opinion: Me. Justice White
  delivered the opinion of the Court with respéct to the validity of the multi-member election district in Marion County, Indiana (Parts I-VI), together with an opinion (Part VII), in which The Chief Justice, Mr. Justice Black, and Me. Justice Black-mun joined, on the propriety of ordering redistricting of the entire State of Indiana, and announced the judgment of the Court.
  We have before us in this case the validity under the Equal Protection Clause of the statutes districting and apportioning the State of Indiana for its general assembly elections. The principal issue centers on those provisions constituting Marion County, which includes the city of Indianapolis, a multi-member district for electing state senators and representatives.
  I
  Indiana has a bicameral general assembly consisting of a house of representatives of 100 members and a senate of 50 members. Eight of the 31 senatorial districts and 25 of the 39 house districts are multi-member districts, that is, districts that are represented by two or more legislators elected at large by the voters of the district. Under the statutes here challenged, Marion County is a multi-member district electing eight senators and 15 members of the house.
  On January 9, 1969,. six residents of Indiana, five of whom were residents of Marion County, filed a suit described by them as “attacking the constitutionality of two statutes of the State of Indiana which provide for multi-member districting at large of General Assembly seats in Marion County, Indiana- . . . Plaintiffs Chavis, Ramsey, and Bryant alleged that the two statutes invidiously diluted the force and .effect of the vote of Negroes and poor persons living within certain Marion .County census tracts constituting what was termed “the ghetto area.” Residents of the area were alleged to have particular demographic characteristics rendering them cognizable as a minority interest group with distinctive interests in specific areas of the substantive law. With single-member districting, it was said, the ghetto area would elect three members of the house and one senator, whereas under the present districting voters in the area “have almost no. political force or control over législators because the effect of their vote is cancelled out by other contrary interest groups” in Marion County. The mechanism of political party organization and the influence of party chairmen in nominating candidates were additional factors alleged to frustrate the exercise of power by .residents of the ghetto area.
  Plaintiff Walker, a Negro resident of Lake County, also a multi-member district but a smaller one, alleged an invidious discrimination against Lake County Negroes because-Marion County Negroes, although no greater in number than Lake County Negroes, had the opportunity to influence the election of more legislators than Lake Gounty Negroes. The claim was that Marion County was one-third larger in population and thus had approximately one-third more assembly seats than Lake County, but that voter influence does not vary inversely with’ population and that permitting Marion County voters to elect 23 assemblymen at large gave them a disproportionate advantage over voters in Lake County. The. : two remaining plaintiffs presented claims not at issue here.
  A three-judge court convened and tried the case on June 17 and 18, 1969. Both documentary evidence and oral testimony were taken concerning the composition and characteristics of the alleged ghetto area, the manner in which legislative candidates were chosen and their residence and tenure, and the performance of Marion County’s delegation in the Indiana general assembly.
  The three-judge court filed its opinion containing its findings and conclusions on July 28, 1969, holding for plaintiffs. Chavis v. Whitcomb, 305 F. Supp. 1364 (SD Ind. 1969). See also 305 F. Supp. 1359 (1969) (pre-trial orders) and 307 F. Supp. 1362 (1969) (statewide reapportionment plan and implementing order). . In sum, it concluded that Marion County’s multi-member district must be disestablished and, because of population disparities not directly related to the phenomena alleged in the complaint, the entire State must be redistricted. More particularly, it first determined that a racial minority group inhabited an identifiable ghetto area in Indianapolis. That area, located in the northern half of Center Township and termed the “Center Township ghetto,” comprised 28'contiguous census tracts and parts of four others. The area contained a 1967 population of 97,000 nonwhites, over 99% of whom were Negro, and 35,000 whites. The court proceeded to compare six of these tracts, representative of the area, with tract 211, a predominantly white, relatively wealthy suburban census tract in Washington Township contiguous to the northwest corner of the court’s ghetto area and with tract 220, also in Washington Township, a contiguous tract inhabited by middle class Negroes. Strong differences were found in terms of housing conditions, income and educational levels, rates of unemployment, juvenile crime, and welfare assistance. The contrasting characteristics between the court’s ghetto area, and its inhabitants on the one hand and tracts 211 and 220 on the other indicated the ghetto’s “compelling interests in such legislative areas as urban renewal and rehabilitation, health care, employment training and opportunities, welfare, and relief of the poor, law enforcement, quality of education, and anti-discrimination measures.” 305 F. Supp., at 1380. These interests were in addition to those the ghetto shared with the rest of the county, such as metropolitan transportation, flood control, sewage disposal, and education
  The court then turned to evidence showing the residences of Marion County’s representatives and senators in each of the five general assemblies elected during the period 1960 through 1968. Excluding tract 220, the middle class Negro district, Washington Township, the relatively wealthy suburban area in which tract 211 was located, with an average of 13.98% of Marion County’s population, was the residence of 47.52% of its senators and - 34.33% of its representatives. The court’s Center Township ghetto area, with 17.-8% of the population, had 4.75% of the senators and 5.97% of the representatives. The nonghetto area of Center Township, with 23.32% of the population, had done little better. Also, tract 220 alone, the middle class Negro district, had only 0.66% of the county’s population but had been the residence of more representatives than had the ghetto area. The ghetto area had been represented in the senate only once — -in 1964 by one senator — and the. house three times — with one representative in 1962 and 1964 and by two representatives in the 1968 general assembly. The court found, the “Negro Center Township Ghetto population” to be sufficiently large to elect two representatives and one senator if the ghetto tracts “were specific single-member legislative districts” in Marion County. 305 F. Supp., at 1385. From these data the court found gross inequity of representation, as determined by residence of legislators, between Washington Township and tract 220 on the one hand and Center Township and the Center Township ghetto area on the other.
  The court also characterized Marion County’s general assembly delegation as tending to coalesce and take common positions on proposed legislation. This was “largely the result of election at large from a common constituency, and obviates representation of a substantial, though minority, interest group within that common constituency.” Ibid. Related findings were that, as a rule, a candidate could not be elected in Marion County unless his party carried the election; county political organizations had substantial influence on the selection and election of assembly candidates (an influence that would be diminished by single-member districting), as well as upon the actions of the county’s delegation in the assembly; and that at-large elections made it difficult for the conscientious voter to make a rational selection.
  The court’s conclusions of law on- the merits may be summarized as follows:
  1. There exists within Marion County an identifiable racial element, “the Negro residents of the Center Township Ghetto,” with special interests in various areas of substantive law, diverging significantly from interests of nonresidents of the ghetto.
  2. The voting strength of this racial group has been minimized by Marion County’s multi-member senate and house district because of the strong control exercised by political parties over the selection of candidates, the inability of the Negro voters to assure themselves the opportunity to vote for prospective legislators of their choice and- the absence of any particular legislators who were accountable for their legislative record to Negro voters.
  3. Party control of nominations, the inability of voters to know the candidate and the responsibility of legislators to their party and the county at large make it diffi-, cult for any legislator to diverge from the majority of his delegation and to be an effective representative of minority ghetto interests.
  4. Although each legislator in Marion County is arguably responsible to all the voters, including those in the ghetto, “[p]artial responsiveness of all legislators is [not] . . . equal [to] total responsiveness and the informed concern of a few specific legislators.”
  5. The apportionment statutes of Indiana ■ as they relate to Marion County operate to minimize and cancel out the voting strength óf a minority racial group, namely Negroes residing in the Center Township ghetto, and to deprive them of the equal protection- of the laws.
  6. As a legislative district, Marion County is large as compared with the total number of legislators, it is not. subdistricted to insure distribution of the legislators over the county and comprises a multi-member district for both the house and the senate. (See Burns v. Richardson, 384 U. S. 73, 88 (1966).)
  7. To redistrict Marion County alone would leave impermissible variations between Marion County districts, and other districts in the State. Statewide redistricting was required, and it could not await the 1970 census figures estimated to be available within a year.
  8. It may not be possible for the Indiana general assembly to comply with the state constitutional require^ ment prohibiting crossing or dividing counties for sena-, torial apportionment and still meet the requirements of the Equal Protection Clause adumbrated in recent cases.
  9. Plaintiff Walker’s claim as a Negro voter resident of Lake County that he was discriminated against because Lake County Negroes could vote for only 16 assemblymen while Marion County Negroes could vote for 23 was deemed untenable. In his second'capacity, as a. general voter in Lake County, Walker “probably has received less effective representation” than Marion County voters because “he votes for fewer legislators and, therefore, has fewer legislators to speak for him,” and, since in theory voting power in multi-member districts does not vary inversely to the number of voters, Marion County voters had greater opportunity to cast tie-breaking or “critical” votes. But the court declined to hold that the latter ground had been proved, absent , more evidence concerning Lake County. In this respect consideration of Walker’s claim was limited to that to be given the uniform districting principle in reapportioning the Indiana general assembly.
