)),
((x1*x2)-x5)*(x3+x4),
((x1*x2)-x5)/(x3+x4),
(x3+x4)/((x1*x2)-x5),
(x5-(x1*x2))*(x3+x4),
(x5-(x1*x2))/(x3+x4),
(x3+x4)/(x5-(x1*x2)),
(x1*x2)/x5+(x3+x4),
(x1*x2)/x5-(x3+x4),
(x3+x4)-(x1*x2)/x5,
x5/(x1*x2)+(x3+x4),
x5/(x1*x2)-(x3+x4),
(x3+x4)-x5/(x1*x2),
#
#Five more with symmetry group <(12)(34), (13)(24)> of order 4
(x1-x2)*(x3-x4)+x5,
(x1-x2)*(x3-x4)-x5,
(x1-x2)*(x3-x4)*x5,
(x1-x2)*(x3-x4)/x5,
x5/((x1-x2)*(x3-x4)),
#
#216 with symmetry group <(12)> of order 2:  (all use x1*x2 or x1+x2)
#
#x1*x2 is added to...
(x3+(x1*x2))*x4+x5,
(x3+(x1*x2))*x4-x5,
x5-(x3+(x1*x2))*x4,
(x3+(x1*x2))*x4/x5,
(x3+(x1*x2))/x4+x5,
(x3+(x1*x2))/x4-x5,
x5-(x3+(x1*x2))/x4,
x4/(x3+(x1*x2))+x5,
x4/(x3+(x1*x2))-x5,
x5-x4/(x3+(x1*x2)),
x4/(x3+(x1*x2))/x5,
(x3+(x1*x2))+x4/x5,
(x3+(x1*x2))-x4/x5,
x4/x5-(x3+(x1*x2)),
(x3+(x1*x2))*(x4-x5),
(x3+(x1*x2))/(x4-x5),
(x4-x5)/(x3+(x1*x2)),
((x1*x2)+x3-x4)/x5,
((x1*x2)+x3-x4)*x5,
x5/((x1*x2)+x3-x4),
(x1*x2)+x3/x4-x5,
((x1*x2)+x3/x4)*x5,
((x1*x2)+x3/x4)/x5,
x5/((x1*x2)+x3/x4),
(x1*x2)+(x3-x4)*x5,
(x1*x2)+(x3-x4)/x5,
(x1*x2)+x5/(x3-x4),
#x1*x2 is subtracted from... 
(x3-(x1*x2))*x4+x5,
(x3-(x1*x2))*x4-x5,
(x3-(x1*x2))*x4/x5,
(x3-(x1*x2))/x4+x5,
(x3-(x1*x2))/x4-x5,
x4/(x3-(x1*x2))/x5,
x4/(x3-(x1*x2))+x5,
x4/(x3-(x1*x2))-x5,
x5-x4/(x3-(x1*x2)),
(x3-(x1*x2))*(x4-x5),
(x3-(x1*x2))/(x4-x5),
(x4-x5)/(x3-(x1*x2)),
(x3-(x1*x2))-x4/x5,
(x3-x4-(x1*x2))*x5, 
(x3-x4-(x1*x2))/x5,
x5/(x3-x4-(x1*x2)),
(x3/x4-(x1*x2))*x5,
(x3/x4-(x1*x2))/x5,
x5/(x3/x4-(x1*x2)),
(x3-x4)*x5-(x1*x2),
(x3-x4)/x5-(x1*x2),
x4/x5+x3-(x1*x2),
x5/(x3-x4)-(x1*x2),
#x1*x2 minus...
((x1*x2)-x3)*x4+x5,
((x1*x2)-x3)*x4-x5,
((x1*x2)-x3)*x4/x5,
((x1*x2)-x3)/x4+x5,
((x1*x2)-x3)/x4-x5,
x4/((x1*x2)-x3)-x5,
x4/((x1*x2)-x3)/x5,
((x1*x2)-x3)-(x4/x5),
((x1*x2)-x3/x4)*x5,
((x1*x2)-x3/x4)/x5,
x5/((x1*x2)-x3/x4),
#x1*x2 multiplies...
(x1*x2)*(x3-x4)+x5,
(x1*x2)*(x3-x4)-x5,
(x1*x2)*(x3-x4)/x5,
(x1*x2)*(x3/x4+x5),
(x1*x2)*(x3/x4-x5),
(x1*x2)*(x5-x3/x4),
#x1*x2 over...
((x1*x2)/x3+x4)*x5,
((x1*x2)/x3+x4)/x5,
((x1*x2)/x3-x4)*x5,
((x1*x2)/x3-x4)/x5,
(x4-(x1*x2)/x3)*x5, 
(x4-(x1*x2)/x3)/x5,
x5/((x1*x2)/x3+x4),
x5/((x1*x2)/x3-x4),
x5/(x4-(x1*x2)/x3),
(x1*x2)/x3+x4/x5,
(x1*x2)/x3-(x4/x5),
x4/x5-(x1*x2)/x3,
(x1*x2)/x3+(x4-x5),
(x4-x5)-(x1*x2)/x3,
(x1*x2)/(x3-x4)+x5,
(x1*x2)/(x3-x4)-x5,
(x1*x2)/(x3-x4)/x5,
(x1*x2)/(x3/x4+x5),
(x1*x2)/(x3/x4-x5),
(x1*x2)/(x5-x3/x4),
#x1*x2 is under...
(x3/(x1*x2)+x4)*x5,
(x3/(x1*x2)+x4)/x5,
x5/(x3/(x1*x2)+x4),
(x3/(x1*x2)-x4)*x5,
(x3/(x1*x2)-x4)/x5,
x5/(x3/(x1*x2)-x4),
(x4-x3/(x1*x2))*x5,
(x4-x3/(x1*x2))/x5,
x5/(x4-x3/(x1*x2)),
x3/(x1*x2)+x4/x5,
x3/(x1*x2)-x4/x5,
x4/x5-x3/(x1*x2),
x3/(x1*x2)+(x4-x5),
(x4-x5)-x3/(x1*x2),
(x3-x4)/(x1*x2)+x5,
(x3-x4)/(x1*x2)-x5,
x3*(x4-x5)/(x1*x2),
x3/(x4-x5)/(x1*x2),
(x3+x4/x5)/(x1*x2),
(x3-x4/x5)/(x1*x2),
(x4/x5-x3)/(x1*x2),
# ================== now the  x1+x2  stuff ======= Not yet prettified!
x5/(x3-(x1+x2))/x4,
(x3-(x1+x2))*x4/x5,
(x1+x2)*x5/(x3-x4),
x3/((x1+x2)-x5)/x4,
((x1+x2)-x4)*x5/x3,
(x5-(x1+x2))*(x3-x4),
(x3-x4)/(x5-(x1+x2)),
(x1+x2)*(x3-x4)*x5,
(x1+x2)/(x3-x5)/x4,
(x1+x2)*(x4-x5)/x3,
x3/(x1+x2)/(x4-x5),
(x3-x4)/(x1+x2)/x5,
(x5-(x1+x2))/(x3-x4),
(x3-x5)*x4/(x1+x2),
x3/((x1+x2)/x4+x5),
(x1+x2)+x3*(x4-x5),
x3+(x4-(x1+x2))/x5,
x4*(x3-x5)-(x1+x2),
(x4-(x1+x2)/x3)/x5,
((x1+x2)/x3+x4)/x5,
x5/(x4-(x1+x2)/x3),
(x3-x5)+(x1+x2)*x4,
(x3-x5)-(x1+x2)*x4,
(x4-(x1+x2)/x5)*x3,
(x3-(x1+x2))/x5-x4,
((x1+x2)/x3+x5)*x4,
((x1+x2)-x5)/x4+x3,
(x4+(x1+x2)*x5)*x3,
(x3-(x1+x2)*x4)*x5,
x4/((x1+x2)/x3-x5),
(x1+x2)/(x4/x3+x5),
(x3-x4/x5)/(x1+x2),
((x1+x2)/x3-x4)/x5,
(x1+x2)*(x4/x3+x5),
((x1+x2)*x3-x5)*x4,
x4/(x3+(x1+x2)*x5),
((x1+x2)/x5-x4)*x3,
(x3+(x1+x2)*x5)/x4,
(x4/x5+x3)/(x1+x2),
((x1+x2)-x3)/x5-x4,   
x5/(x4-(x1+x2)*x3),
((x1+x2)*x4-x3)/x5,
x3/((x1+x2)*x5-x4),
(x1+x2)*(x5/x4-x3),
(x1+x2)/(x4-x3/x5),
(x5-(x1+x2)*x3)/x4,
(x1+x2)/(x4/x3-x5),
(x5/x3-x4)/(x1+x2),
(x1+x2)*(x4-x3/x5),

