POSNER, Circuit Judge.
In 1982 the public school system of South Bend, Indiana laid off 146 teachers. All were white; 48 had more seniority than blacks not laid off; two years later 20 of the 48 had not yet been recalled. In laying off only whites, the school board was acting pursuant to a provision in its collective bargaining agreement with the teachers’ union to the effect that no blacks would be laid off until every white was laid off. The laid-off teachers sued the school system under section 1 of the Civil Rights Act of 1871, 42 U.S.C. § 1983, charging that the racially preferential layoff provision violated the equal protection clause of the Fourteenth Amendment, and seeking reinstatement and damages. The district court, after a bench trial, gave judgment for the board. 593 F.Supp. 1223 (N.D.Ind.1984). The court thought the board’s adoption of the provision a reasonable means toward the board’s goal, which the court also thought reasonable, of raising the percentage of black teachers in the South Bend school system to that of black students. The board had resolved “to increase the percentage of minorities [meaning blacks] in its teaching force until that percentage equals the percentage of minorities in its student body. The Board specifically resolved to increase the percentage of minority pupils [sic —the judge meant ‘teachers’] because it deemed it essential that the student population, both black and white, have a sufficient number of minority teachers to act as role models.” Id. at 1225. “In cases dealing with school corporations, it is proper to compare the percentage of minority faculty with the percentage of minorities in the student body rather than with the percentage of minorities in the relevant labor pool ... because of the vital role teachers play as role-models for their students. This is particularly true in the rise [sic —the judge apparently meant ‘case’] of minority teachers since ‘societal discrimination has often deprived minority children of other role models.’ ” Id. at 1230 n. 3.
The plaintiffs appealed. A divided panel of this court affirmed. 775 F.2d 794 (7th Cir.1985). The full court then granted rehearing en banc. Before the case could be reargued, the Supreme Court decided a similar case in favor of another group of white public school teachers. Wygant v. [768]*768Jackson Board of Education, 476 U.S. 267, 106 S.Ct. 1842, 90 L.Ed.2d 260 (1986). Like the panel in the present case, the Sixth Circuit had upheld the dismissal of the complaint. The Supreme Court reversed. It rejected the “role models” rationale on which the Sixth Circuit, like the district court in the present case, had based its decision. The Supreme Court did not remand for further proceedings to determine whether the plaintiffs' constitutional rights had been violated; it held they had been. When the present case was rear-gued to us, the question no longer was reversal or affirmance; it was whether to reverse outright, holding that the plaintiffs had proved a violation of their constitutional rights and remanding only for the determination of the appropriate remedy; or to remand for further proceedings in which the board would have an opportunity to establish a rationale for racially discriminatory layoffs that would be consistent with the Wygant decision.
The constitutional status of discrimination by public bodies in favor of blacks and other members of minority groups is contentious and unsettled; but with the Supreme Court having spoken so recently to a set of facts so close to those of the present case, the task for us is the interpretation of the Court’s decision rather than the forging of new constitutional law. Wygant came out of the public school system of Jackson, Michigan. In 1968, the year before the Jackson board of education adopted a racially preferential hiring plan, 4 percent of the city’s public school teachers were black, compared to 15 percent of the students. Wygant v. Jackson Board of Education, 746 F.2d 1152, 1156 (6th Cir.1984), rev’d, 476 U.S. 267, 106 S.Ct. 1842, 90 L.Ed.2d 260 (1986). Because Michigan’s civil rights commission believed that the disparity was due to discrimination against black teachers (see 106 S.Ct. at 1854), the board of education agreed to give preference in hiring to blacks until the percentage of black teachers was equal to that of black students. By 1971, 9 percent of the teachers were black. 746 F.2d at 1156. That year it became necessary to lay off some teachers. The board did this in the usual way — reverse order of seniority. A disproportionate number of those laid off were black, because so many blacks had been hired recently and therefore had little seniority. The racial situation in the Jackson public schools soon became even more tense — became, indeed, violent. See 106 S.Ct. at 1859. Expecting that additional layoffs would be necessary in the near future, the board decided it must take measures to make sure that such layoffs would not reduce the number of black teachers disproportionately. The board felt it needed to have as many black teachers as possible in order to quiet the schools and give black students role models. It also feared that the hiring of blacks would be impeded by strict adherence to the principle of laying off teachers in reverse order of seniority, because new teachers would know they would be the first to be laid off if there was a reduction in force.