  Turning to the proper remedy, the court found redistricting of. Marion County essential. Also, although recognizing the complaint was directed only to Marion County, the court thought it must act on the evidence indicating that the entire State required reapportionment. Judgment was withheld in all respects, however, to give the State until October 1, 1969, to enact legislation remedying the improper districting and malappor-tionment found to exist by the court. In so doing the court thought the State “might wish to give consideration to certain principles of legislative apportionment brought out at the trial in these proceedings.” Id., at 1391. First, the court eschewed any indication that Negroes living in the ghetto were entitled to any certain number of legislators-^districts should be drawn with an eye that is color blind, and sophisticated gerrymandering would not be countenanced. Second, the legislature was advised to keep in mind the theoretical advantage inhering in voters in multi-member districts, that is, their theoretical opportunity to cast more deciding votes in any legislative election. Referring to the testimony that bloc-voting, multi-member delegations have disproportionately more power than single-member districts, the court thought that “the testimony has application here.” Also,< “as each member of the bloc delegation is responsible to the voter majority who elected the whole, each Marion County voter has a greater voice in the legislature, having more legislators to speak for him than does a comparable voter” in a single-member district. Single-member districts, the court thought, would equalize voting power among the districts as well as avoiding diluting political or racial groups located in multi-member districts. The court therefore recommended that the general assembly give consideration to the uniform district principle in making its apportionment.
  On October 15, the court judicially noticed that the Indiana general assembly had not been called to redistrict and reapportion the State. Following further hearings and examination of various plans submitted by the parties, the court drafted and adopted a plan based on the 1960 census figures. With respect to Marion County, the court followed plaintiffs’ suggested scheme, which was said to protect “the legally cognizable racial minority group against dilution of its voting strength.” 307 F. Supp. 1362, 1365 (SD Ind. 1969). Single-member districts were employed throughout the State, county lines were • .crossed where necessary, judicial notice was taken of the location of the nonwhite population in establishing district lines in metropolitan areas of the State and the court’s plan expressly aimed at giving “recognition to the cognizable racial minority group whose grievance lead [sic] to this litigation.” Id., at 1366.
  The court enjoined state officials from conducting any elections under the existing apportionment statutes and ordered that the 1970 elections be held in accordance with the plan prepared by the court. Jurisdiction was retained to pass upon any future claims of unconstitutionality .with respect to any future legislative apportionments adopted by the State.
  Appeal was taken following the final judgment by the three-judge court, we noted probable jurisdiction, 397 U. S. 984 (1970), and the State’s motion for stay of judgment was granted pending our final action on this case, 396 U. S. 1055 (1970), thus permitting.the 1970 elections to. be held under the existing apportionment statutes declared unconstitutional by the District Court. On June 1, 1971, wé were advised by the parties that the Indiana Legislature had passed, and the Governor had signed, new apportionment legislation soon to become effective for the 1972 elections and that the new legislation provides for single-member house and senate districts throughout" the State, including Marion County.’
  II
  With the 1970 elections long past and the appearance of new legislation abolishing multi-member districts in Indiana,’ the issue of mootness emerges, ’ Neither party deems the case mooted by recent events. Appellees, plaintiffs below, urge that if the appeal is dismissed as moot and the judgment of the District Court is vacated, as is our practice in such cases, there would be no outstanding judgment invalidating the Marion County multi-member district and that the new apportionment legislation would be in conflict with the state constitutional provision forbidding the division of Marion County for the purpose.of electing senators. If the new sena-’ torial districts were invalidated in the state courts in this respect, it is argued that the issue involved in the present litigation would simply reappear for decision. The attorney general for the State of Indiana, for the appellant, taking a somewhat different tack, urges that the issue of the Marion County multi-member district is not moot since the District Court has retained jurisdiction to pass on the legality of subsequent apportionment statutes for the purpose, among others, of determining whether the alleged discriminátion against a cognizable minority group hgs been remedied, an issue that would not arise if the District Court erred in invalidating multi-member districts in Indiana.
  We agree that the case is not moot and that the central issues before us must be decided. We do not, however, pass upon the details of the plan, adopted by the District Court, since that plan in any event would, have required revision in light of the 1970 census figures.
  HI
  The line of cases from Gray v. Sanders, 372 U. S. 368 (1963), and Reynolds v. Sims, 377 U. S. 533 (1964), to Kirkpatrick v. Preisler, 394 U. S. 526 (1969), and Wells v. Rockefeller, 394 U. S. 542 (1969), recognizes that “representative government is in essence self-government through the medium of elected representatives of the' people, and each and every citizen has ah inalienable right to full and effective participation in the political processes of his State’s legislative bodies.”- ' Reynolds v. Sims, 377 U. S., at 565. Since most citizens find it possible to participate only as qualified voters in electing their representatives, “[f]ull and effective participation- by all citizens in state.government requires, therefore, that each citizen have an equally effective voice in the election of members of his state legislature.” Ibid. Hence,- apportionment schemes “which give the same number of representatives to unequal numbers of constituents,” 377 U. S., at 563, unconstitutionally dilute the value of the votes in the larger districts. And hence the requirement that, “the seats in both Jiouses of a bicameral state legislature must be apportioned on a population basis.” 377 U. S., at 568.
  The question of the constitutional validity of multi-member districts has been pressed in this Court since the first of the modern reapportionment cases. These questions have focused not on population-based apportionment but on the quality of representation afforded by the multi-member district as compared with single-member districts. In Lucas v. Colorado General Assembly, 377 U. S. 713 (1964), decided with Reynolds v. Sims, we noted certain undesirable features of the multi-member district but expressly, withheld any intimation “that apportionment schemes which provide for the at-large election of a number of legislators from a county, or any political subdivision, are constitutionally defective.” 377 U. S., at 731 n. 21. Subsequently, when the validity of the multi-member district, as such, was squarely presented, we held that such a district is not per se illegal under the Equal Protection Clause. Fortson v. Dorsey, 379 U. S. 433 (1965); Burns v. Richardson, 384 U. S. 73 (1966); Kilgarlin v. Hill, 386 U. S. 120 (1967). .See also Bur-nette v. Davis, 382 U. S. 42 (1965); Harrison v. Schaefer, 383 U. S. 269 (1966). That voters in multi-member districts vote for and are represented by more legislators than voters in single-member districts has so far not demonstrated an invidious discrimination against the latter. But w,e have deemed the validity of multi-mem-ber district systems justiciable, recognizing also that they may be subject to challenge where the circumstances of a particular case may “operate to minimize or cancel out the voting strength of racial or political elements of the voting population.” Fortson, 379 U. S., at 439, and Burns, 384 U. S., at 88. Such a tendency, we have said, is enhanced when the district is large and elects a substantial proportion of the seats in éither hoüse of a bicameral legislature, if it is multi-member for both houses of the legislature or if it lacks provision for at-large candidates running from particular geographical sub-districts, as in Fortson. Burns, 384 U. S., at 88. But we have insisted that the challenger carry the Burden of proving that multi-member districts unconstitutionally operate to dilute or cancel the voting strength of racial or political elements. We have not yet sustained .such an attack.
  IV
  Plaintiffs level two quite distinct challenges .to the Marion County district. The first charge is that any multi-member district bestows on its voters several unconstitutional advantages over voters, in single-member districts or smaller multi-member districts. The other allegation is that the Maripn County district, on the record of this case, illegally minimizes and cancels out the voting power of a cognizable racial minority in Marion County. The District Court sustained the latter claim and considered the former sufficiently persuasive to be a substantial factor in prescribing uniform, single-member districts as the basic scheme of the court’s own plan. See 307 F. Supp., at 1366.
  In asserting discrimination against-, voters outside Marion County, plaintiffs recognize that Fortson, Burns, and Kilgarlin proceeded on the assumption that the dilution of voting power suffered by a voter who is placed in a district 10 times the population of another is cured by allocating 10 legislators to the larger district instead of the one assigned to the smaller district. Plaintiffs challenge this assumption at both the voter and legislator level. They demonstrate mathematically that in theory voting power does not vary inversely with the size of the district and that to increase legislative seats in proportion to increased population gives undue voting power, to the voter in the multi-member district since he has more chances to determine election outcomes than does the voter in the single-member district. This consequence obtains wholly aside from the quality or effectiveness of representation later furnished by the successful candidates. The District Court did -not quarrel with plaintiffs’ mathematics, nór do we. But like the District Court we note that the position remains a theoretical one and, as plaintiffs’ witness conceded, knowingly avoids and does “not take into account any political or other factors which might affect the actual voting power of the' residents, which might include party affiliation, race, previous voting characteristics or any other factors which go into the entire political voting situation.” The real-life impact of' multi-memb.er districts on individual voting power has not been sufficiently demonstrated, at least on this record, to warrant departure from prior cases.
