
402 U.S. 1 (1971)
SWANN ET AL.
v.
CHARLOTTE-MECKLENBURG BOARD OF EDUCATION ET AL.
No. 281.
Supreme Court of United States.
Argued October 12, 1970
Decided April 20, 1971[*]
CERTIORARI TO THE UNITED STATES COURT OF APPEALS FOR THE FOURTH CIRCUIT.
*4 Julius LeVonne Chambers and James M. Nabrit III argued the cause for petitioners in No. 281 and respondents in No. 349. With them on the briefs were Jack Greenberg, Norman J. Chachkin, C. O. Pearson, and Anthony G. Amsterdam.
William J. Wagonner and Benjamin S. Horack argued the cause and filed briefs for respondents in No. 281 and petitioners in No. 349.
Solicitor General Griswold argued the cause for the United States as amicus curiae in both cases. With him on the brief was Assistant Attorney General Leonard.
Briefs of amici curiae in No. 281 were filed by Earl Faircloth, Attorney General, Robert J. Kelly, Deputy Attorney General, Ronald W. Sabo, Assistant Attorney General, and Rivers Buford for the State of Florida; by Andrew P. Miller, Attorney General, William G. Broaddus and Theodore J. Markow, Assistant Attorneys General, Lewis F. Powell, Jr., John W. Riely, and Guy K. Tower for the Commonwealth of Virginia; by Claude R. Kirk, Jr., pro se, and Gerald Mager for Claude R. Kirk, Jr., Governor of Florida; by W. F. Womble for the Winston-Salem/Forsyth County Board of Education; by Raymond B. Witt, Jr., and Eugene N. Collins for the Chattanooga Board of Education; by Kenneth W. Cleary for the School Board of Manatee County, Florida; by W. Crosby Few and John M. Allison for the School Board of Hillsborough County, Florida; by Sam J. Ervin, *5 Jr., Charles R. Jonas, and Ernest F. Hollings for the Classroom Teachers Association of the Charlotte-Mecklenburg School System, Inc.; by Mark Wells White, Jr., for Mrs. H. W. Cullen et al., members of the Board of Education of the Houston Independent School District; by Jack Petree for the Board of Education of Memphis City Schools; by Sherwood W. Wise for the Jackson Chamber of Commerce, Inc., et al.; by Stephen J. Pollak, Benjamin W. Boley, and David Rubin for the National Education Association; by William L. Taylor, Richard B. Sobol, and Joseph L. Rauh, Jr., for the United Negro College Fund, Inc., et al.; by Owen H. Page for Concerned Citizens Association, Inc.; by Charles S. Conley, Floyd B. McKissick, and Charles S. Scott for the Congress of Racial Equality; by the Tennessee Federation for Constitutional Government et al.; by William C. Cramer, pro se, and Richard B. Peet, joined by Albert W. Watson et al., for William C. Cramer; by Charles E. Bennett, pro se, James C. Rinaman, Jr., and Yardley D. Buckman for Charles E. Bennett; by Calvin H. Childress and M. T. Bohannon, Jr., for David E. Allgood et al.; by William B. Spong, Jr., and by Newton Collier Estes.
MR. CHIEF JUSTICE BURGER delivered the opinion of the Court.
We granted certiorari in this case to review important issues as to the duties of school authorities and the scope of powers of federal courts under this Court's mandates to eliminate racially separate public schools established and maintained by state action. Brown v. Board of Education, 347 U. S. 483 (1954) (Brown I).
This case and those argued with it[1] arose in States having a long history of maintaining two sets of schools in a *6 single school system deliberately operated to carry out a governmental policy to separate pupils in schools solely on the basis of race. That was what Brown v. Board of Education was all about. These cases present us with the problem of defining in more precise terms than heretofore the scope of the duty of school authorities and district courts in implementing Brown I and the mandate to eliminate dual systems and establish unitary systems at once. Meanwhile district courts and courts of appeals have struggled in hundreds of cases with a multitude and variety of problems under this Court's general directive. Understandably, in an area of evolving remedies, those courts had to improvise and experiment without detailed or specific guidelines. This Court, in Brown I, appropriately dealt with the large constitutional principles; other federal courts had to grapple with the flinty, intractable realities of day-to-day implementation of those constitutional commands. Their efforts, of necessity, embraced a process of "trial and error," and our effort to formulate guidelines must take into account their experience.