In 1972 the board negotiated with the teachers’ union an agreement (which became Article XII of the collective bargaining contract with the union) to deviate from the principle of laying off teachers in reverse order of seniority, but only to the extent necessary to preserve the existing percentage of blacks (and other members of minority groups, but we can ignore that feature of the case) in the teaching force. So if 10 percent of the teachers were black, no more than 10 percent of the teachers laid off could be black.
The collective bargaining contract in Wy-gant was ratified by an overwhelming majority of the Jackson public school teachers, most of whom were white. Nevertheless, in a suit by white teachers laid off because of Article XII, the Supreme Court held that the provision was a denial of equal protection. Although there was no majority opinion in Wygant, a “lowest common denominator” majority position can be pieced together. “When a fragmented Court decides a case and no single rationale explaining the result enjoys the assent of five Justices, ‘the holding of the Court may be viewed as that position taken by those Members who concurred in the judgments on the narrowest grounds.’ ” Marks v. [769]*769United States, 430 U.S. 188, 193, 97 S.Ct. 990, 993, 51 L.Ed.2d 260 (1977).
Justice Powell, writing in Wygant for three Justices, opined that a public body may not use race as a criterion for layoffs unless necessary to protect a proven victim of discrimination, such as a black who if he had not been discriminated against would have had as much seniority as a white. See 106 S.Ct. at 1849-52. Justice White took the same position, only more bluntly. See id. at 1857-58.
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POSNER, Circuit Judge.
In 1982 the public school system of South Bend, Indiana laid off 146 teachers. All were white; 48 had more seniority than blacks not laid off; two years later 20 of the 48 had not yet been recalled. In laying off only whites, the school board was acting pursuant to a provision in its collective bargaining agreement with the teachers’ union to the effect that no blacks would be laid off until every white was laid off. The laid-off teachers sued the school system under section 1 of the Civil Rights Act of 1871, 42 U.S.C. § 1983, charging that the racially preferential layoff provision violated the equal protection clause of the Fourteenth Amendment, and seeking reinstatement and damages. The district court, after a bench trial, gave judgment for the board. 593 F.Supp. 1223 (N.D.Ind.1984). The court thought the board’s adoption of the provision a reasonable means toward the board’s goal, which the court also thought reasonable, of raising the percentage of black teachers in the South Bend school system to that of black students. The board had resolved “to increase the percentage of minorities [meaning blacks] in its teaching force until that percentage equals the percentage of minorities in its student body. The Board specifically resolved to increase the percentage of minority pupils [sic —the judge meant ‘teachers’] because it deemed it essential that the student population, both black and white, have a sufficient number of minority teachers to act as role models.” Id. at 1225. “In cases dealing with school corporations, it is proper to compare the percentage of minority faculty with the percentage of minorities in the student body rather than with the percentage of minorities in the relevant labor pool ... because of the vital role teachers play as role-models for their students. This is particularly true in the rise [sic —the judge apparently meant ‘case’] of minority teachers since ‘societal discrimination has often deprived minority children of other role models.’ ” Id. at 1230 n. 3.
The plaintiffs appealed. A divided panel of this court affirmed. 775 F.2d 794 (7th Cir.1985). The full court then granted rehearing en banc. Before the case could be reargued, the Supreme Court decided a similar case in favor of another group of white public school teachers. Wygant v. [768]*768Jackson Board of Education, 476 U.S. 267, 106 S.Ct. 1842, 90 L.Ed.2d 260 (1986). Like the panel in the present case, the Sixth Circuit had upheld the dismissal of the complaint. The Supreme Court reversed. It rejected the “role models” rationale on which the Sixth Circuit, like the district court in the present case, had based its decision. The Supreme Court did not remand for further proceedings to determine whether the plaintiffs' constitutional rights had been violated; it held they had been. When the present case was rear-gued to us, the question no longer was reversal or affirmance; it was whether to reverse outright, holding that the plaintiffs had proved a violation of their constitutional rights and remanding only for the determination of the appropriate remedy; or to remand for further proceedings in which the board would have an opportunity to establish a rationale for racially discriminatory layoffs that would be consistent with the Wygant decision.