  The District Court was more impressed with the other branch of the claim that multi-member districts inherently discriminate against other districts. This, was the assertion that whatever the individual voting power of Marion County voters in choosing legislators may be, they nevertheless have more effective representation in the Indiana general assembly for two reasons. First, each voter is represented by more legislators and therefore, in theory at least, has more chances to influence critical legislative votes. Second, since multi-member delegations are elected at large and' represent the voters of the entire district, they tend to vote as a bloc, which is tantamount to the district having one representative with several votes. The District Court did not squarely sustain this position, but it appears to have found it sufficiently persuasive to have suggested uniform district-ing to the Indiana Legislature and to have eliminated multi-metiiber districts in the court’s own plan redistricting the State. See 307 F. Supp., at 1368-1383.
  We are not ready, however, to agree that multi-member districts, wherever they exist, overrepresent their voters as compared with voters in single-member districts, even if the multi-member delegation tends to bloc voting. The theory that plural representation itself unduly enhances a district’s power and the influence of its voters remains to be demonstrated in practice and in the day-today operation of the legislature. Neither the findings of the trial court nor the record before us sustains it, even where bloc voting is posited.
  In fashioning relief, the three-judge court appeared to embrace the idea that each member of a bloc-voting delegation has more influence than legislators, from a single-member district. But its findings of fact fail to deal with the actual influence of Marion County’s delegation in the' Indiana Legislature. Nor did plaintiffs’ evidence make such a showing. That bloc voting tended to occur is sustained by the record, and defendants’ own witness thought- it was advantageous for Marion County’s delegation to stick together. But nothing demonstrates that senators and representatives from Marion County counted for more in the legislature than members from single-member "districts or from smaller multi-member districts. Nor is there any thing, in the court’s findings indicating that what might be true of Marion County is also true of other multi-member districts in Indiana or is true of multi-member districts generally. Moreover, Marion County would have no less advantage, if advántage there is, if it elected from individual districts and the elected representatives demonstrated the same bloc-voting tendency, which may also develop among legislators representing single-member districts widely , scattered throughout the State. Of course it is advantageous to start with more than one vote for a bill. But nothing before us shows dr suggests that any legislative skirmish affecting the State of Indiana or Marion County ip particular would have come out differently had Marion County been subdistricted and its delegation elected from single-member districts.
  Rather than squarely finding unacceptable discrimination against out-state voters in favor of Marion County voters, the trial court struck "down Marion County’s multi-member district because it found the scheme worked invidiously against a specific segment of the county’s voters as compared with others. The court identified an area of the city as a ghetto, found it predominantly inhabited, by poor Negroes with distinctive substantive-láw interests and thought this group unconstitutionally underrepresented because the proportion of legislators with residences in the ghetto elected from 1960 to 1968 was less than the ghetto’s proportion of the population, léss than the proportion of legislators elected from Washington Township, a less populous district, and less than the ghetto would likely have elected had the county consisted of single-member districts. We find major deficiencies in this approach.
  First, it needs no emphasis here that the Civil War Amendments were designed to protect the civil rights of Negroes and that the courts have been vigilant in scrutinizing schemes allegedly conceived or operated as purposeful devices to further racial discrimination. There has been no hesitation in striking down those contrivances that can fairly be said to infringe on Fourteenth Amendment rights. Sims v. Baggett, 247 F. Supp. 96 (MD Ala. 1965); Smith v. Paris, 257 F. Supp. 901 (MD Ala. 1966), aff’d, 386 F. 2d 979 (CA5 1967); and see Gomillion v. Lightfoot, 364 U. S. 339 (1960). See also Allen v. State Board of Elections, 393 U. S. 544 (1969). But there is no suggestion here that Marion County’s multi-member district, or similar districts throughout the State, were conceived or operated as purposeful devices to further racial or economic discrimination. As plaintiffs concede, “there was no basis for asserting that the legislative districts in' Indiana were designed to dilute the vote of minorities.” Brief of Appellees (Plaintiffs) 28-29. Accordingly, the circumstances here lie outside the reach of decisions such as Sims v. Baggett, supra.
  Nor does the fact that the number of ghetto residents who were legislators was not in proportion to ghetto, population satisfactorily prove invidious discrimination absent evidence and findings that ghetto residents had less opportunity than did other Marion County residents to participate in the political processes and to elect legislators of their choice.- We have discovered nothing in the record or in the court’s findings indicating that poor Negroes were not allowed to register or vote, to choose the political party they desired to support, to participate in its affairs or to be equally represented on those occasions when legislative candidates were chosen. Nor did the evidence purport to show or the court find that inhabitants of the ghetto were regularly excluded from the slatés of both major parties, thus denying them the chance of occupying legislative seats. It appears reasonably clear that the Republican Party won four of the five elections from 1960 to 1968, that Center Township ghetto voted heavily Democratic and . that ghetto votes were critical to Democratic Party success. Although we cannot-be sure of the facts since the court ignored the question, it seems unlikely that the Democratic Party could afford to overlook.the ghetto in slating its candidates. Clearly, in 1964 — the one election that the Democrats won— the party slated and elected one senator and one representative from Center Township ghetto as well as one senator and four representatives from other parts of Center Township and two representatives from census tract 220, which was- within the ghetto area described by plaintiff. Nor is there any indication that, the party failed to slate candidates satisfactory to the ghetto in. other years. Absent evidence or findings we are not sure, but it seems reasonable to infer that had the Democrats won all of the elections or even most of them, the ghetto would have had no justifiable complaints about representation. The fact is, however, that four of the five elections were won by Republicans, which was not the party of the ghetto and which would not always slate ghetto candidates — although in 1962 it nominated and elected one representative and in 1968 two representatives from that area. If this is the proper view of this case, the failure of the ghetto to have legislative seats in proportion to its population emerges more as a function of losing elections than of built-in bias against poor Negroes. The voting power of ghetto residents may have been "cancelled out” as the District Court held, but this seems a mere euphemism for political defeat at the polls.
  On the record before us plaintiffs’ position comes to' this: that although they have equal opportunity to participate in and influence the selection of candidates and legislators, and although the ghetto votes predominantly Democratic and that party slates candidates satisfactory to the ghetto, invidious discrimination nevertheless results when the ghetto, along with all other Democrats, suffers the disaster of losing too many elections. But typical American legislative elections are district-oriented, head-on races between candidates of two or more parties. As our system has it,- one candidate wins, the others lose. Arguably the losing candidates’ supporters are without representation since the men they voted for have been defeated; arguably they have been denied equal protection of the laws since they have no legislative voice of their own. This is true of both single-member and multi-member districts. But we have not yet deemed it a denial of equal protection to deny legislative seats to losing candidates, even in those so-called “safe” districts where the same party wins year after year.
  Plainly, the District Court saw nothing unlawful about the impact of typical single-member district elections. The court’s own plan created districts giving both Rer publicans and Democrats several predictably safe general assembly seats, with political, racial or economic minorities in those districts being “unrepresented” year after year. But similar consequences flowing from Marion County multi-member district elections were viewed differéntly. Conceding that all Marion County voters could fairly be said to be represented by the entire delegation, just as is each voter in a single-member district by the winning candidate, the District Court thought the ghetto voters’ claim to the partial allegiance of eight senators and 15 representatives was not equivalent to the undivided allegiance of one senator and two representative's; nor was the ghetto voters’ chance of influencing the election of an entire slate as significant as the guarantee of one ghetto, senator and' two ghetto representatives. As the trial court saw it, ghetto voters could ñot be ade--quately and equally represented unless some of Marion County’s general assembly seats were reserved for ghetto residents serving the interests of the ghetto majority. But are poor Negroes of the ghetto any more underrepresented than poor ghetto whites who also voted Democratic and lost, or any more discriminated against than other interest groups or voters in Marion County with allegiance to the Democratic Party, or, conversely, any less represented than Republican areas or voters in years of Republican defeat? We think not. The mere fact that one interest group or another concerned with the outcome of Marion County elections has found itself outvoted and without legislative seats of its own provides no basis for invoking constitutional remedies where, as here, there is no indication that this segment of the population is being denied access to the political system.
  There is another gap in the trial court’s reasoning. As noted by the court, the interest of ghetto residents in certain issues did not measurably differ from that of other voters. Presumably in these respects Marion County’s assemblymen were satisfactorily representative of the ghetto. As to other matters, ghetto residents had unique interests not necessarily shared by others in the community and on these issues the ghetto residents were invidiously underrepresented absent their own legislative voice to further their own policy views.
  Part of the difficulty with this conclusion is that the findings failed to support it. Plaintiffs’ evidence purported to show disregard for the ghetto’s distinctive interests.; r defendants claimed quite the contrary. We see nothing in the findings of the District Court indicating recurring poor performance by Marion County’s delegation with respect to Center Township ghetto, nothing to show what the ghetto’s interests were in particular legislative situations and nothing to indicate that the outcome would have been any different if the 23 assemblymen had been chosen from single-member districts.' Moreover, even assuming bloc voting by the delegation contrary to the wishes of the ghetto majority, it would not follow that the Fourteenth Amendment had been violated unless it is invidiously discriminatory for a county to elect its delegation by majority vote based on party or candidate platforms and so to some extent predetermine legislative votes on particular issues. Such tendencies are inherent in government by elected representatives; and surely elections in single-member districts visit precisely the same consequences on the supporters of losing candidates whose views are rejected at the polls.