I
The Charlotte-Mecklenburg school system, the 43d largest in the Nation, encompasses the city of Charlotte and surrounding Mecklenburg County, North Carolina. The area is large550 square milesspanning roughly 22 miles east-west and 36 miles north-south. During the 1968-1969 school year the system served more than 84,000 pupils in 107 schools. Approximately 71% of the pupils were found to be white and 29% Negro. As of *7 June 1969 there were approximately 24,000 Negro students in the system, of whom 21,000 attended schools within the city of Charlotte. Two-thirds of those 21,000approximately 14,000 Negro studentsattended 21 schools which were either totally Negro or more than 99% Negro.
This situation came about under a desegregation plan approved by the District Court at the commencement of the present litigation in 1965, 243 F. Supp. 667 (WDNC), aff'd, 369 F. 2d 29 (CA4 1966), based upon geographic zoning with a free-transfer provision. The present proceedings were initiated in September 1968 by petitioner Swann's motion for further relief based on Green v. County School Board, 391 U. S. 430 (1968), and its companion cases.[2] All parties now agree that in 1969 the system fell short of achieving the unitary school system that those cases require.
The District Court held numerous hearings and received voluminous evidence. In addition to finding certain actions of the school board to be discriminatory, the court also found that residential patterns in the city and county resulted in part from federal, state, and local government action other than school board decisions. School board action based on these patterns, for example, by locating schools in Negro residential areas and fixing the size of the schools to accommodate the needs of immediate neighborhoods, resulted in segregated education. These findings were subsequently accepted by the Court of Appeals.
In April 1969 the District Court ordered the school board to come forward with a plan for both faculty and student desegregation. Proposed plans were accepted by the court in June and August 1969 on an interim basis *8 only, and the board was ordered to file a third plan by November 1969. In November the board moved for an extension of time until February 1970, but when that was denied the board submitted a partially completed plan. In December 1969 the District Court held that the board's submission was unacceptable and appointed an expert in education administration, Dr. John Finger, to prepare a desegregation plan. Thereafter in February 1970, the District Court was presented with two alternative pupil assignment plansthe finalized "board plan" and the "Finger plan."
The Board Plan. As finally submitted, the school board plan closed seven schools and reassigned their pupils. It restructured school attendance zones to achieve greater racial balance but maintained existing grade structures and rejected techniques such as pairing and clustering as part of a desegregation effort. The plan created a single athletic league, eliminated the previously racial basis of the school bus system, provided racially mixed faculties and administrative staffs, and modified its free-transfer plan into an optional majority-to-minority transfer system.
The board plan proposed substantial assignment of Negroes to nine of the system's 10 high schools, producing 17% to 36% Negro population in each. The projected Negro attendance at the 10th school, Independence, was 2%. The proposed attendance zones for the high schools were typically shaped like wedges of a pie, extending outward from the center of the city to the suburban and rural areas of the county in order to afford residents of the center city area access to outlying schools.
As for junior high schools, the board plan rezoned the 21 school areas so that in 20 the Negro attendance would range from 0% to 38%. The other school, located in the heart of the Negro residential area, was left with an enrollment of 90% Negro.
*9 The board plan with respect to elementary schools relied entirely upon gerrymandering of geographic zones. More than half of the Negro elementary pupils were left in nine schools that were 86% to 100% Negro; approximately half of the white elementary pupils were assigned to schools 86% to 100% white.
The Finger Plan. The plan submitted by the court-appointed expert, Dr. Finger, adopted the school board zoning plan for senior high schools with one modification: it required that an additional 300 Negro students be transported from the Negro residential area of the city to the nearly all-white Independence High School.
The Finger plan for the junior high schools employed much of the rezoning plan of the board, combined with the creation of nine "satellite" zones.[3] Under the satellite plan, inner-city Negro students were assigned by attendance zones to nine outlying predominately white junior high schools, thereby substantially desegregating every junior high school in the system.
The Finger plan departed from the board plan chiefly in its handling of the system's 76 elementary schools. Rather than relying solely upon geographic zoning, Dr. Finger proposed use of zoning, pairing, and grouping techniques, with the result that student bodies throughout the system would range from 9% to 38% Negro.[4]
The District Court described the plan thus:
"Like the board plan, the Finger plan does as much by rezoning school attendance lines as can reasonably *10 be accomplished. However, unlike the board plan, it does not stop there. It goes further and desegregates all the rest of the elementary schools by the technique of grouping two or three outlying schools with one black inner city school; by transporting black students from grades one through four to the outlying white schools; and by transporting white students from the fifth and sixth grades from the outlying white schools to the inner city black school."
Under the Finger plan, nine inner-city Negro schools were grouped in this manner with 24 suburban white schools.
On February 5, 1970, the District Court adopted the board plan, as modified by Dr. Finger, for the junior and senior high schools. The court rejected the board elementary school plan and adopted the Finger plan as presented. Implementation was partially stayed by the Court of Appeals for the Fourth Circuit on March 5, and this Court declined to disturb the Fourth Circuit's order, 397 U. S. 978 (1970).
On appeal the Court of Appeals affirmed the District Court's order as to faculty desegregation and the secondary school plans, but vacated the order respecting elementary schools. While agreeing that the District Court properly disapproved the board plan concerning these schools, the Court of Appeals feared that the pairing and grouping of elementary schools would place an unreasonable burden on the board and the system's pupils. The case was remanded to the District Court for reconsideration and submission of further plans. 431 F. 2d *11 138. This Court granted certiorari, 399 U. S. 926, and directed reinstatement of the District Court's order pending further proceedings in that court.
On remand the District Court received two new plans for the elementary schools: a plan prepared by the United States Department of Health, Education, and Welfare (the HEW plan) based on contiguous grouping and zoning of schools, and a plan prepared by four members of the nine-member school board (the minority plan) achieving substantially the same results as the Finger plan but apparently with slightly less transportation. A majority of the school board declined to amend its proposal. After a lengthy evidentiary hearing the District Court concluded that its own plan (the Finger plan), the minority plan, and an earlier draft of the Finger plan were all reasonable and acceptable. It directed the board to adopt one of the three or in the alternative to come forward with a new, equally effective plan of its own; the court ordered that the Finger plan would remain in effect in the event the school board declined to adopt a new plan. On August 7, the board indicated it would "acquiesce" in the Finger plan, reiterating its view that the plan was unreasonable. The District Court, by order dated August 7, 1970, directed that the Finger plan remain in effect.