The constitutional status of discrimination by public bodies in favor of blacks and other members of minority groups is contentious and unsettled; but with the Supreme Court having spoken so recently to a set of facts so close to those of the present case, the task for us is the interpretation of the Court’s decision rather than the forging of new constitutional law. Wygant came out of the public school system of Jackson, Michigan. In 1968, the year before the Jackson board of education adopted a racially preferential hiring plan, 4 percent of the city’s public school teachers were black, compared to 15 percent of the students. Wygant v. Jackson Board of Education, 746 F.2d 1152, 1156 (6th Cir.1984), rev’d, 476 U.S. 267, 106 S.Ct. 1842, 90 L.Ed.2d 260 (1986). Because Michigan’s civil rights commission believed that the disparity was due to discrimination against black teachers (see 106 S.Ct. at 1854), the board of education agreed to give preference in hiring to blacks until the percentage of black teachers was equal to that of black students. By 1971, 9 percent of the teachers were black. 746 F.2d at 1156. That year it became necessary to lay off some teachers. The board did this in the usual way — reverse order of seniority. A disproportionate number of those laid off were black, because so many blacks had been hired recently and therefore had little seniority. The racial situation in the Jackson public schools soon became even more tense — became, indeed, violent. See 106 S.Ct. at 1859. Expecting that additional layoffs would be necessary in the near future, the board decided it must take measures to make sure that such layoffs would not reduce the number of black teachers disproportionately. The board felt it needed to have as many black teachers as possible in order to quiet the schools and give black students role models. It also feared that the hiring of blacks would be impeded by strict adherence to the principle of laying off teachers in reverse order of seniority, because new teachers would know they would be the first to be laid off if there was a reduction in force.
In 1972 the board negotiated with the teachers’ union an agreement (which became Article XII of the collective bargaining contract with the union) to deviate from the principle of laying off teachers in reverse order of seniority, but only to the extent necessary to preserve the existing percentage of blacks (and other members of minority groups, but we can ignore that feature of the case) in the teaching force. So if 10 percent of the teachers were black, no more than 10 percent of the teachers laid off could be black.
The collective bargaining contract in Wy-gant was ratified by an overwhelming majority of the Jackson public school teachers, most of whom were white. Nevertheless, in a suit by white teachers laid off because of Article XII, the Supreme Court held that the provision was a denial of equal protection. Although there was no majority opinion in Wygant, a “lowest common denominator” majority position can be pieced together. “When a fragmented Court decides a case and no single rationale explaining the result enjoys the assent of five Justices, ‘the holding of the Court may be viewed as that position taken by those Members who concurred in the judgments on the narrowest grounds.’ ” Marks v. [769]*769United States, 430 U.S. 188, 193, 97 S.Ct. 990, 993, 51 L.Ed.2d 260 (1977).
Justice Powell, writing in Wygant for three Justices, opined that a public body may not use race as a criterion for layoffs unless necessary to protect a proven victim of discrimination, such as a black who if he had not been discriminated against would have had as much seniority as a white. See 106 S.Ct. at 1849-52. Justice White took the same position, only more bluntly. See id. at 1857-58. Obviously if either of those opinions had commanded a majority, we would have to reverse outright. But since Justice O’Connor, the fifth and last member of the majority, concurred in the judgment of reversal on the narrowest ground, her opinion is critical to our determining the proper disposition of the present case.