  . Y
  The District Court’s holding, although on the facts of this case limited to guaranteeing one racial group representation, is not. easily contained. It is expressive of the more general proposition that any group with distinctive interests must be represented in legislative halls if it is numerous enough to command at least, one seat and represents a majority living in an area sufficiently compact to constitute a single-member district. This approach would make it difficult to reject claims of Democrats, Republicans, or members of any political organization in Marion County who live in what would be safe districts in a single-member district system but who in one year or another, or year after year, are submerged in a one-sided multi-member district vote. There are also union oriented workers, the university community, religious or ethnic groups occupying identifiable areas of our heterogeneous cities and urban areas. Indeed, it would be difficult for a great many, if not most; multi-member districts to survive analysis under the District Court’s view unless combined with some voting arrangement such as proportional representation or cumulative voting aimed at providing representation for minority parties or interests. At the very least, affirmance of the District Court would spawn endless litigation concerning the multi-member district systems now widely employed in this country.
  We are not insensitive to the objections long voiced to multi-member district plans. Although not as prevalent as they were in our early history, they have been with us since colonial times and were much in evidence both before and after the adoption of the Fourteenth Amendment. Criticism is rooted in their winner-take-all aspects, their tendency to submerge minorities and to overrepresent the winning party as compared with the party’s statewide electoral position, a general preference for legislatures reflecting community interests as closely as possible and disenchantment with political parties and elections as devices to settle policy differences between contending interests. The chance of winning or significantly influencing intraparty fights and issue-oriented elections has seemed to some inadequate protection to minorities, political, racial, or economic; rather, their voice, it is said, should also be heard in the legislative forum where public policy is finally fashioned. In our view, however, experience and insight have not yet demonstrated that multi-member districts are inherently invidious and violative of the Fourteenth Amendment. Surely the findings of the District Court do not demonstrate it. Moreover, if the problems of multi-member districts are unbearable or even unconstitutional it is not at all clear that the remedy is a single-member district system with its lines carefully drawn to ensure representation to sizable racial, ethnic, economic, or religious groups and with its. own capacity for overrepresenting and underrepresenting parties and interests and even for permitting a minority of the voters to control the legislature and government of a State. The short of it is that we are unprepared to hold that district-based elections decided by plurality vote are unconstitutional in either single- or multi-member districts simply because the supporters of losing candidates have no legislative seats assigned to them. As presently advised we hold that the District Court misconceived the Equal Pro tec-, tion Clause in applying it to invalidate the Marion County multi-member district.
  VI
  Even if the District Court was correct in finding, unconstitutional discrimination against poor inhabitants of the ghetto, it did not explain why it was constitutionally compelled to disestablish the entire county district and to intrude upon state policy any more than necessary to ensure representation of ghetto interests. The court entered judgment without expressly putting aside on supportable grounds the alternative of creating single-member districts in the ghetto and leaving the district otherwise intact, as well as the possibility that the Fourteenth Amendment could be satisfied by a simple requirement that some of the at-large candidates each year must reside in the ghetto.. Cf. Fortson v. Dorsey, supra.
  We are likewise at a loss to understand how on; the court’s own findings of fact and conclusion^ of law it was justified in eliminating every multi-member district in the State óf Indiana.. It did not forthrightly sustain the theory that multi-member districts always overr represent their voters to the invidious detriment of single-member residents. Nor did it examine any multi-mem-ber district aside> from Marion County for possible intradistrict discrimination.
  The remedial powers of an equity court must be adequate to the task, but they are not unlimited. Here the District Court erred in so broadly brushing aside state apportionment policy without solid constitutional or equitable grounds for doing so.
  VII
  At the same time, however, we reject defendant’s suggestion that the court was wrong- in ordering statewide reapportionment. After determining that Marion County required reapportionment, the court concluded that “it becomes clear beyond question that the evidence adduced in this case and the additional apportionment requirements set forth by the Supreme Court call for a redistricting of the entire state as to both houses of the General Assembly.” 305 F. Supp., at 1391. This evidence, based on 1960 census figures, showed that Senate district 20, with one senator for 80,496, was overrepresented by 13.68% while district 5, with one senator for 106,790, was underrepresented by 14:52%, for a total variance of 28.20% and a ratio between the largest and smallest districts of 1.327 to 1. The house figures were similar. The variation ranged from one representative for 41,449 in district 39 to one for 53,003 in district 35, for a variance of 24.78% and a ratio of 1.279 to l. These variations were in excess of, or very nearly equal to, the variation of 25.65% and the ratio of 1.30 to 1 which we held excessive for state legislatures in Swann v. Adams, 385 U. S. 440 (1967). Even with this convincing showing of malapportionment, the court refrained from action in .order to allow the Indiana Legislature to call a special session for the purpose of redistricting. ' When the legislature ignored the court’s findings and suggestion, it was not improper for the court to order, statewide redistricting, as district courts have done from the time Reynolds v. Sims, 377 U. S. 533 (1964), and its companion cases were decided. And see Maryland Committee for Fair Representation v. Tawes, 377 U. S. 656, 673 (1964).
  Nor can we accept defendant’s argument |hat the statutory plan was beyond attack because the District Court had held in 1965 that at that time the plan met the “substantial equality” test of Reynolds. Stout v. Bot- torff, 249 F. Supp. 488 (SD Ind. 1965).. Defendant does not argue that the 1969 variances were acceptable under the Reynolds test, which has been considerably refined since that decision, see Swann v. Adams, supra. Rather, he contends that because Reynolds indicated that decennial reapportionment would be a “rational approach” to the problem, a State cannot be compelled to reapportion itself more than once in a 10-year period. Such a reading misconstrues the. thrust of Reynolds in this respect. Decennial reapportionment was suggested as a presumptively rational method to avoid “daily, monthly, annual or biennial reapportionment” as population shifted throughout the State. Here, the District Court did not order reapportionment as a result of population shifts since the 1965 Stout decision, but only because the disparities among districts which were thought to be permissible at the time of that decision had been shown by intervening decisions of this Court to be excessive.
  We therefore reverse the judgment of the District Court and remand the case to that court for further proceedings consistent with this opinion.
  It is so ordered.
  [For Appendix to opinion of the Court, see post, p. 164.]
  Mr. Justice Stewart joins in Part I through VI of the Court’s opinion, holding that the multi-member district-ing scheme here in issue did not violate the Equal Protection Clause of the Fourteenth Amendment. He dissents from Part VII of the opinion for the reasons expressed in his dissenting opinion in Lucas v. Colorado General Assembly, 377 U. S. 713, 744.
  Separate opinion of
  Me. Justice Hablan.
  Earlier this Term I remarked on “the evident malaise among the members of the Court” with prior, decisions in the field of voter qualifications and reapportionment. Oregon v. Mitchell, 400 U. S. 112, 218 (1970) (separate opinion of this writer).
  Today’s opinions in this and two other voting cases now decided confirm that diagnosis.
  I
  Past decisions have held that districting in local governmental units must approach equality of voter population “as far as is practicable,” Hadley v. Junior College District, 397 U. S. 50, 56 (1970), and that the “as nearly as is practicable” standard of Wesberry v. Sanders, 376 U. S. 1, 7-8 (1964), for congressional districting forbade a maximum variation of 6%'. Kirkpatrick v. Preisler, 394 U. S. 526 (1969). Today the Court sustains a local governmental apportionment scheme with a 12% variation. Abate v. Mundt, post, p. 182.
  Other past decisions have suggested that multi-member constituencies would be unconstitutional-if they could be shown “under the circumstances of a particular case . . . to minimize or cancel out the voting strength of racial or political elements of the voting population.” Fortson v. Dorsey, 379 U. S. 433, 439 (1965); Burns v. Richardson, 384 U. S. 73, 88 (1966). Today the Court holds that a three-judge District Court, which struck down an apportionment scheme for just this reason, “misconceived the Equal Protection Clause.” Ante, at 160.
  Prior opinions stated that “once the class of voters is chosen and their qualifications specified, we see no constitutional way by which equality of voting power may be evaded.” Gray v. Sanders, 372 U. S. 368, 381 (1963); Hadley v. Junior College District, 397 U. S. 50, 59 (1970). Today the Court sustains a" provision that gives opponents of school bond issues half again the voting power of proponents. Gordon v. Lance, ante, p. 1.