II
Nearly 17 years ago this Court held, in explicit terms, that state-imposed segregation by race in public schools denies equal protection of the laws. At no time has the Court deviated in the slightest degree from that holding or its constitutional underpinnings. None of the parties before us challenges the Court's decision of May 17, 1954, that
"in the field of public education the doctrine of `separate but equal' has no place. Separate educational facilities are inherently unequal. Therefore, *12 we hold that the plaintiffs and others similarly situated. . . are, by reason of the segregation complained of, deprived of the equal protection of the laws guaranteed by the Fourteenth Amendment. . . .
"Because these are class actions, because of the wide applicability of this decision, and because of the great variety of local conditions, the formulation of decrees in these cases presents problems of considerable complexity." Brown v. Board of Education, supra, at 495.
None of the parties before us questions the Court's 1955 holding in Brown II, that
"School authorities have the primary responsibility for elucidating, assessing, and solving these problems; courts will have to consider whether the action of school authorities constitutes good faith implementation of the governing constitutional principles. Because of their proximity to local conditions and the possible need for further hearings, the courts which originally heard these cases can best perform this judicial appraisal. Accordingly, we believe it appropriate to remand the cases to those courts.
"In fashioning and effectuating the decrees, the courts will be guided by equitable principles. Traditionally, equity has been characterized by a practical flexibility in shaping its remedies and by a facility for adjusting and reconciling public and private needs. These cases call for the exercise of these traditional attributes of equity power. At stake is the personal interest of the plaintiffs in admission to public schools as soon as practicable on a nondiscriminatory basis. To effectuate this interest may call for elimination of a variety of obstacles in making the transition to school systems operated in accordance with the constitutional principles set forth in our May 17, 1954, decision. Courts of *13 equity may properly take into account the public interest in the elimination of such obstacles in a systematic and effective manner. But it should go without saying that the vitality of these constitutional principles cannot be allowed to yield simply because of disagreement with them." Brown v. Board of Education, 349 U. S. 294, 299-300 (1955).
Over the 16 years since Brown II, many difficulties were encountered in implementation of the basic constitutional requirement that the State not discriminate between public school children on the basis of their race. Nothing in our national experience prior to 1955 prepared anyone for dealing with changes and adjustments of the magnitude and complexity encountered since then. Deliberate resistance of some to the Court's mandates has impeded the good-faith efforts of others to bring school systems into compliance. The detail and nature of these dilatory tactics have been noted frequently by this Court and other courts.
By the time the Court considered Green v. County School Board, 391 U. S. 430, in 1968, very little progress had been made in many areas where dual school systems had historically been maintained by operation of state laws. In Green, the Court was confronted with a record of a freedom-of-choice program that the District Court had found to operate in fact to preserve a dual system more than a decade after Brown II. While acknowledging that a freedom-of-choice concept could be a valid remedial measure in some circumstances, its failure to be effective in Green required that:
"The burden on a school board today is to come forward with a plan that promises realistically to work . . . now . . . until it is clear that state-imposed segregation has been completely removed." Green, supra, at 439.
*14 This was plain language, yet the 1969 Term of Court brought fresh evidence of the dilatory tactics of many school authorities. Alexander v. Holmes County Board of Education, 396 U. S. 19, restated the basic obligation asserted in Griffin v. School Board, 377 U. S. 218, 234 (1964), and Green, supra, that the remedy must be implemented forthwith.
The problems encountered by the district courts and courts of appeals make plain that we should now try to amplify guidelines, however incomplete and imperfect, for the assistance of school authorities and courts.[5] The failure of local authorities to meet their constitutional obligations aggravated the massive problem of converting from the state-enforced discrimination of racially separate school systems. This process has been rendered more difficult by changes since 1954 in the structure and patterns of communities, the growth of student population,[6] movement of families, and other changes, some of which had marked impact on school planning, sometimes neutralizing or negating remedial action before it was fully implemented. Rural areas accustomed for half a century to the consolidated school systems implemented by bus transportation could make adjustments more readily than metropolitan areas with dense and shifting population, numerous schools, congested and complex traffic patterns.


