Copyright ©2005 by Paul Niquette. All rights reserved. |
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Copyright ©2005 by Paul Niquette. All rights reserved. |
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For people in a hurry,
the solution to the
puzzle is an Okie named...
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Nollyn |
Suppose there to be two groups of individuals named Group 1 and Group 2, each having some number of members, respectively N1 and N2. Assume that every member of both groups can be characterized by some numerical parameter of interest, whether height in centimeters, as in the Yaw of Averages puzzle, or their weight in pounds or their respective test scores or the currency in their pockets or -- well, their intelligence, as in what has become known as the "Will Rogers Phenomenon." To determine group averages, one simply divides the sums of the parameters in each group by the number of members in that group. The sums might be represented by the symbols S1 and S2. The respective averages are then given as S1 / N1 and S2 / N2.Inspired by Will Rogers, we must set about to show that the averages in both groups can be increased by moving a member from one group to the other. To do that, we simply postulate the movement of a member, whose parameter is X, from Group 1 to Group 2. The new averages will be given as (S1-X)/(N1-1) and (S2+X)/(N2+1). What we hope to make true are the following two inequalities: [1] (S1-X)/(N1-1) > S1/N1 andA little algebra music please. From [1] we find that X < S1/N1, and from [2] we find that X > S2/N2. Taken together these expressions say that to increase the average of Group 1, we most remove a member whose value is lower than the average of Group 1, and to increase the average for Group 2 at the same time, we must add a member whose value is higher than the average for Group 2. We can chain the expressions to read... S2/N2 < X < S1/N1...which tells us that if we move a member whose parametric value lies between the averages of the two groups, the averages of both groups will increase. It says something else, too. For both averages to increase, S1 / N1 > S2 / N2. In words, the starting average for the from-group must be higher than the starting average for the to-group. Kind of obvious, when you think about it. If so, then the to-group must grow in number, and the from-group must shrink. Indeed, here is how the inequalities would look in a swap... [3] (S1-X+Y)/N1 > S1/N1 andFrom [3] we find that X > Y, and from [4] we find that X < Y. Both cannot be simultaneously true, which mandates what might be called the Yaw of Averages (a pretentious pun invoking a nautical term, which has been appropriated by aviation).
Back in our classroom, suppose the teacher had crossed out two names of Okies -- Ravyn and Jenwyn -- and moved them both to the Calies list. The average height of those two Okies just happens to be 135 centimeters, same as Nollyn alone. Accordingly, the averages of both groups would increase by moving Ravyn and Jenwyn instead of -- or in addition to -- Nollyn. In general, other combinations will be possible, each calling for the movement of various numbers of names.Intuition seems to insist that, in moving sets between groups, there must exist some combination which will allow one-for-one swaps, thereby setting aside the Yaw of Averages. Here is how the inequalities would look in swaps of averages... [5] (S1-Sx/Nx+Sy/Ny)/(N1-Nx+Ny) > S1/N1 and...where Sx/Nx and Sy/Ny represent the respective parametric averages of the sets being swapped. For constant group sizes, we must set Nx = Ny. Alas, from [5] we find that Sx > Sy, and from [6] we find that Sx < Sy, and since both cannot be simultaneously true, we have confirmed the Yaw of Averages for all time. Exclamation point optional. The classroom exercise was obviously cooked up for Puzzles with a Purpose, the only purpose being mere aesthetics (for large values of "mere"). Sophisticated solvers may want to experiment with the Yaw of Averages by cooking up groups with parametric distributions that illustrate other features, answering questions like...
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ou
probably observe that out of the 26 students in the
class, there are nine
Okies who are each taller than the average of the
Calies (Nollyn, Pailwyn,
Qualyn, Ravyn, Soulyn, Ulewyn, Wailyn, Xelwyn,
Zalyn) and three Calies
who are each taller than the average of the Okies
(Irwyn, Oldewyn, Terryn).
Any or all of them moving over to the opposite group
would increase the
latter's average height. The name must have
been selected by the
teacher from just those twelve. Is it
necessary to move each of those
names one at a time, re-calculating both averages
fully a dozen times?
Maybe not. Let's figure out what the teacher
might have done while
the students were at recess...
h,
but wait. Throughout our analysis, we have been
making a secret assumption
-- that X is the parametric value
for one solitary member.
Same thing for
Y. We have learned 
s
we have seen in this puzzle, the
movement of one
or more members of a given group into another can
increase the average
for some common parameter in both
groups. In another puzzle,