  II
  The Court justifies the wondrous results in these cases by relying on different combinations of factors. Abate v. Mundt relies on the need for flexibility in local governmental arrangements, the interest in preserving the integrity of political subdivisions, and the longstanding tradition behind New York's practice in the latter respect. This case finds elementary probability theory too simplistic as a guide to resolution of what is essentially-a practical question of political power; the opinion relies on the long history of multi-member districts in this country and the fear that “affirmance of the District Court would spawn endless .litigation.” Ante, at 157. Gordon v. Lance relies heavily on the “federal analogy” and the prevalence of similar, anti-majoritarian elements in the constitutions of, the several States;
  To my mind the relevance of such considerations as the foregoing is undeniable and their cumulative' effect is unanswerable. I can only marvel, therefore, that they were dismissed, singly and in combination, in a line of cases which began with Gray v. Sanders, 372 U. S. 368 (1963), and ended with Hadley v. Junior College District, 397 U. S. 50 (1970).
  That line of cases can best be understood, I think, as reflections of deep personal commitments by some members of the Court to the principles of pure majoritarian democracy. This majoritarian strain and its nonconsti-tutional soürces aré most clearly revealed in Gray v. Sanders, supra, at 381, where my Brother Douglas, speaking for the Court, said: “The conception of political equality from the Declaration of Independence, to Lincoln’s Gettysburg Address, to the Fifteenth, Seventeenth, and Nineteenth Amendments can mean only one thing — one person, one vote.” If this philosophy of majoritarianism had been given jits head, it would have led to different results in each of the cases decided today, for it is in the very nature of the principle that it regards majority rule as an imperative of social organization, not subject to compromise in furtherance of merely political ends. It is a philosophy which ignores or overcomes the fact that the scheme of the Constitution' is one not of majoritarian democracy, but of federal republics, with equality of representation a value subordinate to many others, as both the body of the Constitution, and the Fourteenth Amendment itself show on their face. See generally Baker v. Carr, 369 U. S. 186, 297-324 (1962) (Frankfurter, J., dissenting).
  Ill
  If majoritarianism is to be rejected as a rule of decision, as the Court implicitly rejects it today, then an alternative principle must be supplied if this earlier line ■ of cases just referred to is still to be regarded as good law. The reapportionment opinions of this Court provide little help. They speak in conclusory terms of “debasement” or “dilution” of the “voting power” or “representation” of citizens without explanation of what these concepts are. The answers are hardly apparent, for as the Court observes today:
  “As our. system has it, one candidate wins, the others lose. Arguably the losing candidates’ supporters are without representation since the men they voted for have been defeated; arguably they have been denied equal protection of the laws since they have no legislative voice of their own. . . . But we have-not yet deemed it a denial of- equal protection to deny legislative seats to losing candidates,even in those so-called 'safe’ districts where the same party wins year after year.” Ante, at 153.
  A coherent and realistic notion of what is meant by “voting power” might have restrained some of the extreme lengths to which this Court has gone in pursuit of the will-o’-the-wisp of “one man, one vote.”
  An interesting illustration of the light. which a not, implausible definition of “voting power” can shed on reapportionment doctrine is provided by the theoretical model created by Professor Banzhaf, to which the Court refers, ante, at 144-146. This model uses as a measure of voting power the probability that a given voter will cast a tie-breaking ballot in an election. Two further assumptions are made: first, that the voting habits of. all members of the electorate are alike; and second, that each voter is equally likely to vote for' either candidate beforé him. On these assumptions, and taking the voting population in Marion County as roughly 300,000, it can be shown that the probability of an individual voter’s casting a decisive vote in a given election is approximately .00146. This provides a standard to which “voting-power” of residents in other districts may be compared. See generally Banzhaf, Multi-Mem-ber Electoral restricts — Do They Violate the “One Man, One Vote”, Principle, 75 Yale L. J. 1309 (1966).
  However, Professor Banzhaf’s model also reveals that minor variations in assumptions can lead to major variations in-results. For instance, if the temper of the electorate changes by one-half of one percent, each individual’s voting power is reduced by a factor of approximately 1,000,000. Or if . a few of the 300,000 voters are committed — say 15,000 to candidate A and 10,000 to candidate B — the probability of any individual’s casting a tie-breaking vote is reduced by a factor on' the rough order of 120,000,000,000,000,000,000. Obviously in comparison with the astronomical differences in voting power which can result from such minor variations in political characteristics, the effects of the 12% and 28% population variations considered in Abate v. Mundt and in this case are de minimis, and even' the extreme deviations from the norm, presented in Baker v. Carr, 369 U. S. 186 (1962), and Avery v. Midland County, 390 U. S. 474 (1968), pale into insignificance.
  It is not surprising therefore that the Court, in this case declines to embrace the measure of voting power suggested by Professor Banzhaf; .But it neither suggests an alternative nor considers the consequences of its inability to measure what it purports to be equalizing. See n. 2, supra. Instead it becomes enmeshed in the haze of slogans and numerology which for. 10 years has obscured its vision in this field, and finally remands the case “for further proceedings consistent with [its] opinion.” Ante, at 163. This inexplicit mandate is at least subject to the interpretation that the court below is to inquire into such matters as “the actual influence of Marion County’s delegation in the Indiana Legislature,” ante, at 147, and the possibility of “recurring poor performance by Marion County’s delegation with respect to Center Township ghetto,” ante, at 155, with a view to determining whether “any legislative skirmish affecting the State of Indiana or Marion County in particular would have come out differently had Marion County been subdistricted and its delegation elected from single-member districts.” Ante, at 148. If there are less appropriate subjects for federal judicial inquiry, they do not come readily to mind. The suggestion implicit in the Court’s opinion that appellees may ultimately prevail if they can make their record in these and other like respects should be recognized for what it is!: a manifestation of frustration by a Court that has become trapped in the “political thicket” and is looking for the way out.
  This case is nothing short of a complete vindication of Mr. Justice Frankfurter’s warning nine years ago “of the mathematical quagmire (apart from divers judicially inappropriate and elusive determinants) into which this Court today catapults- the lower courts of the country.” Baker v. Carr, 369 U. S. 186, 268 (1962) (dissenting opinion). With all respect, it also bears witness to the morass into, which the Court has gotten itself by departing from sound constitutional'principle in the electoral field. See the dissenting opinion of Mr. Justice Frankfurter in Baker v. Carr, supra, and my separate opinions in Reynolds v. Sims, 377 U. S. 533, 589 (1964), and in Oregon v. Mitchell, 400 U. S. 112, 152 (1970). I hope the day will come when the Court will frankly recognize the. error of its ways in ever having undertaken to restructure state electoral processes.
  I would reverse the judgment below and remand the case to the District Court with directions to dismiss the complaint.
  As later indicated, shortly before announcement of this opinion, the Court was informed that the statutes at issue here will soon be superseded by new apportionment legislation recently adopted by the Indiana Legislature and' signed by the Governor. That legislation provides for single-member districts throughout the State including Marion County.- For the reasons stated below the controversy is not moot, and, as will be evident, this opinion proceeds ' as though the state statutes before us remain undisturbed by new legislation.
  The provisions attacked, contained in Acts 1965 (2d Spec. Sess.), c. 5, § 3, and c. 4, § 3, and appearing in Ind. Ann. Stat. §§ 34 — 102 and 34-104 (1969) were as follows:
  “34-102. Apportionment of representatives. — Representatives shall be elected from districts comprised of one [1] or more counties and having one [1] or more representatives, as follows: . . , Twenty-sixth District Marion County: fifteen [15] representatives . . . .”
  “34-104. Apportionment of senators. — Senators shall be elected from districts, comprised of one or more counties and having one or inore senators; as follows: . . . Nineteenth District — Marion County: eight [8] senators, two [2] to be elected in 1966.”
  The District Court denied plaintiffs’ motion to have the suit declared a class action under Fed. Rule Civ. Proc. 23 (b). 305 F. Supp. 1359, 1363 (SD Ind. 1969). See n. 17, infra.
  Plaintiffs in the trial court are appellees here and defendant Whitcomb is the appellant. We shall refer to the parties in this opinion as they-stood in the trial court.
  Walker also alleged that “in both Lake and Marion County, Indiana there are. a sufficient number of negro [sic] voters and inhabitants for a bloc vote by 'the said inhabitants to change the result of any election recently held.”
  The mathematical basis- for the assertion was set out in detail in the complaint. See also n. 23, infra. It was also alleged that “[b]oth Marion County . . . and Lake County . -. . are the sole matter for consideration before two separate state legislative committees, one directed .to the affairs of each county. The laws enacted . . . which directly effect [sic] Marion or Lake County typically apply to only <Jne county or the other.” App. 15.
  Plaintiff Marilyn Hotz, a Republican and a resident of what she described as the white suburban belt of Marion County lying outside the city of Indianapolis, alleged that malapportionment of precincts in party organization together with multi-member district-ing invidiously diluted her vote.
  Plaintiff Rowland Allan (spelled “Allen” in the District Court’s opinion), an independent voter, alleged that multi-member districting deprived him of any chance to make meaningful judgments on the merits of individual candidates because he was confronted with a list of 23 candidates of each party.
  In their final arguments and proposed findings of fact and conclusions Of law plaintiffs urged that the Center Township ghetto was largely inhabited by Negroes who had distinctive interests and whose bloc voting potential was canceled out by opposing interest groups in the at-large elections held in Marion County’s multi-member district, that the few Negro legislators, including the three then serving the general assembly from Marion County, were chosen by white voters and were unrepresentative of ghetto Negroes, and that Negroes should be given the power and opportunity to choose their own assemblymen. It was also urged that the power of political as well as racial elements was canceled out in that in every assembly election since 1922, one party or the other had won all the seats with two minor exceptions; hence many voters, in numbers large enough and geographically so located as to command control over one or more general assembly seats if Marion County were subdistricted, were wholly without representation whichever - way an assembly election turned out.
  The defendants argued that Marion County’s problems were countywide and that its delegation could better represent the various interests in the county if elected at large and responsible to the county as a whole rather than being elected in single-member districts and thus fragmented by parochial interests and jealousies. They also urged that the 1960 census figures were an unreliable basis for redistricting Marion County and opposed the court’s suggestion that the apportionment of the whole State was an issue properly before the court on the pleadings and the evidence.
  A ghetto was defined as a residential area with a higher density of population and greater proportion of. substandard housing than in the overall metropolitan area and inhabited primarily by racial or other minority groups with lower than average socioeconomic status and whose residence in the area is often the result of a social, legal, or economic restriction or custom. 305 F. Supp., at 1373.
  The court's ghetto area was not congruent with that alleged in the complaint. It included five census tracts and parts of four others not within the ghetto area alleged in the complaint, but it omitted census tract 220 which the complaint had included. 305 F. Supp., at 1379-1381. That district, which was contiguous to both tract 211 and the ghetto area, was inhabited primarily by Negroes but was found to be a middle class district differing substantially in critical elements from the remainder of the ghetto. The court also made it unmistakably clear that its ghetto 'area “does not represent the entire ghettoized portion of Center Township but only the portion which is predominantly inhabited by Negroes and which was alleged in the complaint.” 305 F. Supp., at 1380-1381. Although census tract 563, a tract “randomly selected to typify tracts . . . within the predominantly white ghetto portion of Center Township,” id., at 1374, was shown to have characteristics very similar to the tracts in the court’s ghetto area except for the race of its inhabitants, the size and configuration of the white ghetto area were not revealed by the findings.
  See Appendix to opinion, post, p. 164.
  A striking but typical example of the importance of party affiliation and the “winner take all” effect is shown by the 1964 House of Representatives election. .
  Democrats Votes
  Neff .'. 151,822
  Bridwell. 151,756
  Murphy. 151,746
  Dean . 151,702
  Creedon .151,573
  Jones . 151,481
  DeWitt. 151,449
  Logan. 151,360
  Roland. 151,343
  Walton. 151,282
  Huber. 151,268
  Costeño . 151,153
  Fruits. 151,079
  Lloyd . 150,862
  Ricketts. 150,797
  Republicans Votes
  Cox .;. 144,336
  Hadley. 144,235
  Baker. 144,032
  Burke. 143,989
  Borst . 143,972
  Madinger. 143,918
  Clark .'. 143,853
  Bosma . 143,810
  Brown . 143,744
  Durnil . 143,588
  Gallagher. 143,553
  Cope.:. 143,475
  Elder . 143,436
  Zerfas. 143,413
  Allen . 143,369
  Though nearly 300,000 Marion County voters cast nearly 4% million votes for the House, the high and low candidates within each party varied by only about a thousand votes. ' And, as these figures show, the Republicans lost every seat though they received 48.69% of the vote. Plaintiffs’ Exhibit 10.
  “The first requirement implicit in Fortson v. Dorsey and Burns v. Richardson, that of an identifiable racial or political element within the multi-member district, is met by the Negro residents of the Cénter Township Ghetto. These Negro residents have .interests in areas of substantive law such as housing regulations, sanitation, welfare programs (aid to families with dependent children, medical care, etc.),, garnishment statutes, and unemployment compensation, among others, which diverge significantly from the interests of nonresidents of the Ghetto.” 305 F. Supp., at 1386.
  Ibid. The District Court implicitly, if not expressly, rejected the testimony of defendants’ witnesses, including a professor of political science, to the effect that Marion County’s problems and all its voters would be better served by a delegation sitting and voting as a team and responsible to the district at large, than by a delegation elected from single-member districts and split into groups representing special interests.
  Article 4, § 6, of the Indiana Constitution provides:
  “A Senatorial or Representative district, where more than one county shall constitute a district, shall be composed of contiguous counties; and no county, for Senatorial apportionment, shall ever be divided.” (Emphasis added.)
  See part VII, infra. ,
  “In his second status, we find that plaintiff Walker is a voter of Indiana who resides outside Marion County. Applying the uniform district principle, discussed infra in the remedy section, we find that he probably has received less effective representation than Marion County voters. It has been shown that he votes for fewer legislators and, therefore, has fewer legislators to speak for him. He also, theoretically, casts fewer critical votes than Marion County voters, but we decline to so hold in the absence of sufficient evidence as to other factors such as bloc and party voting in Lake County. We hold that, in the absence of stronger evidence of dilution, his remedy is limited to the consideration which should be given to the uniform district principle in any subsequent reapportionment of the Indiana General Assembly.” 305 F. Supp., at 1390.
  The court found a failure of proof on behalf of plaintiff Hotz, a resident of the white suburban belt, and on behalf of plaintiff Allan, an independent voter. • Two other plaintiffs • were entitled to no relief, plaintiff Chavis because he resided outside the Center Township ghetto and. plaintiff Ramsey because he failed to show that he was a resident of that area. Only plaintiff Bryant, in addition to the qualified recognition given Walker, was found to have standing to sue and to be entitled to the relief prayed for.
  See part VII, infra.
  The Governor appealed here following this opinion. Since at that time no judgment had been entered and no injunction had been granted or denied, we do not have jurisdiction of that appeal and it is therefore dismissed. Gunn v. University Committee, 399 U. S. 383 (1970).
  The trial court’s discussion of this subject may be fqund in 305 F. Supp., at 1391-1392. ■
  The court also provided for the possibility that the legislature would fail to redistrict in time for the 1972 elections:
  “The Indiana constitutional provision for staggering the terms of senators, so that one-half of the Senate terms expire every two years, is entirely proper and valid and would be mandatory in a legislatively devised redistricting plan.
  “However, the plan adopted herein is provisional in nature and probably will be applicable for only the 1970 election and the subsequent 2-year period. This is true since the 1970 census will have been completed in the interim, and the legislature can very well redistrict itself prior to the 1972 election. On the other hand, it is conceivable that the legislature may fail to redistrict before the 1972 elections. In such event, all fifty senatorial seáts shall be up for election- every two years until such time as the legislature properly redistricts itself. It will then properly be the province of the legislature in redistrieting to determine which senatorial districts shall elect senators to 4-year terms.and which shall elect senators to 2-year terms to reinstate the staggering of terms.” 307 F. Supp., at 1367.
  In Fortson, the' Court reversed a three-judge District Court which found a violation of the Equal Protection Clause in that voters in single-member districts were allowed to “select their own senator” but that voters in multi-member districts were not. The statutory scheme in Fortson provided for subdistricting within the county, so that each subdistrict was the residence of exactly one senator. However, each. senator was elected by the county at large. The Court said, “Each [sub] district’s senator must be a resident of that [sub] district, but since his tenure depends upon the county-wide electorate he must be vigilant to serve the interests of all the people in the county, and not meiely those of people in his home [sub]district; thus in fact he is the county’s and not merely the [sub] district’s senator.” 379 U. S., at 438. The question of whether the scheme “operate[d] te minimize or cancel out the voting ■strength of racial or political elements of the voting population!’ was hot presented.
  In Burnette, we summarily affirmed a three-judge District Court ruling, Mann v. Davis, 245 F. Supp. 241 (ED Va. 1965),' which upheld a multi-member district consisting of the city of Richmond; Va., and suburban Henrico-County over the objections of both urban Negroes and suburban whites. Since the urban Negroes did not appeal here, the affirmance is of no weight as to them, but as to the suburbanites it represents an adherence to Fortson. Similarly, Harrison summarily affirmed a District Court reapportionment plan, Schaefer v. Thomson, 251 F. Supp. 450 (Wyo. 1965), where multi-member districts in Wyoming were held necessary to keep county splitting at a minimum.
  Burns vacated a three-judge court decree which required single-member districts except in extraordinary circumstances. The Court in Burns noted that “the demonstration that a particular multi-member scheme effects an invidious result must appear from evidence in the record.” 384 U. S., at 88.
  In Kilgarlin, the Court affirmed, per curiam, a district court ruling
  “insofar as it held that appellants had not proved their allegations that [the Texas House of Representatives reapportionment plan] was a racial or political gerrymander violating the Fourteenth Amendment, that it unconstitutionally deprived Negroes of their franchise and that because of its utilization of single-member, multi-member- and floterial districts it was an unconstitutional ‘crazy quilt.’ ” 386 U. S., at 121.
