The Proposed Solution

(trial by mathematical combat)
Since there is no agreed or even good definition of best solution, this method guarantees a good (i.e.equal compact districts respecting local borders) determined by a nonpolitical process.

Redistricting to be based upon a reasonable and rigid mathematical formula which combines equality of population, compactness, and respect for local city, town, and county boundaries.

Best map wins and determination of best is clear

The proposed amendment uses this bottom equation to reach the same result as the positive top one.

Equality

Compactness

Respect
for
Localities

High
Number
Is Best

=

This equation maximizing good things can be more conveniently be realized by minimizing bad things

Inequality

Sprawl

Disrespect
for
Localities

=

Low
Number
Is Best

The amendment in math

Proposed Constitutional Amendment

[Section 1] A state must choose its congressional, subject to having at least five congressional districts, and legislative districts, from among duly submitted maps, the map which has the lowest product of the following three factors:

a. The population of citizens in the district with the largest number of citizens divided by the population of citizens in the district with the smallest number of citizens minus the gauge percentage,

b. The sum of the shortest interior public road distances between the longest geographic distance between two points in each district minus the gauge percentage of the lowest submitted road distance sum under this subsection (before this subtraction), and

c. The number of crossings where a district border crosses a local jurisdiction border, but at least one.

[Section 2] Any citizen of a state may propose a map showing clearly the district borders with the above calculations.The state legislature will define the relevant local jurisdictions and the gauge percentages (between 0.88 and 0.99) as well as the procedures for map submission and discernment of the chosen map. A two-thirds majority of both houses of Congress may permit a state to use a specific alteration of paragraph one on petition from that state.

[Section 3] This section shall be inoperative unless it shall, prior to December 31, 2031, be ratified by majority vote at conventions in three-fourths of the several states, where delegates are selected by popular election.Each current congressional district shall elect five convention delegates at the next state-wide election after submission to the states by Congess.

The point of section 1a:

The population of citizens in the district with the largest number of citizens divided by the population of citizens in the district with the smallest number of citizens minus one

To ensure that the districts have good equivalence of population.
Subtracting a gauge percentage ensures that this factor will be the most powerful. (Subtracting one would have the theoretical possibility of making this factor equal to zero and thus disabling the other factors)
If the gauge percentage is .99, this factor will be most powerful. I would hazard a guess that a percentage of 0.96 would generally be appropriate.

The point of 1b

The sum of the shortest interior public road distances between the longest geographic distance between two points for each district minus the gauge percentage of the lowest submitted distance under this subsection before this subtraction

To engender compactness.
Note that it makes no difference whether the unit is in parsecs or angstroms as the relative size between submitted maps is all that matters. I expect most jurisdictions will measure to a tenth of a mile, with some sparse juridictions using miles and dense ones hundredths of a mile or even yards.
Using road distance will ensure that geographic barriers like lakes, rivers, and mountain ranges are respected.
Interior distance will prevent connecting the horns of a crescent – think the original gerrymander. It will also forbid noncontiguous districts.
Note that a map submitter must calculate both the as-the-crow-flies geographic distance and the road distance
I prefer this method to alternatives such as the shortest perimeter which will disadvantage river borders
“Public road distance” refers to public roads plus any other common mode of public scheduled transportation such as train, ferry or regular air route. See the Roosevelt Island Example
The combination of road distance and geographic distance makes the entire system more difficult to play games with.

An Example of Road Distance

Assume this impossible imaginary district in New Orleans (a square ex the SE corner) Geographic distance is diagonal NE to SW. Note the lengthy road distance in red which must detour to cross the Mississippi and only can get near the southwest corner as there are no roads at the corner

Roosevelt Island

is an island with about 12 thousand people in the East River between Manhattan and Queens, but legally part of the County of New York (Manhattan) If Roosevelt Island were at a geographic extreme of a district, how could you calculate the road distance?

There are many ways to reach Roosevelt Island. Not all of them can be used for public road distance under the proposed amendment.

Queens

59th St. Bridge

Manhattan (County of New York)

The point of 1c:

The number of crossings where a district border crosses a local jurisdiction border, but at least one.

To give advantage to maps which respect local jurisdictions.
Considered using number of split local jurisdictions. This fails to take notice of local jurisdictions split in three or artificially jagged borders.
A similar method would be number of jursdictions in each district with or without subtracting the total number of local jurisdictions.
I prefer counting crossings because it disadvantages cuteness.

Section Two – the process

The implementation is left to each state. This will permit local differences to be taken into account as each state determines along the process and procedures, units of distance, deadlines, the gauge percentages and local jurisdictions.
I envision a process where a special geek-ish master is appointed to receive proposed maps.
Each map must lay out all of the calculations and be submitted by a deadline. The master verifies the calculations of the map with the lowest product. If they are correct or lower, then that map wins. If he finds an error that increases the product above another map, then he begins the process again with the now-winning map. Note the special master might also need to check another map if it is being used for one of the gauge percentages.
I think the leading map should be made public during the vetting process so the special master can gain helpful commentary from interested parties
Reasonable special masters would include accounting and actuarial firms as well as others. Public accounting is as seasonal as strawberries and the late spring lull is about the right time to be judging maps.
States may offer prizes for winning maps. I did not include prizes in the amendment because this is trivia, but I think they should be offered. The amounts could vary with the complexity of the job and the wealth of the state.
Judicial review is still possible if a losing map submitter can show error or corruption by the special master, but I would expect that to be rare.
I would estop a map submitter from claiming a lower product in court than submitted to the master.

Section Two – the local jurisdiction – this will vary by state.

Both New York and Massachusetts have both counties and towns. I would expect Massachusetts to use towns and New York to use counties. A Massachusetts resident will always know his town, but frequently be unable to name his county. The reverse is true in Westchester and Rockland Counties in New York.
New Jersey might conceivably use counties for congressional districts but townships for legislative districts.
Virginia and Maryland will probably use counties and independent cities.
Each state can choose its own set of local jurisdictions to honor the way that state operates.

Section Two – the escape hatch

So long as the resulting formula were party-neutral, I think Congress would easily defer to a petition from a state.

Some state may have a problem with this formula peculiar to itself. My experience is all on the Atlantic seaboard and may work less well in western or even mid-western states. I worry most about Alaska and Hawaii with their islands and sparse roads and vast distances where people get about by air, boat, or dogsled.

A possible Ratification Section 3:

This section shall be inoperative unless it shall, prior to December 31, 2031, be ratified by majority vote at conventions in three-fourths of the several states, where delegates are selected by popular election. Each current congressional district shall elect five convention delegates at the next state-wide election after submission to the states by Congress.

This section differs from standard ratification language such as this for the 22nd amendment:
- This section shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several states within seven years from the date of its submission to the States by the Congress
The reason for the difference is to overcome a projected unwillingness of state legislatures to give up their gerrymandering power. This wording forces a popular vote in each state. Unlike national convention proposals, it does not open the United States Constitution to wider amendment.
The reason for the shortened date is that this amendment should be used for the next decennial redistricting. If it cannot pass in that time, perhaps it is not a sufficiently good idea.
This section is inessential, and could be omitted without harm to the central thesis.

Why multiply and not some other operation?

Multiplication is easy to understand. It forces recognition of each of the three factors.

Addition would make the result dependent on the unit of distance. Maps with distances in angstroms would only depend on factor b.

The Proposed Solution (trial by mathematical combat) Since there is no agreed or even good definition of best solution, this method guarantees a good (i.e.equal compact districts respecting local borders) determined by a nonpolitical process.