((x1+x2)-x4/x3)*x5,
x4/(x5/x3+(x1+x2)),
((x1+x2)-x3)*x4-x5,
x5+(x3-(x1+x2))*x4,
(x4-x3)/x5+(x1+x2),
x4/((x1+x2)-x5/x3),
(x3/x4+(x1+x2))/x5,
(x4/x5-(x1+x2))/x3,
(x5/x4+(x1+x2))*x3,
(x3-(x1+x2))*x4-x5,
x3-x5-(x1+x2)/x4,
x4/(x3/x5-(x1+x2)),
x5+((x1+x2)-x3)*x4,
x3-(x1+x2)*x5/x4,
(x5/x3-(x1+x2))*x4,
(x1+x2)*x3/x5-x4,
(x1+x2)*x3/x4+x5,
(x3-x5)+(x1+x2)/x4,
((x1+x2)-x4/x5)/x3,
(x4-x5)/x3-(x1+x2),
x3/x5-(x1+x2)*x4,
x4/x5-(x1+x2)/x3,
(x1+x2)*x4-x5/x3,
x3/x5+(x1+x2)*x4,
x4-x5/x3-(x1+x2),
(x3-x5)/(x1+x2)-x4,
(x3/(x1+x2)+x5)/x4,
(x1+x2)/(x4-x5)+x3,
x4/x5+x3-(x1+x2),
x3/(x4/(x1+x2)+x5),
(x1+x2)/x4+x3/x5,
(x5/(x1+x2)+x3)*x4,
(x1+x2)-(x4/x5+x3),
(x4-x3/(x1+x2))*x5,
x3/(x5-x4/(x1+x2)),
(x1+x2)/x4-x3/x5,
(x3-x4)/(x1+x2)+x5,
x5/(x4/(x1+x2)-x3),
(x5-x3/(x1+x2))/x4,
(x5/(x1+x2)-x3)*x4,
(x1+x2)+x5/x3-x4,
(x1+x2)/(x3-x5)-x4,
(x4/(x1+x2)-x3)/x5,
x5+(x1+x2)*(x3-x4),
(x1+x2)*(x3-x4)-x5,
x3/x5+x4/(x1+x2),
#
x4/(x1+x2)-x5/x3,
x5/x3-x4/(x1+x2),
x3/x5/(x1+x2)-x4,
x3/x4/(x1+x2)+x5,
x4-x3/x5/(x1+x2),
x4/(x3-(x1+x2))+x5,
x3/(x5-(x1+x2))-x4,
x3/((x1+x2)-x4)-x5,
x4-x3/(x5-(x1+x2)),
x5/(x3-x4)+(x1+x2),
x3/(x1+x2)+(x4-x5),
(x4-x5)-x3/(x1+x2),
x5/(x3-x4)-(x1+x2),
#
#Ten more with symmetry group <(12)(34)> also of order 2:
(x1-x2)/(x3-x4)+x5,
(x1-x2)/(x3-x4)-x5,
(x1-x2)/(x3-x4)*x5,
(x1-x2)/(x3-x4)/x5,
(x1/x3+x2/x4)+x5,
(x1/x3+x2/x4)-x5,
x5-(x1/x3+x2/x4),
(x1/x3+x2/x4)*x5,
(x1/x3+x2/x4)/x5,
x5/(x1/x3+x2/x4),
#
#and 75 with symmetry group of order 1 (Not yet sorted, sorry)
((x1-x2)*x3+x4)-x5,
((x1-x2)*x3+x4)*x5,
((x1-x2)*x3-x4)*x5,