  The mathematical backbone of this theory is as follows: In a . population of n voters, where each voter has a choice between .two alternatives (candidates), there are 2n possible voting combinations. For example, with a population of three voters, A, B, and C, and two candidates, X and Y, there áre eight combinations:
  ABC
  #1. X X X
  #2. X X Y
  #3. X Y X
  #4. X Y Y
  #5. Y X X
  #6. Y -X Y
  #7. Y Y X
  #8. Y Y Y
  The theory hypothesizes that the true test of voting power is the ability to cast a tie-breaking, or “critical” Vote. In . the population of three voters as shown above, any voter can cast a critical vote in four situations; in the other four situations, the vote is not critical since it cannot change the outcome of the election: For example, C can cast a tie-breaking vote only in situations 3, 4, 5, and 6. The number of- combinations in which a voter can
  (« — 1)!
  cast a tie-breaking vote is 2 • -:-, where n is
  n— 1 ■ 7i — 1
  -• |-j '
  2 ' 2
  the number of voters. Dividing this result (critical votes) by 2n (possible combinations), one arrives at that fraction of possible combinations in which a voter can cast a critical vote. This is the theory’s measure of voting power. In District K with three voters, the fraction is Vs, or 50%. In District L with nine voters, the fraction is.14%i2, or 28%. Conventional wisdom would give District K one representative and District L three. But under the theory, a voter in District L is not % as powerful as the voter in District K, but more than half' as powerful. District L deserves only two representatives, and by giving it three the State causes •voters therein to be overrepresented. For a fuller explanation of this theory, see Banzhaf, Multi-Member Electoral Districts — Do They Violate the “One Man, One Vote” Principle, 75 Yale L. J. 1309 (1966).
  Tr. 39. Plaintiffs’ brief in this Court recognizes the issue: “The obvious question which the foregoing presentation gives rise to is that of whether the fact that a voter in a large multi-member district has a greater mathematical chance to cast a crucial vote has any practical significance.” Brief of Appellees' (Plaintiffs) 14.
  Cf. Banzhaf, Weighted Voting Doesn’t Work: A Mathematical Analysis, 19 Rutgers L. Rev. 317 (1965).
  It is apparent that the District Court declined to rule as a matter of law that a multi-member district, was per se illegal as giving an invidious advantage to multi-member district voters over voters in single-member districts or smaller multi-member districts. See 305 F. Supp., at 1391-1392.
  The so-called urban-rural division has been much talked about. . Antagonistic bloc voting .by the two Camps may occur but it has perhaps been overemphasized. See White & Thomas, Urban and Rural Representation and State Legislative Apportionment, 17 W. Pol. Q. 724 (1964). Legislation dealing with uniquely urban problems may be routinely approved when urban delegations are in agreement but encounter insuperable difficulties when the delegations are split internally. See Kovach, Some Lessons of Reapportionment, 37 Reporter 26, 31 (Sept. 21, 1967).
  See Appendix to opinion, post, p. 164.
  It does not appear that the Marion County multi-member district always operated to exclude Negroes or the poor from the legislature. In the five general assemblies' from 1960-1968, the county’s Center Township ghetto had one senator and four representatives. The remainder of the township, which includes a white ghetto, elected one senator and eight representatives. Census tract 220, inhabited predominantly by Negroes but having different economic and social characteristics according to the trial court, elected one senator and- five representatives. Ibid. Plaintiffs’ evidence indicated that Marion County as a whole elected two Negro senators and seven representatives in those years. Plantiffs’ Exhibit 10.
  Plaintiffs’ Exhibit 10 purported to list the names and race- of both parties’ general assembly candidates from 1920 through 1968. .For the 1960-1968 period whieh concerned the District Court, the exhibit purported to show that the Democratic Party slated one Negro representative in I960; one in 1962; one senator and two representatives in 1964; three representatives in 1966; and one senator and two representatives in 1968. The Republican Party slated one Negro senator in 1960; two representatives in 1966; and three representatives in 1968. The racial designations on the exhibit, however, were excluded as hearsay.
  The Brief of Appellees (Plaintiffs), at 23 n. 7, indicates that in. the 1970 elections:
  “[O]ne of the .major political parties in Marion County held district ‘mini-slating conventions’ for purposes of determining its legislative candidates.- All of the slated candidates were subsequently nominated in the primary. Black candidates filed in .the slating conventions in six of the fifteen Marion County ‘districts’ including the five that contain parts of the ghetto area. Only two black candidates were slated and nominated including one in the district that contains only a very small part of the ghetto area where the black candidate overwhelmingly defeated the white candidate in a head-on race notwithstanding a very substantial white voting majority. In a district that was almost entirely ghetto a white candidate won almost all. of the vote in a head-on race against a black candidate who campaigned primarily on the basis of skin-color. All five of the candidates in the ‘ghetto districts,’ however, avowed a substantial commitment to the substantive interests of black people and the' poor.”
  The record shows that plaintiff Chavis was slated by the Democratic Party and elected to the state senate in 1964. Exhibit 10. Also, plaintiffs Ramsey and Bryant were both slated by the same party as candidates for the House of Representatives in 1968 but were defeated in the general election. Ibid.; see also Tr. 131 (Ramsey) ,Tr. 133 (Bryant).
  One of plaintiffs’ witnesses, an attorney and political figure in the Republican Party, testified as follows:
  ■ “Q. In your experience, Mrs. Allen, aren’t tickets put together by party organization to appeal [to] the various interest groups throughout Marion County?
  “A. Yes.
  “Q. Among these interest groups are economic groups, racial groups and others?
  . “A. Yes.
  “Q. I show you exhibit 5B that is in evidence, showing the location of the elected Republican representatives’ homes at the time they filed in the party primary, does it to you somehow reflect an interest in making an appeal .to each conceivable faction in the family, in the county area, each geographical interest?
  “A. Yes, it does, if I can explain.
  “Q. Yes, you may.
  “A. Back in 1966, as I stated, we had a.real primary fight and at the time we selected our candidates in the primary Republican Action Committee was not real, real strong in some, geographical areas, and we felt that necessary to come up with a 15 man slate, many of the people who lived in Center Township including myself did not feel ready to run for public office and therefore there was a hiatus in Center Township residents. However, many of the Washington Township residents, I believe at least two Washington Township residents had a number of family and historical ties in , this Center Township Area, even though they did not live there and' to the best of the Committee’s ability they tried to achieve racial, geographical, economical and social diversity on the ticket. • I can’t say they were entirely successful, but they made a real good ' attempt and this is a result of their attempts.'
  “Q. And the real hard driving effort to put the Action Committees through did take place by the residents of Center Township; did it not?
  “A. It was an over-all drive. Center Township, having the population it has, could not be ignored.” Tr. 145-148.
  Plaintiffs’ lawyer was at. the time of the- trial the Marion County Democratic chairman, Tr. 256; plaintiff Chavis was a ward chairman and a longtime precinct committeeman, Tr. 77.
  See Appendix to opinion, p. 164.
  See ibid. In addition, the Republicans nominated and elected one senator (1960), and three representatives (1960, 1966, 1968) from census tract 220, and four representatives (three in 1962, one in 1966) from the nonghetto area of Center Township. Ibid.
  Although plaintiffs asserted it, there was no finding by the. District Court that Republican legislators residing in the ghetto were not representative of the area or had failed properly to represent ghetto interests in the general assembly.
  The comparative merits of the two approaches to metropolitan representation has been much mooted and is still in contention. See the authorities, cited in n. 38, infra, particularly the piece by Kovach • and the series of studies by Collins, Dauer, David, Lacy, & Mauer. And, of course, witnesses in the trial court differed on this very issue. E. g., Tr. 209-214, 223-229, 235-238, 256-258. David & Eisenberg in their study, infra, n: 38, concluded that the case for rigid insistence on single-member districting has not been proved. They would prefer á system of small multi-member districts in metropplitan areas to either the larger multi-member district or the single-member district, thereby minimizing the acknowledged shortcomings of each. More generally, still in suspense is definitive judgment about the long-range impact of voting systems and malapportionment on legislative output. Sokolow, After Reapportionment: Numbers or Policies?, 19 W. Pol. Q. Supp. 21 (1966); T. Dye, Politics, Economies, and the Public 260-277 (1966); D. Lockard, The Politics of State and Local Government 290-293 (2d ed. 1969).
  Interestingly enough,- in Wright v. Rockefeller, 376 U. S. 52 (1964), challenge was to a single-member district plan with districts allegedly drawn on racial lines and designed to limit Negroes to voting for their own candidates in safe Negro districts. We rejected the challenge for failure of proof, but noted in passing.that “some of these voters . . . would prefer a more even distribution of minority groups among the four congressional districts,, but others, like the intervenors in this case, would argue strenuously that the kind of districts for which appellants contended would be undesirable, and, because based on race or place of origin, would themselves be unconstitutional.” 376 U. S., at 57-58.