*15 III
The objective today remains to eliminate from the public schools all vestiges of state-imposed segregation. Segregation was the evil struck down by Brown I as contrary to the equal protection guarantees of the Constitution. That was the violation sought to be corrected by the remedial measures of Brown II. That was the basis for the holding in Green that school authorities are "clearly charged with the affirmative duty to take whatever steps might be necessary to convert to a unitary system in which racial discrimination would be eliminated root and branch." 391 U. S., at 437-438.
If school authorities fail in their affirmative obligations under these holdings, judicial authority may be invoked. Once a right and a violation have been shown, the scope of a district court's equitable powers to remedy past wrongs is broad, for breadth and flexibility are inherent in equitable remedies.
"The essence of equity jurisdiction has been the power of the Chancellor to do equity and to mould each decree to the necessities of the particular case. Flexibility rather than rigidity has distinguished it. The qualities of mercy and practicality have made equity the instrument for nice adjustment and reconciliation between the public interest and private needs as well as between competing private claims." Hecht Co. v. Bowles, 321 U. S. 321, 329-330 (1944), cited in Brown II, supra, at 300.
This allocation of responsibility once made, the Court attempted from time to time to provide some guidelines for the exercise of the district judge's discretion and for the reviewing function of the courts of appeals. However, a school desegregation case does not differ fundamentally from other cases involving the framing of *16 equitable remedies to repair the denial of a constitutional right. The task is to correct, by a balancing of the individual and collective interests, the condition that offends the Constitution.
In seeking to define even in broad and general terms how far this remedial power extends it is important to remember that judicial powers may be exercised only on the basis of a constitutional violation. Remedial judicial authority does not put judges automatically in the shoes of school authorities whose powers are plenary. Judicial authority enters only when local authority defaults.
School authorities are traditionally charged with broad power to formulate and implement educational policy and might well conclude, for example, that in order to prepare students to live in a pluralistic society each school should have a prescribed ratio of Negro to white students reflecting the proportion for the district as a whole. To do this as an educational policy is within the broad discretionary powers of school authorities; absent a finding of a constitutional violation, however, that would not be within the authority of a federal court. As with any equity case, the nature of the violation determines the scope of the remedy. In default by the school authorities of their obligation to proffer acceptable remedies, a district court has broad power to fashion a remedy that will assure a unitary school system.
The school authorities argue that the equity powers of federal district courts have been limited by Title IV of the Civil Rights Act of 1964, 42 U. S. C. § 2000c. The language and the history of Title IV show that it was enacted not to limit but to define the role of the Federal Government in the implementation of the Brown I decision. It authorizes the Commissioner of Education to provide technical assistance to local boards in the preparation of desegregation plans, to arrange "training institutes" *17 for school personnel involved in desegregation efforts, and to make grants directly to schools to ease the transition to unitary systems. It also authorizes the Attorney General, in specified circumstances, to initiate federal desegregation suits. Section 2000c (b) defines "desegregation" as it is used in Title IV:
" `Desegregation' means the assignment of students to public schools and within such schools without regard to their race, color, religion, or national origin, but `desegregation' shall not mean the assignment of students to public schools in order to overcome racial imbalance."
Section 2000c-6, authorizing the Attorney General to institute federal suits, contains the following proviso:
"nothing herein shall empower any official or court of the United States to issue any order seeking to achieve a racial balance in any school by requiring the transportation of pupils or students from one school to another or one school district to another in order to achieve such racial balance, or otherwise enlarge the existing power of the court to insure compliance with constitutional standards."
On their face, the sections quoted purport only to insure that the provisions of Title IV of the Civil Rights Act of 1964 will not be read as granting new powers. The proviso in § 2000c-6 is in terms designed to foreclose any interpretation of the Act as expanding the existing powers of federal courts to enforce the Equal Protection Clause. There is no suggestion of an intention to restrict those powers or withdraw from courts their historic equitable remedial powers. The legislative history of Title IV indicates that Congress was concerned that the Act might be read as creating a right of action under the Fourteenth Amendment in the situation of so-called "de facto segregation," where racial imbalance exists in the *18 schools but with no showing that this was brought about by discriminatory action of state authorities. In short, there is nothing in the Act that provides us material assistance in answering the question of remedy for state-imposed segregation in violation of Brown I. The basis of our decision must be the prohibition of the Fourteenth Amendment that no State shall "deny to any person within its jurisdiction the equal protection of the laws."

