Inspect geography and population before judging a map
The same voters can elect different delegations under different equal-sized district partitions.
By the end: Explain geographic aggregation while separating boundary effects, malapportionment, and evidence of intent.
Keep voters fixed and change the partition
The map experiment has twenty-five voters on a five-by-five grid: twelve prefer A and thirteen B. Five contiguous row districts each contain five voters. Five contiguous column districts also contain five. Row districts elect two A representatives; column districts elect four.
Both partitions preserve every voter and use the same plurality rule. This demonstrates sensitivity to grouping. It does not determine which partition is legally permitted, what communities it respects, or whether any real boundary maker acted with partisan intent. The cells are hypothetical voters, not a real population map.
Count the same electorate under both partitions
A district map groups voters before a counting rule selects representatives. Here each of five districts elects one member by plurality, with only A and B on the hypothetical ballot. Each of the twenty-five cells contains one voter who votes for the printed party. That makes a majority three of five within any district. The experiment asks how these five local decisions compose a legislature. Pooling all twenty-five ballots to choose one officeholder would be a different institution.
Read the row districts first, counting A cells across each horizontal line. The five A totals are 3, 3, 2, 2, and 2. Their sum is twelve; the remaining thirteen cells support B. A therefore wins the first two row districts and loses the other three. Each row contains five voters, so the A shares are 60%, 60%, 40%, 40%, and 40%. Those local percentages average to 48% because the district vote denominators are equal. The resulting A seat share is two of five, or 40%.
Now count vertically without moving a single cell. The A totals in the five column districts are 3, 3, 3, 3, and 0. A wins four columns and B wins the final column with all five votes. The combined A total remains twelve, yet its seat share becomes four of five, or 80%. You can check the totals by summing either rows or columns. Both sums count every cell once. If either partition produced a different national vote total, it would not be the fixed-electorate comparison the lesson claims to make.
Identical voters, different five-voter districts
Outlines are actual district memberships in this schematic grid, not proportional land or population areas. Each district is connected and elects one member. The two panels repeat the same twenty-five voters; rows elect A twice and columns elect A four times.
Why does the result change? A district awards one seat to its local winner and does not carry the winning margin into another district. B's five votes in the final column elect one representative, just as A's three votes in the first column elect one. The four two-vote B minorities in the other columns add eight more B votes but no further B seats. Calling those votes unrepresented by a winning party is a description of this party-seat measure. It does not mean their ballots were invalid or omitted from the count.
The row-to-column comparison changes A's seat share by forty percentage points while leaving its vote share at 48%. Compare this with a vote change, which would require changing one or more cells. No persuasion, abstention, nomination, or strategic reporting occurs between these panels. A ranking of candidates, approval cutoff, or score report is also unnecessary here because each ballot supplies the one party choice the plurality rule uses. Inferring later preferences or intensity from these letters would add information that the grid does not contain.
Both partitions are contiguous in the elementary sense that each district's cells connect along shared edges. A row and a column are each a straight strip. They also contain equal numbers of voters. These properties remove unequal voter group size and disconnected districts as explanations for this particular contrast. They do not certify either map under a real legal specification. The grid omits roads, settlements, population records, and communities. Its purpose is to demonstrate the conversion of a fixed spatial vote pattern into different delegations through different groupings.
Five local contests, not a pooled winner
District number
Row votes A/B
Row winner
Column votes A/B
Column winner
1
3 / 2
A
3 / 2
A
2
3 / 2
A
3 / 2
A
3
2 / 3
B
3 / 2
A
4
2 / 3
B
3 / 2
A
5
2 / 3
B
0 / 5
B
All districts
12 / 13
A gets 2 seats
12 / 13
A gets 4 seats
Each numbered row contains two separate five-voter contests, one from each partition. Do not add the two panels as though they contained fifty people. They depict the same twenty-five voters.
Population inequality is a different mechanism
Malapportionment means unequal representation relative to the population basis being used. For example, districts with very different populations may each elect one member. Equal numbers of votes cast do not establish equal populations because eligibility and turnout can differ.
The grid deliberately gives each district the same number of voters so that unequal group size is not the cause of the illustrated result. Real districting rules may refer to residents, citizens, or another legal population measure. They may also require contiguity and consider communities, geography, or minority representation.
Identify the population behind a representation ratio
Equal numbers of votes cast per district do not establish equal population per representative. Votes cast are an observed election count. Residents are a population count, eligible voters are another population, and registered voters may be another subset. A legal apportionment rule chooses its population basis. To assess that rule, divide the specified population by seats and compare like quantities. The grid stipulates one voter per cell. It does not model people who cannot vote, choose not to vote, or are missing from registration records.
