Range Voting Algorithm Cuts Gerrymandering Out of US Redistricting

CGP Grey

Recursive geometric formulas split voter populations strictly by mathematical ratios, eliminating partisan human manipulation from electoral district design.

AI-assisted summary of The Shortest-Splitline Algorithm: a Gerrymandering Solution [Bonus Video] — original source. Check the original for full context.

Source published: 2011-07-12

Key ideas

The shortest-splitline algorithm provides an objective, geometric method to partition a territory into voting districts without partisan manipulation. (0:00)
When the district count is odd, dividing by two and rounding up and down establishes the precise population ratio needed for the split line (such as a 4:3 ratio for seven districts). (1:00)
The Center for Range Voting (RangeVoting.org) created the algorithm and offers side-by-side comparisons of algorithmic maps versus existing gerrymandered US congressional districts. (3:10)

Detailed notes

## Background This bonus video was published on YouTube by CGP Grey on July 12, 2011. According to the page description, it offers a brief explanation of the shortest split-line redistricting method as a geometric solution to gerrymandering.

## TL;DR CGP Grey breaks down the mechanics of the shortest-splitline algorithm, an objective mathematical approach to redistricting that eliminates partisan boundary manipulation. For even powers of two, the algorithm recursively divides populations in half using the shortest possible straight lines until the desired number of equal-sized districts is reached. When handling odd district counts, it divides the target by two and rounds both up and down to create an exact population ratio, repeating this recursion until every sector is properly allocated. Grey concludes by pointing viewers to RangeVoting.org to view real-world comparisons of actual U.S. congressional maps against algorithmically generated districts.

## Atmosphere & style The presentation has the crisp, brisk, and intellectually tidy feel characteristic of early CGP Grey educational shorts. Delivered with focused, rapid narration, it unpacks an intricate geometric problem step-by-step with clean, methodical diagrams that make the recursive logic click effortlessly into place. The tone is practical, rational, and quietly triumphant, leaving the viewer with a satisfying sense of clarity about how simple math can circumvent convoluted political manipulation.

## Bulletin - [00:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=0s) The shortest-splitline algorithm provides an objective, geometric method to partition a territory into voting districts without partisan manipulation. - [01:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=60s) When the district count is odd, dividing by two and rounding up and down establishes the precise population ratio needed for the split line (such as a 4:3 ratio for seven districts). - [03:10](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=190s) The Center for Range Voting (RangeVoting.org) created the algorithm and offers side-by-side comparisons of algorithmic maps versus existing gerrymandered US congressional districts.

## Key takeaways - [00:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=0s) The shortest-splitline method replaces political boundary drawing with an objective, recursive mathematical formula. - [00:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=0s) When dividing into a power of two, the algorithm cuts the population into equal halves using the geometrically shortest straight line available. - [00:35](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=35s) Sub-regions are treated as independent areas and repeatedly bisected until reaching single-district allocations. - [01:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=60s) Odd numbers of districts are handled by dividing the target count by two and rounding both up and down to form an integer ratio. - [01:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=60s) The derived integer ratio dictates both the proportion of population allocated to each side and the count of congressional seats assigned. - [01:20](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=80s) In a 42-voter, 7-district example, the first split creates a 4:3 ratio of 24 voters to 18 voters. - [02:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=120s) Each sub-region recursively repeats the rounding and shortest-line rule until all districts contain equal numbers of voters. - [02:50](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=170s) The algorithm produces contiguous, mathematically balanced districts completely free of human political interference. - [03:10](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=190s) RangeVoting.org developed the algorithm and provides real-world models applying official census data to all 50 US states.

## Detailed walkthrough ### [00:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=0s) The Basics of the Shortest-Splitline Method

In his primary video on gerrymandering, CGP Grey touched upon the shortest-splitline algorithm only in passing due to strict time constraints, but he returns in this bonus video to explore the mechanics behind this geometric approach to boundary drawing. The method serves as an objective, mathematical solution for carving a territory into separate voting districts without partisan manipulation. To illustrate how it works under straightforward conditions, Grey uses a sample population of 40 voters that needs to be partitioned into eight distinct, equal-sized voting districts.

When the target number of districts is an easily divisible power of two, the procedure is simple and recursive. The algorithm first identifies a straight line across the region that cuts the total population into two exactly equal halves of 20 voters each; if multiple lines can accomplish this 20-to-20 split, it selects whichever line is geometrically shortest. Next, the algorithm treats each half as an independent area and repeats the procedure, finding the shortest line that cuts each 20-voter group into two sub-groups of 10. A third and final iteration divides each group of 10 in half, leaving eight distinct voting ranges containing five voters apiece. As Grey summarizes: “And that's the basics of the shortest split line method: just always use the shortest line you can to divide the two groups.”

- Sample population: 40 voters partitioned into 8 equal districts. - First division: One shortest line divides 40 voters into two groups of 20. - Second division: Shortest lines divide each group of 20 into groups of 10. - Third division: Shortest lines divide each group of 10 into final districts of 5 voters each.

