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[]
[null,19,null,10,null,8,null,14]
[]
[null,null,"E(X) =",null,{"$f":"(B2+D2+F2+H2)/4","v":12.75}]
[]
[null,null,"Var(X) =",null,{"$f":"(B2^2+D2^2+F2^2+H2^2)/4-E4^2","v":17.6875}]
[]
[null,"Can you find positive integer values "]
[null,"for the four sides of a fair tetrahedral"]
[null,"dice that are not all even "]
[null,"so that E(X) = Var(X)?"]
[]
[null,null,null,"Answers on Sheet2, Search program on Sheet3"]

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["Answers"]
[]
[]
[]
[]
["E(X) = (a + b + c + d)/4,                    Var (X) = (a2 + b2 + c2 + d2)/4 - ((a + b + c + d)/4)2"]
["If we put these equal to each other, then we get the equation:  "]
[]
["4(a2 + b2 + c2 + d2) =  4(a + b + c + d) + (a + b + c + d)2    *"]
[]
["There are plenty of solutions to this for positive whole number a, b, c and d. "]
["(See Sheet3 for a comprehensive search.)"]
["All of these solutions, however, seem to have a, b, c and d all even. Why? "]
[]
["If we complete the square on the right hand side of * and divide by 4 we get;"]
["a2 + b2 + c2 + d2 + 1 =  ((a + b + c + d)/2 + 1)2  "]
["If one or three of a, b, c and d are odd, then the right hand side is not an integer. "]
["If two of a, b, c and d are odd then the left hand side is congruent to 3 (mod 4) and so cannot be a square."]
["If four of a, b, c and d are odd then the left hand side is congruent to 5 (mod 8) and so cannot be a square."]
["(With many thanks to Bernard Murphy for this elegant solution.) "]
[]
["Start on Sheet1"]

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[4,10,10,12,"is highest"]
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