Consider a separate two-district arithmetic example, not an alternative setting of the grid. Each district elects one representative and records 10,000 valid votes. District X has 20,000 residents, while Y has 40,000. Their residents per seat are therefore 20,000 and 40,000. On a resident basis, Y has twice as many people per representative. Equal observed vote totals conceal that difference. Dividing the 10,000 votes by residents gives 50% and 25%, but those fractions are not eligible-voter turnout rates because the eligible populations have not yet been supplied.
Now supply 12,500 eligible people in X and 25,000 in Y. If exactly those same 10,000 votes represent participating eligible people in each district, the eligible-voter participation rates are 80% and 40%. Each participating person supplies one valid vote. The example shows how equal participation counts can coexist with unequal rates and unequal residents per seat. It does not explain those rates. Access, motivation, eligibility, and registration require evidence rather than an inference from the equal totals.
Equal vote counts can conceal unequal population ratios
Measure
District X
District Y
Residents
20,000
40,000
Seats
1
1
Residents per seat
20,000
40,000
Eligible people
12,500
25,000
Participating people and valid votes
10,000
10,000
Participation / eligible people
80%
40%
Valid votes / residents
50%
25%
Separate invented example, not grid data. Each participating person supplies one valid vote. Resident fractions are not eligible-voter participation rates. No partisan preferences or causal explanation of participation is supplied.
Malapportionment concerns unequal representation against the chosen population base. Boundary effects concern which people share a district even when each district contains the same relevant population. The mechanisms can coexist, but the grid isolates the latter among its twenty-five stipulated voters. The two-district example isolates a population-per-seat difference without supplying any partisan vote pattern. A cannot receive a calculated seat bonus in that second example because no A or B choices have been provided. Keeping the examples separate prevents one mechanism from supplying evidence that belongs to the other.
Seat weighting also matters if districts elect different numbers of members. Suppose a hypothetical region with 40,000 residents elects two members while another with 20,000 elects one. Both then have 20,000 residents per seat despite different total populations. Their district magnitudes are two and one. Magnitude counts seats in a district, not residents, land area, or turnout. The original grid deliberately keeps magnitude one everywhere. Changing magnitude would require a new allocation rule for the extra members, not just drawing a larger outline around the same plurality winner.
When reading a claim of equal representation, write down the numerator and denominator before deciding whether the evidence matches it. Equal ballots per seat answers an election-participation question. Equal residents per seat answers a resident-apportionment question. Equal party vote and seat shares answers a partisan proportionality question. These measures can disagree without an arithmetic error. Which equality should constrain a real map is partly a legal and normative question. Establishing compliance requires the applicable population records and rules, not the historical handbook's terminology alone.
Change the population ratio or change district membership?
Comparison
Population or voter measure
What changes
Supported conclusion
Grid rows versus columns
Five voters per seat throughout
Which voters share a district
A seats change from 2 to 4
X versus Y, one seat each
20,000 versus 40,000 residents per seat
Residents represented per member
Y has twice the residents per seat
40,000 residents / 2 seats versus 20,000 / 1
20,000 residents per seat in both
Magnitude is 2 versus 1
Equal resident ratio, different magnitude
These are separate comparisons. The resident examples have no party data. Equality on one measure does not establish equality on another or compliance with a current legal rule.
Work through the mechanism
Keep voters fixed, redraw equal-sized groups
A has 48% of votes in both panels. Row districts give A 2 of 5 seats; column districts give A 4 of 5. Selected district 3: row A 2, B 3; column A 3, B 2.
Each circle is one voter, labeled A or B as well as colored. Outlines enclose five voters per district. Presets change only the highlighted district, not voters or boundaries. Both partitions are connected on this hypothetical grid. Equal voter counts here do not stand in for a real jurisdiction’s population requirements.
Partition
A voters
B voters
A seats out of 5
Rows
12
13
2
Columns
12
13
4
Every district contains five voters. The support pattern is identical in both panels.
Inspect local margins without changing the map
Use the lab's first-district preset to check what its selector actually does. It highlights the first row in one panel and the first column in the other. Both contain three A voters and two B voters, so both elect A. The selector does not turn rows into columns, move voters, or generate a third partition. The two full partitions remain visible together. Distinguish a change of focus from a change in the underlying data.
Before selecting the middle district, count its five cells in each panel. The third row contains A at its first and third positions, so it has two A and three B voters. The third column contains A in the first, third, and fifth rows, so it has three A and two B voters. The row elects B while the column elects A. Different district membership reverses the winner under the same one-seat rule and five-voter denominator.
Select the last district next. The fifth row has two A voters and three B voters, electing B by one vote. The fifth column has zero A voters and five B voters, electing B by five votes. Each produces one B seat despite the different margin. Relative to the three votes needed to win a five-voter two-party contest, the final column has two additional B votes. This is one way concentrated support can produce fewer seats per vote. It is not a claim that those additional votes lack any political value outside this count.
What each district-inspection preset reveals
Preset
Selected row
Selected column
Mechanism to notice
First district
A 3, B 2: A wins
A 3, B 2: A wins
Same local result in both partitions
Middle district
A 2, B 3: B wins
A 3, B 2: A wins
Different membership reverses the winner
Last district
A 2, B 3: B wins
A 0, B 5: B wins
Different margins, same one-seat award
Presets only highlight existing districts. They neither alter the partition nor edit voters. All these contests have five voters, one seat, and no tie.
To inspect the full mechanism, list all districts rather than extrapolating from the selected one. Under rows, B wins three districts by three votes to two and loses two by two votes to three. Under columns, B loses four districts by two votes to three and wins one by five to zero. In both descriptions B has thirteen votes. Its concentration in one column leaves too few B voters in each of the other columns to win there. The words packing and cracking describe concentration and division of support; using them does not by itself prove intentional boundary manipulation.
A useful hand calculation tests sensitivity without pretending the lab has a voter-editing control. Suppose the top-left A voter instead votes B. In the row partition the first district changes from A 3, B 2 to A 2, B 3. In the column partition the first district changes in exactly the same way. A's totals become eleven of twenty-five votes, with one row seat and three column seats. This is a different electorate. Its results must not replace the original fixed-grid totals when explaining the lab's two-versus-four comparison.
Return your attention to the unchanged lab and state what the original result establishes. The row and column alternatives produce a range containing at least two and four A seats among these two admissible schematic partitions. They do not reveal the minimum or maximum over every contiguous partition. Nor does the lab sample many maps or estimate a probability of any result. These examples expose a mechanism. A claim about typical or extreme outcomes would require a defined collection of valid maps and a method for examining that collection.
A paper counterfactual changes one specified ballot
Electorate
First row A/B
First column A/B
A votes / all votes
A seats: rows / columns
Original fixed grid
3 / 2
3 / 2
12 / 25 = 48%
2 / 4
Top-left A changes to B
2 / 3
2 / 3
11 / 25 = 44%
1 / 3
Only the first row and first column change in this hand calculation. All other districts retain their counts. This is not a lab control and not the original fixed-electorate map comparison.
No single map statistic settles fairness
Compactness measures geometry, not all dimensions of representation. A compact map can produce a large seat bonus when support is geographically clustered. A less compact district can sometimes reflect a particular geographic or community constraint. Assess the actual criteria and feasible alternatives.
A stronger investigation compares many valid maps using the same population and voting data, checks sensitivity across elections, and identifies the legal and substantive objectives. A large vote-seat gap alone does not identify intent. Conversely, equal population alone does not prove the absence of other representational problems.
Build a counterfactual before attributing an effect
A map comparison needs a counterfactual, an explicit alternative against which to compare the observed result. The grid compares the reference rows against the alternative columns. Voters and their ballots, the seat count, and the plurality rule stay fixed. This supports a narrow mechanical statement that changing this partition changes the delegation. It cannot establish how a real redistricting reform changes future votes, because it includes no model of campaigns, turnout, migration, or candidate entry after boundaries change.
For a real investigation, define which alternatives are allowed before comparing their results. A partition that ignores a binding population constraint is not automatically a meaningful alternative to one that satisfies it. Contiguity, population tolerance, and specified community requirements can reduce the feasible set. A collection of computer-generated maps also needs a stated generation procedure. Sampling many maps does not make the sample uniform over every legally possible map. The position of one map within that collection describes the chosen comparison, not an assumption-free probability of wrongdoing.
Compactness adds geometric information but does not replace this specification. Both grid partitions consist of five equal strips. Rotating the entire square by a quarter turn converts horizontal strip shapes into vertical strip shapes, so a rotation-invariant shape measure would give the same assessment of their district geometry. Yet their party seat outcomes differ. This is a direct reason that a shape measure alone cannot recover partisan results. Conversely, an irregular boundary may reflect a river or settlement pattern. Its shape alone does not identify which objective produced it.
Which change does each comparison isolate?
Comparison
Held fixed
Changed input
A seat result
Original rows versus original columns
25 ballots, 5 seats, plurality
District membership only
2 versus 4
Original rows versus changed top-left ballot in rows
Row boundaries, 5 seats, plurality
One A ballot becomes B
2 versus 1
Original columns versus changed top-left ballot in columns
Column boundaries, 5 seats, plurality
One A ballot becomes B
4 versus 3
Rows and columns use identical strip shapes under rotation, yet have different seat results. The table identifies a map-only counterfactual versus a changed-ballot counterfactual; neither supplies future behavior or a distribution over all valid maps.
Distinguish an effect from a purpose. The fixed grid demonstrates a partisan seat effect of grouping. To attribute deliberate advantage in a real case, seek evidence about the mapmaking process, available alternatives, stated criteria, and the data the decision makers used. A large vote-seat gap can prompt investigation without proving intent. Absence of such a gap in one election does not prove the process was neutral either. Different election patterns can interact with the same boundaries differently, so evidence across elections should retain its own ballots and denominators.
The historical boundary illustration in International IDEA's handbook compares alternative groupings of a town and countryside under simplifying assumptions. It supports examining how geography and district membership interact, not a current statutory test or an empirical estimate for today's elections. Our fixed grid supplies different invented data and makes every voter visible. Neither exercise identifies social-group representation from party letters. To assess that separate question, an investigation would need relevant group data, legal criteria where applicable, and a defined representational outcome rather than treating party proportionality as its substitute.
Finish an audit with a sentence that names the comparison and its limits. For example, these twenty-five fixed voters elect two A representatives in row districts and four in column districts under the same local plurality rule. Then state what remains untested, such as resident population equality or whether either partition satisfies an actual jurisdiction's requirements. A recommendation needs priorities among distinct criteria such as population equality and partisan proportionality. The grid helps you reason about one conversion mechanism without deciding which combination of criteria a legislature should adopt.
A map effect does not settle every map claim
Question
Evidence in this lesson
Further evidence needed
Can grouping change seats?
Same voters, rows give A 2, columns give A 4
None for this bounded example
Is population representation equal?
Five voting cells per district
Relevant legal population and seats
Was partisan advantage intended?
A mechanical seat effect
Mapmaking process and feasible alternatives
Does a map represent social groups?
Only party labels A and B
Relevant group data and outcome criteria
Would reform change future results?
Fixed ballots only
Behavioral evidence and a causal comparison
A computed seat difference, legal compliance, intention, and a future causal effect are distinct propositions. Selected historical design passages support the mechanism, not a contemporary legal verdict.
Exercises
A minority of votes, a majority of seats
In the grid, A has twelve of twenty-five votes but wins four column districts. Calculate its vote and seat shares.
Hint
Do not change the vote total when the district boundaries change.
Worked answer
The same twelve A voters and thirteen B voters form different district majorities.
Measure
Rows
Columns
A vote share
48%
48%
A seat share
40%
80%
The fixed electorate makes the comparison interpretable.
A has 12/25 = 48% of votes and 4/5 = 80% of seats under column districts. Under row districts it still has 48% of votes but 40% of seats. The example shows a grouping effect, not a change in underlying preferences.
Equal turnout is not equal population
Two districts each cast 10,000 votes and elect one representative. Does that prove equal population per representative?
Hint
Turnout is a subset of a population, not the population itself.
Worked answer
Observed
Not established
10,000 votes in each district
Equal resident populations
One seat each
Equal population per representative without population data
The denominator must match the representational claim.
No. Eligible populations or turnout rates may differ, and legal apportionment may count residents rather than eligible voters. Equal ballots cast establish equal observed turnout counts only. To assess malapportionment, identify the relevant population denominator and compare it per seat.
Sources and further reading
The explanations, counts, and worked answers are complete without these optional links. Reference annotations identify the selected material reviewed. Historical country examples are not statements of current law, and the bounded models are not certified election-counting software. Mechanical results, behavioral evidence, and judgments about desirable representation remain separate.
Selected explanations of system families, district magnitude, thresholds, proportional formulas, mixed systems, and design objectives. Country examples and legal thresholds are historical, not statements of current law.
Overview of how magnitude and other design choices affect the translation of votes into seats. Not evidence that one institutional objective is universally preferable.