((x2 - x1)/x5 + x4)*x3,
((x1 - x2)/x5 - x4)*x3,
((x4 - x2)*x1 +x3)/x5,
(x3 - x4)*(x2/x5 + x1),
(x4 - x5)*(x1 - x2/x3),
x4/(x3 + x2*(x5 - x1)),
x1/((x3 - x2)*x5 - x4),
(x4*(x5 - x1) - x3)/x2,
x5/(x3/x4 + x2 - x1),
x5 +(x4 - x1)/x3  - x2,
((x3 - x5) - x2/x4)*x1,
x3 + (x2 - x1)*x4/x5,
(x3 - x1)*x5/x2 - x4,
((x1 - x4) + x5/x3)*x2,
x2*(x3 - x4) - x5/x1,
(x5 - x4/x3)/x2 + x1,
x3/x4 + (x2 - x1)*x5,
(x1 + x2/x4)*x5 + x3,
x1 + (x3/x2 - x4)*x5,
(x4/x1 + x3) - x2,
(x3  - x4/x1)*x5 -x2,
x2 + x3*(x4 - x1/x5),
x4 - (x1/x2 + x3)*x5,
(x3/(x2 - x4) - x5)/x1,
x4*(x1/x3 - x5) - x2,
( x1/(x3 - x5) - x4)*x2,
(x5 + x4/(x1 - x2))*x3,