IV
We turn now to the problem of defining with more particularity the responsibilities of school authorities in desegregating a state-enforced dual school system in light of the Equal Protection Clause. Although the several related cases before us are primarily concerned with problems of student assignment, it may be helpful to begin with a brief discussion of other aspects of the process.
In Green, we pointed out that existing policy and practice with regard to faculty, staff, transportation, extra-curricular activities, and facilities were among the most important indicia of a segregated system. 391 U. S., at 435. Independent of student assignment, where it is possible to identify a "white school" or a "Negro school" simply by reference to the racial composition of teachers and staff, the quality of school buildings and equipment, or the organization of sports activities, a prima facie case of violation of substantive constitutional rights under the Equal Protection Clause is shown.
When a system has been dual in these respects, the first remedial responsibility of school authorities is to eliminate invidious racial distinctions. With respect to such matters as transportation, supporting personnel, and extracurricular activities, no more than this may be necessary. Similar corrective action must be taken with regard to the maintenance of buildings and the distribution of equipment. In these areas, normal administrative *19 practice should produce schools of like quality, facilities, and staffs. Something more must be said, however, as to faculty assignment and new school construction.
In the companion Davis case, post, p. 33, the Mobile school board has argued that the Constitution requires that teachers be assigned on a "color blind" basis. It also argues that the Constitution prohibits district courts from using their equity power to order assignment of teachers to achieve a particular degree of faculty desegregation. We reject that contention.
In United States v. Montgomery County Board of Education, 395 U. S. 225 (1969), the District Court set as a goal a plan of faculty assignment in each school with a ratio of white to Negro faculty members substantially the same throughout the system. This order was predicated on the District Court finding that:
"The evidence does not reflect any real administrative problems involved in immediately desegregating the substitute teachers, the student teachers, the night school faculties, and in the evolvement of a really legally adequate program for the substantial desegregation of the faculties of all schools in the system commencing with the school year 1968-69." Quoted at 395 U. S., at 232.
The District Court in Montgomery then proceeded to set an initial ratio for the whole system of at least two Negro teachers out of each 12 in any given school. The Court of Appeals modified the order by eliminating what it regarded as "fixed mathematical" ratios of faculty and substituted an initial requirement of "substantially or approximately" a five-to-one ratio. With respect to the future, the Court of Appeals held that the numerical ratio should be eliminated and that compliance should not be tested solely by the achievement of specified proportions. Id., at 234.
*20 We reversed the Court of Appeals and restored the District Court's order in its entirety, holding that the order of the District Judge
"was adopted in the spirit of this Court's opinion in Green . . . in that his plan `promises realistically to work, and promises realistically to work now.' The modifications ordered by the panel of the Court of Appeals, while of course not intended to do so, would, we think, take from the order some of its capacity to expedite, by means of specific commands, the day when a completely unified, unitary, nondiscriminatory school system becomes a reality instead of a hope. . . . We also believe that under all the circumstances of this case we follow the original plan outlined in Brown II . . . by accepting the more specific and expeditious order of [District] Judge Johnson . . . ." 395 U. S., at 235-236 (emphasis in original).
The principles of Montgomery have been properly followed by the District Court and the Court of Appeals in this case.
The construction of new schools and the closing of old ones are two of the most important functions of local school authorities and also two of the most complex. They must decide questions of location and capacity in light of population growth, finances, land values, site availability, through an almost endless list of factors to be considered. The result of this will be a decision which, when combined with one technique or another of student assignment, will determine the racial composition of the student body in each school in the system. Over the long run, the consequences of the choices will be far reaching. People gravitate toward school facilities, just as schools are located in response to the needs of people. The location of schools may thus influence *21 the patterns of residential development of a metropolitan area and have important impact on composition of inner-city neighborhoods.
In the past, choices in this respect have been used as a potent weapon for creating or maintaining a state-segregated school system. In addition to the classic pattern of building schools specifically intended for Negro or white students, school authorities have sometimes, since Brown, closed schools which appeared likely to become racially mixed through changes in neighborhood residential patterns. This was sometimes accompanied by building new schools in the areas of white suburban expansion farthest from Negro population centers in order to maintain the separation of the races with a minimum departure from the formal principles of "neighborhood zoning." Such a policy does more than simply influence the short-run composition of the student body of a new school. It may well promote segregated residential patterns which, when combined with "neighborhood zoning," further lock the school system into the mold of separation of the races. Upon a proper showing a district court may consider this in fashioning a remedy.
In ascertaining the existence of legally imposed school segregation, the existence of a pattern of school construction and abandonment is thus a factor of great weight. In devising remedies where legally imposed segregation has been established, it is the responsibility of local authorities and district courts to see to it that future school construction and abandonment are not used and do not serve to perpetuate or re-establish the dual system. When necessary, district courts should retain jurisdiction to assure that these responsibilities are carried out. Cf. United States v. Board of Public Instruction, 395 F. 2d 66 (CA5 1968); Brewer v. School Board, 397 F. 2d 37 (CA4 1968).


*22 V
The central issue in this case is that of student assignment, and there are essentially four problem areas:
(1) to what extent racial balance or racial quotas may be used as an implement in a remedial order to correct a previously segregated system;
(2) whether every all-Negro and all-white school must be eliminated as an indispensable part of a remedial process of desegregation;
(3) what the limits are, if any, on the rearrangement of school districts and attendance zones, as a remedial measure; and
(4) what the limits are, if any, on the use of transportation facilities to correct state-enforced racial school segregation.

(1) Racial Balances or Racial Quotas.