  Plaintiffs’ final arguments in the District Court asserted political ■ as well as racial and economic discrimination in the workings of the Marion County district, iri that the “political minority,” whether Republicans or Democrat, is "always shut out” when the opposing party wins. Tr. 254. See n. 11, supra.
  For discussions of voting systems designed to achieve minority representation, see Dixon, infra, n. 38, at 516-527; Black, The Theory of Elections in Single-member Constituencies, 15 Can. J. of Economics and Pol. Sci. 158 (1949); Silva, Relation of Representation and the Party System to the Number of Seats Apportioned to a Legislative District, 17 W. Pol. Q. 742, 744 et seq. (1964); S. Bedford, The Faces of Justice (1961); E. Lakeman & J. Lambert, Voting in Democracies (1959); Blair, Cumulative Voting: An Effective Electoral Device in Illinois Politics, 45 Ill. Studies in the Social Sciences (1960).
  As of November 1970, 46% of the upper houses and 62% of the lower houses in the States contained some multi-member districts. National Municipal League, Apportionment in the Nineteen Sixties (Rev. Nov. 1970). In 1955, it was reported that the figures were 33% and 75%, respectively.' Klain, A New Look at the Constituencies: The Need for a Recount and a Reappraisal, 49 Am. Pol. Sci. Rev. 1105 (1955). Though the overall effect of the reapportionment eases on this phenomenon is necessarily somewhat speculative, there is no doubt that some States switched to multi-member districts ■ as ' a result of those decisions. Prior to the decisions, for example', Vermont’s lower house was composed entirely of single-member districts. Id., at 1109. This resulted in the colorful situation of one representative for a town of 33,155 and another for a town of 38. in 1962. National Municipal League, Apportionment in the Nineteen Sixties, pt. I (b). Reapportioned and redistrieted in light of Reynolds, Vermont’s lower house now has 36 multi-member and .36 single-member districts. Buckley v. Hoff, 243 F. Supp. 873 (Vt. 1965). Reapportionment has also been credited with abolishing Maryland’s tradition of single-member districts in its senate. Burdette, Maryland Reapportionment, in Apportionment in the Nineteen Sixties, supra.
  The relative merits of multi-member and single-member plans have been much debated and the general- preference for single-member districts has not gone unchallenged. For representative treatment of the subject see:
  R. Dixon, Democratic Representation: Reapportionment in Law and Politics 461-463, 470-472, 476-490, 503-507 (1968); P. David & R. Eisenberg, State Legislative Redistricting: Major Issues in the Wake of Judicial Decision (1962); Barnett, Unitary-Multiple Election Districts, 39 Am. Pol. Sci. Rev. 65 (1945); Silva, Compared Values of the Single- and the Multi-member Legislative District, 17 W. Pol. Q. 504 (1964); Hamilton, Legislative Constituencies: Single-member Districts, Multi-member Districts, and Floterial Districts, 20 W. Pol. Q. 321 (1967) (includes a discussion of districting in Indiana); Silva, Relation of Representation and the Party System to the Number of Seats Apportioned to a Legislative District, 17 W. Pol. Q. 742 (1964); Lindquist, Socioeconomic Status and Political Participation, 17 W. Pol. Q. 608 (1964); Klain, A New Look at the Constituencies: The Need for a Recount and. a Reappraisal, 49 Am. Pol. Sci. Rev. 1105 (1955); Kovach, Some Lessons of Reapportionment, 37 Reporter 26 (Sept. 21, 1967); and M. Collins, M. Dauer, P. David, A. Lacy, & G. Mauer, Evolving Issues and Patterns of State Legislative Redistricting in Large Metropolitan Areas (1966).
  Interesting material with respect to the relative merits of single- and multi-member districts may be found in the congressional debates surrounding the passage in 1842 of the statute requiring representatives to be elected in single-member districts. See n. 39, infra. Though the racial considerations present here were, not surprisingly, absent in these pre-Civil War Amendments debates, the concern voiced by congressmen over the submergence of minorities, bloc voting, and party control shows, at least, that the plaintiffs’ apprehensions are not entirely new ones. See, e. g., Cong. Globe, 27th Cong., 2d Sess., 445-448, 452-453, 463-464.
  In colonial days, “[mjultiple districts were the rule, single ones the exception,” and “[f]or nearly a century and a half after the Declaration of Independence the American states elected by far the greater part of their lawmakers in multiple constituencies.” Klain, supra, n. 38, at 1112, 1113. Although a trend toward single-member districts began long ago, multiple districts are still much in evidence. See n. 37, supra. See also David & Eisenberg, supra, n. 38, at 20; Dixon, supra, n. 38, at 504.
  In 1842, Congress by statute required single-member districts for congressional elections. Act of June 25, 1842, § 2, 5 Stat. 491. The substance of the restriction was continued in Rev. Stat. § 23 and in apportionment legislation in this century until 1929. In 1941, Congress enacted a law that required that until a State is redistricted in a manner provided by law after decennial reapportionment of the House, representatives were to be elected from the districts prescribed by the law'of-the State, and that “if any of them are elected from the State at large they shall continue to be so elected,” provided that if reapportionment of the House following a census shows that a State is entitled to an increase in the number of representatives, the additional representatives shall be elected at large until the State is redistricted, and if there is a decrease in the number of representatives and the number of districts in the State exceeds the number of representatives newly apportioned, all representatives shall be elected at large. Act of Nov. 15, 1941, 55 Stat. 762, amending § 22 (c) of the Act of June 18, 1929, 46 Stat. 27, 2 U. S. C. § 2a (c). In 1967, Congress reinstated the single-member district requirement, “except that a State which is entitled to more than one Representative and which has .in all previous elections elected its Representatives at Large may elect its Representatives at Large to the Ninety-first Congress.” 81 Stat. 581, 2 U. S. C. § 2c (1964 ed., Supp. V). Hawaii was the only State to take advantage of this exception. It has districted for the 92d Congress. Hawaii Rev. Stat. § 12-32.5 (Supp. 1969).
  Congress has not purported to exercise Fourteenth Amendment powers to regulate or prohibit multi-member districts in state elections.
  The. court was also impressed by the 1967 Indiana Board of Health Vital Statistics population estimates which showed a senate variance of 36.83% and a house variance of 37.30%. It did not base its order on these interim figures, however. See 307 F. Supp. 1362, 1366.
  See also Kirkpatrick v. Preisler, 394 U. S. 526 (1969), and Wells v. Rockefeller, 394 U. S. 542 (1969), in which the Court held that variances of 5.97% and 13.096%, respectively, were impermissible for congressional redistricting.
  In redistricting the State, the District Court divided some counties into several districts, and defendant attacks this as an unwarranted violation of Indiana Const., Art. 4, § 6, which says “no county, for Senatorial apportionment, shall ever be divided.” Defendant concedes that “[t]he error ... is not the per se violation” of the constitution, but rather that the court drew its plan “without having meaningfully considered” the dictates of the constitution. Brief for Appellant (Defendant) 49. But the contrary appears to us to be true. The court announced that it “would strive to preserve the integrity of county and township lines” wherever possible, 307 F. Supp., at 1364, though it ultimately concluded that the “difficulty of devising . . . compact and contiguous . . . districts within that framework [of mathematical equality] has in large part precluded preservation of county lines.” Id., at 1366. We note that none of the statewide redistricting plans that were submitted for the court’s consideration, including those of the house and senate minority leaders and the chairman of the senate majority caucus committee, followed the state constitution in this respect. R. 57-137, 198-228.
  In any event, the Court was careful to note that “we do not mean to intimate that more frequent reapportionment would not be constitutionally permissible, or practicably desirable.” 377 U. S., at 584.
  Abate v. Mundt, post, p. 182; Gordon v. Lance, ante, p. 1.
  The Court, though stating that it does “not quarrel with plaintiffs’ mathematics,” nevertheless implies that it may be ignored because “the position remains a theoretical one . . . and does ‘not take into account any -political or other factors which might affect the actual. voting power of the residents, which might include party affiliation, race, previous voting characteristics or any other factors which go into the entire political voting situation.’ ” Ante, at 145, 146. Precisely the same criticism applies, with even greater force, to the “one man, one vote” opinions of this Court. The only relevant difference between the elementary arithmetic on which the Court relies and the elementary probability theory on which Professor Banzhaf relies is that calculations in the latter field cannot be done on one’s fingers.
  More precisely, the result follows if the second of Professor Banzhaf’s assumptions is altered so that the probability of each voter’s selecting candidate A over candidate B is 50.5% rather than 50%.
  The text assumes that- each of the remaining. 275,000 voters is equally likely to vote for A or. for IS.
  “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact:” "Mark Twain, Life on the Mississippi 109 (Harper & Row, 1965).
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