(x3/x4 - x5)*x2/x1,
x2/(x3 - x5/x1)/x4,
(x5/x4 + x1)*x3/x2,
((x4 - x2)/x5 + x3)/x1,
x4/(x2/x5 + x3)/x1,
(x1 - x4/x2)*x5/x3,
(x3 - x2)/(x5 - x1/x4),
((x2 - x5)/x1 - x4)/x3,
(x2 - x1/x3)/(x4 - x5),
x2/((x4 - x3)/x1 + x5),
x5/((x2 - x1)/x4 - x3),
(x3 - x2)/(x1/x5 + x4),
(x4/x1 + x3)/(x5 - x2),
(x3/x5 - x1/x2)*x4,
((x2 - x1) - x5/x4)/x3,
(x5/x4 + (x1 - x2))/x3,
x5/((x4 - x2) - x3/x1),
(x3/x4 + x1)/x5 + x2,
(x3/x5 - x1)/x2 + x4,
x4 - (x1 + x2/x5)/x3,
(x5/x2 - x1)/x4 - x3,
(x1/(x4 - x5) + x2)/x3,
x1/(x4 - x2/(x3 - x5)),
x1/x2 + (x5 - x4)/x3,
(x1 + x2/x5)/x3 - x4,
(x5 - x1/x2)/x3 - x4,
x1/(-x4 + x2/(x3-x5)),
x3/x1 - x5/x4 + x2,
x5/x1 - x3/x2 - x4,
(x2- x1)/x5 - x3/x4,
x3/(x2 - x4) - x1/x5,
x3/x4 - x1/(x5 - x2),
x4/x5/(x1 - x3) + x2,
x1/x2/(x5 - x4) - x3,

x1/(x2/x3 - x5)/x4,
(x3/x1 - x5/x2)/x4,
x1/(x4/x2 - x3/x5),

x3 + x2/(x4/x1 + x5),
x3 - x5/(x1/x2 + x4),
x3/(x4/x2 + x5) - x1,
x2/(x1/x4 - x3) + x5,
x2/(x4/x5 - x3) - x1,
x3/(x2 - x1/x4) + x5,
x3/(x4 - x5/x1) - x2,

(x5-x4)-x2/(x3-x1)
]:



#Here are the routines which generate the longer list of permutated values.

with(combinat):P:=permute(5):
H:=[seq({seq(subs({seq(x||i=x||(P[j][i]),i=1..5)},G[k]),j=1..120)},k=1..500)]:
K:=factor(H):                  
# That part is slow! But it's necessary to avoid duplications in  L:
L:={seq(op(K[i]),i=1..500)}: 
#   ...though if I worked at it a little I could probably limit the
#      factoring just to a few formulas for which it's really necessary...
LD:={op(denom(L))}:
lprint(`Copied`,nops(G),` formulas; expected 500`);
lprint(`Computed`,nops(L),` values; expected 27142`);
lprint(`Computed`,nops(LD),` denominators; expected 1001`);

#There is a separate file at www.math.niu.edu/~rusin/uses-math/games/krypto
#containing these denominators (and other information)