The constant theme and thrust of every holding from Brown I to date is that state-enforced separation of races in public schools is discrimination that violates the Equal Protection Clause. The remedy commanded was to dismantle dual school systems.
We are concerned in these cases with the elimination of the discrimination inherent in the dual school systems, not with myriad factors of human existence which can cause discrimination in a multitude of ways on racial, religious, or ethnic grounds. The target of the cases from Brown I to the present was the dual school system. The elimination of racial discrimination in public schools is a large task and one that should not be retarded by efforts to achieve broader purposes lying beyond the jurisdiction of school authorities. One vehicle can carry only a limited amount of baggage. It would not serve the important objective of Brown I to seek to use school desegregation cases for purposes beyond their scope, although desegregation of schools ultimately will have *23 impact on other forms of discrimination. We do not reach in this case the question whether a showing that school segregation is a consequence of other types of state action, without any discriminatory action by the school authorities, is a constitutional violation requiring remedial action by a school desegregation decree. This case does not present that question and we therefore do not decide it.
Our objective in dealing with the issues presented by these cases is to see that school authorities exclude no pupil of a racial minority from any school, directly or indirectly, on account of race; it does not and cannot embrace all the problems of racial prejudice, even when those problems contribute to disproportionate racial concentrations in some schools.
In this case it is urged that the District Court has imposed a racial balance requirement of 71%-29% on individual schools. The fact that no such objective was actually achievedand would appear to be impossible tends to blunt that claim, yet in the opinion and order of the District Court of December 1, 1969, we find that court directing
"that efforts should be made to reach a 71-29 ratio in the various schools so that there will be no basis for contending that one school is racially different from the others . . . , [t]hat no school [should] be operated with an all-black or predominantly black student body, [and] [t]hat pupils of all grades [should] be assigned in such a way that as nearly as practicable the various schools at various grade levels have about the same proportion of black and white students."
The District Judge went on to acknowledge that variation "from that norm may be unavoidable." This contains intimations that the "norm" is a fixed mathematical *24 racial balance reflecting the pupil constituency of the system. If we were to read the holding of the District Court to require, as a matter of substantive constitutional right, any particular degree of racial balance or mixing, that approach would be disapproved and we would be obliged to reverse. The constitutional command to desegregate schools does not mean that every school in every community must always reflect the racial composition of the school system as a whole.
As the voluminous record in this case shows,[7] the predicate for the District Court's use of the 71%-29% ratio was twofold: first, its express finding, approved by the Court of Appeals and not challenged here, that a dual school system had been maintained by the school authorities at least until 1969; second, its finding, also approved by the Court of Appeals, that the school board had totally defaulted in its acknowledged duty to come forward with an acceptable plan of its own, notwithstanding the patient efforts of the District Judge who, on at least three occasions, urged the board to submit plans.[8] As the statement of facts shows, these findings are abundantly *25 supported by the record. It was because of this total failure of the school board that the District Court was obliged to turn to other qualified sources, and Dr. Finger was designated to assist the District Court to do what the board should have done.
We see therefore that the use made of mathematical ratios was no more than a starting point in the process of shaping a remedy, rather than an inflexible requirement. From that starting point the District Court proceeded to frame a decree that was within its discretionary powers, as an equitable remedy for the particular circumstances.[9] As we said in Green, a school authority's remedial plan or a district court's remedial decree is to be judged by its effectiveness. Awareness of the racial composition of the whole school system is likely to be a useful starting point in shaping a remedy to correct past constitutional violations. In sum, the very limited use made of mathematical ratios was within the equitable remedial discretion of the District Court.

(2) One-race Schools.

The record in this case reveals the familiar phenomenon that in metropolitan areas minority groups are often found concentrated in one part of the city. In some circumstances certain schools may remain all or largely of one race until new schools can be provided or neighborhood patterns change. Schools all or predominately *26 of one race in a district of mixed population will require close scrutiny to determine that school assignments are not part of state-enforced segregation.
In light of the above, it should be clear that the existence of some small number of one-race, or virtually one-race, schools within a district is not in and of itself the mark of a system that still practices segregation by law. The district judge or school authorities should make every effort to achieve the greatest possible degree of actual desegregation and will thus necessarily be concerned with the elimination of one-race schools. No per se rule can adequately embrace all the difficulties of reconciling the competing interests involved; but in a system with a history of segregation the need for remedial criteria of sufficient specificity to assure a school authority's compliance with its constitutional duty warrants a presumption against schools that are substantially disproportionate in their racial composition. Where the school authority's proposed plan for conversion from a dual to a unitary system contemplates the continued existence of some schools that are all or predominately of one race, they have the burden of showing that such school assignments are genuinely nondiscriminatory. The court should scrutinize such schools, and the burden upon the school authorities will be to satisfy the court that their racial composition is not the result of present or past discriminatory action on their part.
An optional majority-to-minority transfer provision has long been recognized as a useful part of every desegregation plan. Provision for optional transfer of those in the majority racial group of a particular school to other schools where they will be in the minority is an indispensable remedy for those students willing to transfer to other schools in order to lessen the impact on them of the state-imposed stigma of segregation. In order to be effective, such a transfer arrangement must grant *27 the transferring student free transportation and space must be made available in the school to which he desires to move. Cf. Ellis v. Board of Public Instruction, 423 F. 2d 203, 206 (CA5 1970). The court orders in this and the companion Davis case now provide such an option.

(3) Remedial Altering of Attendance Zones.

The maps submitted in these cases graphically demonstrate that one of the principal tools employed by school planners and by courts to break up the dual school system has been a frankand sometimes drasticgerry-mandering of school districts and attendance zones. An additional step was pairing, "clustering," or "grouping" of schools with attendance assignments made deliberately to accomplish the transfer of Negro students out of formerly segregated Negro schools and transfer of white students to formerly all-Negro schools. More often than not, these zones are neither compact[10] nor contiguous; indeed they may be on opposite ends of the city. As an interim corrective measure, this cannot be said to be beyond the broad remedial powers of a court.
*28 Absent a constitutional violation there would be no basis for judicially ordering assignment of students on a racial basis. All things being equal, with no history of discrimination, it might well be desirable to assign pupils to schools nearest their homes. But all things are not equal in a system that has been deliberately constructed and maintained to enforce racial segregation. The remedy for such segregation may be administratively awkward, inconvenient, and even bizarre in some situations and may impose burdens on some; but all awkwardness and inconvenience cannot be avoided in the interim period when remedial adjustments are being made to eliminate the dual school systems.
No fixed or even substantially fixed guidelines can be established as to how far a court can go, but it must be recognized that there are limits. The objective is to dismantle the dual school system. "Racially neutral" assignment plans proposed by school authorities to a district court may be inadequate; such plans may fail to counteract the continuing effects of past school segregation resulting from discriminatory location of school sites or distortion of school size in order to achieve or maintain an artificial racial separation. When school authorities present a district court with a "loaded game board," affirmative action in the form of remedial altering of attendance zones is proper to achieve truly nondiscriminatory assignments. In short, an assignment plan is not acceptable simply because it appears to be neutral.
In this area, we must of necessity rely to a large extent, as this Court has for more than 16 years, on the informed judgment of the district courts in the first instance and on courts of appeals.
We hold that the pairing and grouping of noncontiguous school zones is a permissible tool and such action is to be considered in light of the objectives sought. Judicial *29 steps in shaping such zones going beyond combinations of contiguous areas should be examined in light of what is said in subdivisions (1), (2), and (3) of this opinion concerning the objectives to be sought. Maps do not tell the whole story since noncontiguous school zones may be more accessible to each other in terms of the critical travel time, because of traffic patterns and good highways, than schools geographically closer together. Conditions in different localities will vary so widely that no rigid rules can be laid down to govern all situations.

(4) Transportation of Students.

The scope of permissible transportation of students as an implement of a remedial decree has never been defined by this Court and by the very nature of the problem it cannot be defined with precision. No rigid guidelines as to student transportation can be given for application to the infinite variety of problems presented in thousands of situations. Bus transportation has been an integral part of the public education system for years, and was perhaps the single most important factor in the transition from the one-room schoolhouse to the consolidated school. Eighteen million of the Nation's public school children, approximately 39%, were transported to their schools by bus in 1969-1970 in all parts of the country.
The importance of bus transportation as a normal and accepted tool of educational policy is readily discernible in this and the companion case, Davis, supra.[11] The *30 Charlotte school authorities did not purport to assign students on the basis of geographically drawn zones until 1965 and then they allowed almost unlimited transfer privileges. The District Court's conclusion that assignment of children to the school nearest their home serving their grade would not produce an effective dismantling of the dual system is supported by the record.
Thus the remedial techniques used in the District Court's order were within that court's power to provide equitable relief; implementation of the decree is well within the capacity of the school authority.
The decree provided that the buses used to implement the plan would operate on direct routes. Students would be picked up at schools near their homes and transported to the schools they were to attend. The trips for elementary school pupils average about seven miles and the District Court found that they would take "not over 35 minutes at the most."[12] This system compares favorably with the transportation plan previously operated in Charlotte under which each day 23,600 students on all grade levels were transported an average of 15 miles one way for an average trip requiring over an hour. In these circumstances, we find no basis for holding that the local school authorities may not be required to employ bus transportation as one tool of school desegregation. Desegregation plans cannot be limited to the walk-in school.
An objection to transportation of students may have validity when the time or distance of travel is so great as to either risk the health of the children or significantly *31 impinge on the educational process. District courts must weigh the soundness of any transportation plan in light of what is said in subdivisions (1), (2), and (3) above. It hardly needs stating that the limits on time of travel will vary with many factors, but probably with none more than the age of the students. The reconciliation of competing values in a desegregation case is, of course, a difficult task with many sensitive facets but fundamentally no more so than remedial measures courts of equity have traditionally employed.

VI
The Court of Appeals, searching for a term to define the equitable remedial power of the district courts, used the term "reasonableness." In Green, supra, this Court used the term "feasible" and by implication, "workable," "effective," and "realistic" in the mandate to develop "a plan that promises realistically to work, and . . . to work now." On the facts of this case, we are unable to conclude that the order of the District Court is not reasonable, feasible and workable. However, in seeking to define the scope of remedial power or the limits on remedial power of courts in an area as sensitive as we deal with here, words are poor instruments to convey the sense of basic fairness inherent in equity. Substance, not semantics, must govern, and we have sought to suggest the nature of limitations without frustrating the appropriate scope of equity.
At some point, these school authorities and others like them should have achieved full compliance with this Court's decision in Brown I. The systems would then be "unitary" in the sense required by our decisions in Green and Alexander.
It does not follow that the communities served by such systems will remain demographically stable, for in a growing, mobile society, few will do so. Neither *32 school authorities nor district courts are constitutionally required to make year-by-year adjustments of the racial composition of student bodies once the affirmative duty to desegregate has been accomplished and racial discrimination through official action is eliminated from the system. This does not mean that federal courts are without power to deal with future problems; but in the absence of a showing that either the school authorities or some other agency of the State has deliberately attempted to fix or alter demographic patterns to affect the racial composition of the schools, further intervention by a district court should not be necessary.
For the reasons herein set forth, the judgment of the Court of Appeals is affirmed as to those parts in which it affirmed the judgment of the District Court. The order of the District Court, dated August 7, 1970, is also affirmed.
It is so ordered.
NOTES
[*]  Together with No. 349, Charlotte-Mecklenburg Board of Education et al. v. Swann et al., also on certiorari to the same court.
[1]  McDaniel v. Barresi, No. 420, post, p. 39; Davis v. Board of School Commissioners of Mobile County, No. 436, post, p. 33; Moore v. Charlotte-Mecklenburg Board of Education, No. 444, post, p. 47; North Carolina State Board of Education v. Swann, No. 498, post, p. 43. For purposes of this opinion the cross-petitions in Nos. 281 and 349 are treated as a single case and will be referred to as "this case."
[2]  Raney v. Board of Education, 391 U. S. 443 (1968), and Monroe v. Board of Commissioners, 391 U. S. 450 (1968).
[3]  A "satellite zone" is an area which is not contiguous with the main attendance zone surrounding the school.
[4]  In its opinion and order of December 1, 1969, later incorporated in the order appointing Dr. Finger as consultant, the District Court stated:

"Fixed ratios of pupils in particular schools will not be set. If the board in one of its three tries had presented a plan for desegregation, the court would have sought ways to approve variations in pupil ratios. In default of any such plan from the school board, the court will start with the thought . . . that efforts should be made to reach a 71-29 ratio in the various schools so that there will be no basis for contending that one school is racially different from the others, but to understand that variations from that norm may be unavoidable." 306 F. Supp. 1299, 1312.
[5]  The necessity for this is suggested by the situation in the Fifth Circuit where 166 appeals in school desegregation cases were heard between December 2, 1969, and September 24, 1970.
[6]  Elementary public school population (grades 1-6) grew from 17,447,000 in 1954 to 23,103,000 in 1969; secondary school population (beyond grade 6) grew from 11,183,000 in 1954 to 20,775,000 in 1969. Digest of Educational Statistics, Table 3, Office of Education Pub. 10024-64; Digest of Educational Statistics, Table 28, Office of Education Pub. 10024-70.
[7]  It must be remembered that the District Court entered nearly a score of orders and numerous sets of findings, and for the most part each was accompanied by a memorandum opinion. Considering the pressure under which the court was obliged to operate we would not expect that all inconsistencies and apparent inconsistencies could be avoided. Our review, of course, is on the orders of February 5, 1970, as amended, and August 7, 1970.
[8]  The final board plan left 10 schools 86% to 100% Negro and yet categorically rejected the techniques of pairing and clustering as part of the desegregation effort. As discussed below, the Charlotte board was under an obligation to exercise every reasonable effort to remedy the violation, once it was identified, and the suggested techniques are permissible remedial devices. Additionally, as noted by the District Court and Court of Appeals, the board plan did not assign white students to any school unless the student population of that school was at least 60% white. This was an arbitrary limitation negating reasonable remedial steps.
[9]  In its August 3, 1970, memorandum holding that the District Court plan was "reasonable" under the standard laid down by the Fourth Circuit on appeal, the District Court explained the approach taken as follows:

"This court has not ruled, and does not rule that `racial balance' is required under the Constitution; nor that all black schools in all cities are unlawful; nor that all school boards must bus children or violate the Constitution; nor that the particular order entered in this case would be correct in other circumstances not before this court." (Emphasis in original.)
[10]  The reliance of school authorities on the reference to the "revision of . . . attendance areas into compact units," Brown II, at 300 (emphasis supplied), is misplaced. The enumeration in that opinion of considerations to be taken into account by district courts was patently intended to be suggestive rather than exhaustive. The decision in Brown II to remand the cases decided in Brown I to local courts for the framing of specific decrees was premised on a recognition that this Court could not at that time foresee the particular means which would be required to implement the constitutional principles announced. We said in Green, supra, at 439:

"The obligation of the district courts, as it always has been, is to assess the effectiveness of a proposed plan in achieving desegregation. There is no universal answer to complex problems of desegregation; there is obviously no one plan that will do the job in every case. The matter must be assessed in light of the circumstances present and the options available in each instance."
[11]  During 1967-1968, for example, the Mobile board used 207 buses to transport 22,094 students daily for an average round trip of 31 miles. During 1966-1967, 7,116 students in the metropolitan area were bused daily. In Charlotte-Mecklenburg, the system as a whole, without regard to desegregation plans, planned to bus approximately 23,000 students this year, for an average daily round trip of 15 miles. More elementary school children than high school children were to be bused, and four- and five-year-olds travel the longest routes in the system.
[12]  The District Court found that the school system would have to employ 138 more buses than it had previously operated. But 105 of those buses were already available and the others could easily be obtained. Additionally, it should be noted that North Carolina requires provision of transportation for all students who are assigned to schools more than one and one-half miles from their homes. N. C. Gen. Stat. § 115-186 (b) (1966).
