An aircraft devotes almost its entire lifetime aloft in level flight.

Level flight requires a balance of aerodynamic forces -- especially in pitch.  Stability requires that vertical perturbations will be correctable by aerodynamic control forces.  Moreover, in the event of an engine failure, the aircraft must not stall by default.  That means the center of gravity for an aircraft is located forward of the center of lift as shown in the puzzle.
Nota bene, elevator is a misleading term.  It controls pitch.  Same for the canard, by the way.  Pitch determines the angle-of-attack between the wing and the relative wind. Thus either the elevator or the canard controls angle-of-attack.  It is thrust that really controls the "elevating" of an aircraft, whether climbing, descending, or in level flight.  Loss of thrust results in loss of altitude but must not result in an increase in angle-of-attack toward a stall condition.  Thus, for safety, center of gravity must be located forward of center of lift.
The resulting nose-heaviness in level flight must be balanced either by a downward force provided by the elevator located in the empennage or by an upward force provided by a canard located near the nose of the aircraft.  In the Conventional Configuration, the required forces obey the following equation: Lift = Weight + Elevator.  For the Canard Configuration, the equation must be changed to read Lift = Weight - Canard
• For an aircraft of a given Weight, a reduction in requisite Lift and therefore drag will save fuel, which is an important consideration, as addressed in Green Flight.
• Inasmuch as the canard provides an upward force, it shares the effort of carrying the aircraft's Weight.  Thus, the Canard Advantage can take the form of greater useful load -- passengers or freight (payload) or fuel (range).
From the puzzle formulation, solvers can directly calculate that for a given Weight of an aircraft in level flight, the difference in the required Lift between the two configurations would be equal to the sum of the respective balancing forces, Elevator + Canard.  Thus, the Canard Advantage is greater useful load which can be expressed as a percentage of the aircraft Weight by this simple equeation: 100 (Elevator + Canard) / Weight %.
Balancing forces, Elevator and Canard, are provided by air flowing over and under the respective control surfaces.  The expression 'pitch authority' can be used to characterize the  'torques' on the fuselage resulting from those two forcesTorque is proportional to the product of the surface area of a control and its moment arm.
Solvers are invited to use the sketch below, which depicts the respective moment arms for the aerodynamic forces in reference to the aircraft's center of gravity....

Sophisticated solvers might make the assumption that xWL is the same for the two configurations then calculate the balancing torques with respect to each aircraft's center of gravity for each configuration.
• Clockwise torques are the same for both configurations, as given by Weight xWL
• Counter-clockwise torques are given by Elevator xWE and by Canard xWC respectively.
Accordingly, Weight xWL = Elevator xWE = Canard xWC, and by using elementary algebra, we derive the following relationship: Canard / Elevator = xWE / xWC.

The ratio of the required balancing force on a canard compared to an elevator varies inversely with the length of their respective moment arms.  Here are three interesting cases...

Case 1.  xWE = xWC, Canard = Elevator

If the aircraft's center of gravity is located longitudinally in the middle of the fuselage, the canard can have the same control surface as the elevator.

Case 2.  xWE > xWC, Canard > Elevator

If the aircraft's center of gravity is located closer to the nose of the aircraft, the canard must have a larger control surface than the elevator to overcome its shorter moment arm.

Case 3.  xWE < xWC, Canard < Elevator

If the aircraft's center of gravity is located closer to the tail of the aircraft, the canard can have a smaller control surface than the elevator because of its longer moment arm.

In all three cases, there is a Canard Advantage as derived above.  Having the largest value for Canard, an up-lifting force, Case 2 would seem to offer the greatest reduction in required Lift; however, Case 2 requires the largest control area and therefore the highest incremental weight for the canard itself.  Solvers of the Tin-Can Mystery puzzle will recall that surface area of an object varies as the square of a linear dimension, while volume and therefore weight varies as the cube.  One can estimate that the weight of the canard varies with surface area according as the 3/2-power, (Canard / Elevator)3/2.

Solvers who study the historical collection of canard aircraft listed in the puzzle will find configurations with a wide variety of moment arms and control surfaces
Let us conclude by analyzing the Boeing 767 canard configuration postulated in the puzzle.  The sketch below applies the parameters defined above.  The moment arm for the center of lift with respect to the center of gravity xWL is assumed to be the same for both configurations

One might estimate that the moment arm for the Elevator xWE to be about twice the moment arm for the Canard  xWC, which means the required balancing force Canard > Elevator by a factor of two, which corresponds to Case 2.  There would be a penalty of about 83% of the weight of the elevator in the Canard Configuration, which is small compared to the Canard Advantage resulting from the reduction of requisite Lift, which solvers will find described in Wages of Flight.

Illustrated below is a postulated lengthening the fuselage for the Boeing 767, which exploits the
Canard Advantage by applying the increase in useful load as accommodations for a larger number of passengers or space for freight...

'Stretching' the fuselage forward of the aircraft's center of gravity does increase Weight and xWL.  Meanwhile, moving the nose and its Canard forward increases the moment arm xWC, thus assuring 'pitch authority' without necessitating an increase its control area and an incremental weight penalty.  Clearly these design changes can be accomplished with no increase in requisite Lift or wing loading.

Finally, the Canard Advantage
includes a solution to the problem of ground effect under the elevator.
During its take-off roll, the pilot of a fully loaded transport aircraft must perform a rotation maneuver at a specified speed V-R, lifting the plane's nose to increase the angle-of-attack on the wings for lift-off.  In the Conventional Configuration that means the elevator is being brought low to the ground such that pressure increases underneath tending to prevent further rotation.  If the pilot delays rotation until after accelerating to a higher speed, the aircraft may not be able to take off before the end of the runway.

The puzzle asks the question,
Why would anyone want to do that? (place the horizontal stabilizer and elevator at the nose of an aircraft rather than in the empennage back at the tail), we might offer a reversal of the question, based on the Canard Advantage...

 Why would anyone want to do otherwise?

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