---

[01:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=60s) Handling Odd Numbers of Districts

The core challenge arises when a mapmaker must divide a region into an odd number of districts, such as seven instead of eight. Grey explains that this scenario highlights the true mathematical ingenuity of the technique: “This is where the cleverness of the shortest split line method comes up is if you take 7 divided by 2 you don't get a round number you get 3.5 but what you do is you take that 3.5 and you round it up to get 4 and down to get 3 and this tells you the ratio that you want to draw the line at for the population.” This 4:3 split dictates both the population balance across the initial cut and the exact number of congressional seats assigned to each side.

Applying this rule to a map of 42 voters, the algorithm locates the shortest possible line that splits the electorate into a 4:3 ratio. This places 24 voters on the left and 18 on the right, matching the target proportion exactly (24 divided by 18 yields 1.333, just as 4 divided by 3 yields 1.333). From there, the two sectors are resolved independently based on their allocated district counts:

```mermaid flowchart TD accTitle: Shortest-Splitline Algorithm for 7 Districts accDescr: Flowchart showing how 42 voters are split into 7 districts of 6 voters each using recursive shortest split lines.

subgraph Step1 ["Initial Split: 7 Districts (42 Voters)"] A["Total: 42 Voters Target: 7 Districts"] -->|"Round 7 / 2 = 3.5 Ratio 4:3"| B["Left: 24 Voters (4 Districts)"] A -->|"Ratio 4:3"| C["Right: 18 Voters (3 Districts)"] end

subgraph Step2 ["Left Sub-Region: 4 Districts"] B -->|"Split 2:2"| D["Two Groups of 12"] D -->|"Split 1:1"| E["4 Districts (6 Voters Each)"] end

The left region requires four districts. Because four is an even number, it is halved with a shortest split line into two sections of 12 voters, and each of those is bisected again into groups of six. For the right region, which requires three districts, the rounding trick is applied once more: dividing 3 by 2 gives 1.5, which rounds up to 2 and down to 1. The shortest line is drawn to divide the 18 voters in a 2:1 ratio, placing 12 voters in the upper section and 6 voters in the lower section (12 divided by 6 equals 2). The single-district region of 6 voters is finished immediately, while the two-district region of 12 voters is bisected with one final shortest line into two districts of 6. The process concludes with seven contiguous, equally populated districts of six voters each, generated purely by geometric optimization without human political interference.

- Initial calculation: 7 ÷ 2 = 3.5, rounding up to 4 and down to 3 (a 4:3 target ratio). - Initial division: 42 total voters split into 24 on the left and 18 on the right (24 ÷ 18 = 1.333; 4 ÷ 3 = 1.333). - Left-side recursion: 24 voters divided by halving (4 → 2 and 2 → 1, 1, 1, 1), creating 4 districts with 6 voters each. - Right-side recursion: 3 ÷ 2 = 1.5, rounding to a 2:1 ratio, splitting 18 voters into 12 (top) and 6 (bottom). - Final right-side resolution: The bottom group of 6 is complete as 1 district; the top group of 12 is split in half into 2 districts of 6 voters each.

---

[03:10](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=190s) Resources and Applications at RangeVoting.org

Grey directs viewers interested in fair representation to the original proponents of the concept, pointing them toward RangeVoting.org (the Center for Range Voting). He notes that the group developed the algorithm and maintains detailed real-world demonstrations of how the formula behaves when applied to actual geography: “If you're interested in this and other stuff relating to electoral fairness I highly recommend you go to range voting org they're the group that as far as I can tell came up with a split line method and they have a lot of really interesting information there such as maps of all of the states in America drawn using this method and comparing them to the districts that currently exist.”

By inputting official census population data into the algorithm, the site showcases side-by-side comparisons of all fifty U.S. states, contrasting bizarrely shaped, gerrymandered congressional districts with clean, algorithmically generated borders. Grey closes the bonus explanation by thanking the audience for watching.

## Notable quotes > [00:00](https://www.youtube.com/watch?v=kUS9uvYyn3A&t=0s) And that's the basics of the shortest split line method: just always use the shortest line you can to divide the two groups. — CGP Grey

## Mentioned CGP Grey, Center for Range Voting, RangeVoting.org, United States

## Video description A brief description of the shortest split-line redistricting method.

[Source](https://www.youtube.com/watch?v=kUS9uvYyn3A)

中文简报

值域投票算法消除美国选区重划中的杰利蝾螈现象递归几何公式严格按数学比例分割选民人口，消除选区划分中的党派人为操纵。 CGP Grey详细解析了“最短分割线算法”的运作机制，这是一种旨在消除党派边界操纵的客观数学重划选区方法。对于2的整次幂，该算法利用尽可能短的直线递归地将人口一分为二，直至达到所需数量且规模均等的选区。在处理奇数选区数量时，它将目标数除以二并分别向上和向下取整以建立精确的人口比例，重复该递归直至每个区域都得到妥善分配。Grey在最后引导观众访问RangeVoting.org，查看实际的美国国会选区地图与算法生成选区的真实对比。最短分割线算法提供了一种客观的几何方法来将某一区域划分为投票选区，不受党派操纵。 (0:00) 当选区数量为奇数时，除以二并分别向上和向下取整，即可确定分割线所需的精确人口比例（例如划分为七个选区时的4:3比例）。 (1:00) 值域投票中心（RangeVoting.org）创建了该算法，并提供了算法地图与现有经过杰利蝾螈操纵的美国国会选区的并排对比。 (3:10)

Source quotes

[0:00] CGP Grey: And that's the basics of the shortest split line method: just always use the shortest line you can to divide the two groups.

Original transcript

[0:00] in my main video I mentioned something [0:01] called the shortest split line method I [0:03] really had to zoom past that because of [0:05] time constraints but it's a very [0:07] interesting way of dividing up voters [0:08] into boundaries into different districts [0:10] so I wanted to spend a little bit more [0:12] time going into depth with it here so [0:14] this is the sample population I used in [0:16] the main gerrymandering video there [0:18] about 40 voters here and I want to [0:20] divide them up into eight different [0:21] groups now when you have an easy number [0:23] it's a pretty simple thing to do you [0:25] find the line that divides them into two [0:27] equally populous groups and use the [0:29] shortest line if you have multiple lines [0:31] that will do the same thing there are 20 [0:33] voters on the left and there's 20 voters [0:34] on the right then you take each of those [0:36] two groups and you do the same thing [0:38] again you find the shortest line that [0:40] will divide the two populations into [0:41] equal sized groups so now there's 10 in [0:44] each group and ultimately we want eight [0:46] voting ranges here so we divide them one [0:48] more time [0:48] then you end up with eight equally sized [0:50] ranges each with five votes and that's [0:53] the basics of the shortest split line [0:55] method just always use the shortest line [0:57] you can to divide the two groups but the [1:00] question that naturally comes up is what [1:01] if you have an odd number of districts [1:02] you want to divide them into seven [1:04] groups instead of into eight groups this [1:07] is where the cleverness of the shortest [1:08] split line method comes up is if you [1:10] take 7 divided by 2 you don't get a [1:12] round number you get 3.5 but what you do [1:15] is you take that 3.5 and you round it up [1:17] to get 4 and down to get 3 and this [1:21] tells you the ratio that you want to [1:22] draw the line app for the population so [1:24] you want to draw a line that divides the [1:26] population into a 4:3 ratio for every 4 [1:29] voters in one group you want 3 voters in [1:31] the other group so as best I can tell [1:33] this is the shortest line that will [1:35] divide this population of voters into a [1:36] 4:3 ratio unless there's double check [1:39] there's 24 voters on the left there are [1:41] 18 voters on the right 24 divided by 18 [1:44] gives us 1 point 3 and 4 divided by 3 [1:46] gives us 1 point 3 we've drawn this line [1:49] in the right spot the next step is to [1:51] take that ratio that 4:3 ratio and that [1:53] actually tells you how many districts [1:56] you then want to divide each of these [1:57] separate groups into so on the left hand [1:59] side we want to take that group of [2:01] voters and divide them into four equally [2:03] sized groups now that's really easy to [2:05] do because we have an even number so we [2:07] find the shortest line to divide them [2:09] into two equal sized groups and then do [2:12] it once again [2:13] now we have our four groups on the Left [2:14] each of which has six voters inside of [2:17] it now to divide up the group that's on [2:19] the right we do the same thing that we [2:21] did with the number seven take three [2:22] divided by two you get one point five [2:25] round one point five up you get to [2:27] around one point five down you get one [2:29] so we want to divide that group into a [2:31] 2:1 ratio finding the shortest line that [2:33] will do that should be this line we can [2:35] just double-check our math again there [2:37] are 12 voters in the top six voters in [2:38] the bottom 12 divided by 6 gives us 2 2 [2:41] divided by 1 gives us 2 so we know we've [2:43] drawn that line in the correct spot and [2:45] again that ratio tells us how many [2:47] districts we actually want to draw among [2:49] these remaining areas so that bottom one [2:50] is 1 we're done there right that has six [2:53] voters there's nothing more to do with [2:54] that bottom district the district that's [2:56] two it's an even number so that's easy [2:58] we want to make two districts out of [2:59] here just divide it in half [3:00] there's the shortest line that will give [3:02] us 6 voters on each side and now at the [3:04] end of that method we have 7 equally [3:06] populous districts that were drawn with [3:08] the shortest lines possible to create [3:09] those districts if you're interested in [3:13] this and other stuff relating to [3:14] electoral fairness I highly recommend [3:15] you go to range voting org they're the [3:18] group that as far as I can tell came up [3:19] with a split line method and they have a [3:21] lot of really interesting information [3:22] there such as maps of all of the states [3:24] in America drawn using this method and [3:26] comparing them to the districts that [3:27] currently exist thank you very much for [3:30] watching