#Summary: 10 types of symmetry among the 500 formulas and 27142 values:
#
#   2  with orbits of size   1 (Sym(5))      --      2 formulas
#  10  with orbits of size   5 (Sym(4))      --     50 formulas
#  22  with orbits of size  10 (Sym(3)xZ2)   --    220 formulas
#  12  with orbits of size  15 (Dih(8))      --    180 formulas
#  46  with orbits of size  20 (Sym(3))      --    920 formulas
# 102  with orbits of size  30 (Z2^2)        --   3060 formulas
#   5  with orbits of size  30 (diag(Z2^2))  --    150 formulas
# 216  with orbits of size  60 (Sym(2))      --  12960 formulas
#  10  with orbits of size  60 (diag(Z2))    --    600 formulas
#  75  with orbits of size 120 ( 1 )         --   9000 formulas


#Are we really sure these values are distinct? YES! They give
#distinct values for some sets of inputs. For example,

#  samp:=[13^2,17^2,19^2,29^2,43^2]:
#  L0:=subs({seq(x||i=samp[i],i=1..5)},L): samp, nops(L0);  
#shows that all 27142 values are different.

#I tried varying these numbers and got other, smaller good sets,
#e.g. [13^2,nextprime(240),nextprime(332),nextprime(380),nextprime(460)]:
#                         [169, 241, 337, 383, 461], 0
#and a random search yielded
#                         [127, 174, 223, 262, 293], 0

#Here -- you can keep playing looking for more small "witnesses" to the total
#Just type    search():   to Maple, and wait...

kernelopts(printbytes=false):       randomize():
r:=proc() floor(rand()/10^12*220)+1:end:

search:=proc()  local samp,L0,i:
while true do samp:=sort([r(),r(),r(),r(),r()]):
if not member(0, subs({seq(x||i=samp[i],i=1..5)},LD)) then
 L0:=subs({seq(x||i=samp[i],i=1..5)},L): 
 if nops(L0)>27140 then print(samp,27142-nops(L0)) fi:
fi:od:
end:

#That random-search combination of values for the  x_i gives 27142 different 
#values, all in  [ -882992138, 882992138], with
#the 11973rd smallest = -4927299/127, next=102114/293 ; includes
#1559 negative and 2153 positive integers, inc. -39, -31, 31, 39, 47, ...
#Others are fractions with denominators as large as  2978674332 .

#Other "small" sets recently found by random searches: 
#[106, 173, 202, 228, 239],[116, 135, 156, 231, 238],[118, 145, 199, 222, 237],
#[45, 74, 173, 211, 236],[129, 158, 199, 222, 233],[61, 124, 157, 214, 227]    
#[55, 131, 193, 202, 218],[92, 143, 167, 179, 188],  

#This last prompted an exhaustive search for lower top-two terms, with
#the find that this hand works:
#          [92, 143, 167, 177, 182]   <-- current winner! 
#There are no smaller triples retaining these bottom 2 #s
#There are no other triples using the same top three.
        
#(Hm, there are about 1.5 billion or so combinations with all terms < 182...
#But it seems like many of them cause division-by-zero errors.)
#When is an exhaustive search OK? Can check all combinations with terms
#under 68 with 10million cases; under 106 with 100million; under 167 
#with 1billion; ...

#Also e.g. [47, 55, 126, 166, 186] gives just a single coincidence (27141 vals)
#Smaller sets, e.g.  [74, 105, 121, 125, 127]  fall somewhat further short
#of the total (25 short here).


#Open questions: How do these formulas relate to the actual Krypto
#games, where the five inputs and the target are drawn together from a
#deck of 56 cards: 3 each of number 1-6, 4 each of 7-10, 2 each of 11-17,
#and one each of 18-25. 
#Questions:
#What is the largest number of possible values from a given hand of 5? average?
#What is the probability that a 5+1 hand is unsolvable? "~1/3000" is in rules
#What about hands which have a unique solution? average number of solutions?
#...


#Here is the routine to check possibilities:
play:=proc(a,b,c,d,e,f): local samp,l,dd,nn,i:
samp:=[a,b,c,d,e]:
for l in L do 
 dd:=denom(l):nn:=numer(l): if
  subs({seq(x||i=samp[i],i=1..5)},dd)*f=subs({seq(x||i=samp[i],i=1..5)},nn)
 then lprint(l) fi:
od:
end: