Find a set of six numbers that has a range of eight and a median that is equal to the mean and half the third quartile

Question

Quantitative Reasoning

Answer 1

This problem is difficult to prove without some assumptions. Assume "numbers" mean the natural numbers plus zero or 0,1,2,3,4,5 , ... and represent the solution as six non-unique numbers a,b,c,d,e,f. Also assume a=b=0 and f=8 and . Then a solution can be found as [0, 0, 4, 4, 8, 8]. Which has a mean of 4, a median of 4, and a third quartile and range of 8 satisfying the constraints. If someone can find a solution with less constraint let me know. Actually from the constraints one can derive three equations 2a+b-e+8=0, c+d=e, f=a+8, in six unknowns. Then one can arbitrarily choose any three and solve for the other three.