She reserved the question whether a racially preferential layoff plan might ever be a constitutionally permissible measure “to correct apparent prior employment discrimination against minorities while avoiding further litigation,” id. at 1854 (see also id. at 1857), and she noted in this connection that the Jackson school board had “reasoned that without the layoff provision, the remedial gains made under the ongoing hiring goals contained in the collective bargaining agreement could be eviscerated by layoffs,” id. at 1854. The fact that there had been no authoritative determination of hiring discrimination and that the layoff provision would not merely benefit victims of such discrimination did not in her view automatically condemn the plan. Nevertheless she agreed that the plan was unconstitutional and that outright reversal was the proper disposition of the appeal, because the plaintiffs had
met their burden of establishing that this layoff provision is not “narrowly tailored” to achieve its asserted remedial purpose by demonstrating that the provision is keyed to a hiring goal that itself has no relation to the remedying of employment discrimination.
Id. at 1857. That is,
the hiring goal that the layoff provision was designed to safeguard was tied to the percentage of minority students in the school district, not to the percentage of qualified minority teachers within the relevant labor pool. The disparity between the percentage of minorities on the teaching staff and the percentage of minorities in the student body is not probative of employment discrimination.... Because the layoff provision here acts to maintain levels of minority hiring that have no relation to remedying employment discrimination, it cannot be adjudged “narrowly tailored” to effectuate its asserted remedial purpose.
Id. (citation omitted). The hiring goal in the present case was likewise “tied to the percentage of minority students in the school district.”
Justice Marshall, the author of the principal dissenting opinion in Wygant (which Justices Brennan and Blackmun joined), made two points that are particularly relevant to the present case. First, he noted that an alternative to a racially proportional layoff provision — such as Article XII, which merely preserved the percentage of black teachers achieved before the layoffs —“would have been a freeze on layoffs of minority teachers. This measure ... would have been substantially more burdensome than Article XII, not only by necessitating the layoff of a greater number of white teachers, but also by erecting an absolute distinction between the races, one to be benefited and one to be burdened, in a way that Article XII avoids.” Id. at 1865. That hypothetical “substantially more burdensome” measure is the one the South Bend school board adopted.
Second, Justice Marshall took exception to the majority’s refusal to remand the case for findings on possible justifications for Article XII other than those the majority had rejected. The district court had granted summary judgment for the Jackson board of education because the court found, on the basis of evidence that a much higher percentage of students than of faculty was black, that favoring blacks in layoffs was necessary both to give black students adequate “role models” and to rectify “societal discrimination” against black [770]*770teachers (“societal discrimination” meaning a racial imbalance not caused by the defendants’ own discriminatory acts). The defendants in Wygant, perhaps foreseeing rejection of these grounds, submitted evidence relevant to other possible justifications to the Supreme Court. The submission had no standing as evidence, but it provided a reason for remanding the case to give the lower courts a chance to consider it. The rejection of Justice Marshall’s suggestion that the case be remanded has implications for the present case, which the defendants have asked us to remand.
South Bend, Indiana, like Jackson, Michigan, had a lower percentage of black teachers in its public schools than of black students. In 1978, on the eve of adopting a racially preferential hiring plan, the percentages were 10 and 22. Although the 10 percent figure is more than twice the percentage of black teachers in the Jackson public schools at the corresponding period in the evolution of its program of racial preferences, the South Bend school board was not satisfied, and resolved to raise the percentage of black teachers until it equaled that of black students. The layoff plan ensured that if layoffs were necessary they would not impede achievement of the board’s goal of racial parity between teachers and students. Indeed, since no blacks could be laid off if any whites had not yet been laid off, the layoff plan (unlike the one in Wygant) was calculated to increase rather than just maintain the percentage of black teachers in the event that any layoffs became necessary. By 1981, 13 percent of the teachers (and 25 percent of the students) were black. As a result of the layoff provision, the percentage of black teachers rose — to 14 percent — when it became necessary to lay off teachers, since all of those laid off were white.
No one doubts that the signatories of the plurality opinion in Wygant, plus Justice White (a total of four Justices), would invalidate South Bend’s racially preferential layoff plan. The plan goes further than the one struck down in Wygant) unlike Wygant there is no background of racial violence; as in Wygant there is no evidence that any of the black teachers who have benefited from the plan are victims of racial discrimination that deprived them of seniority they would otherwise have had. Conceivably Justice O’Connor might approve a racially preferential layoff plan of some sort (a critical qualification, as we shall see) if she were convinced that the purpose of the plan was to correct previous hiring discrimination by the school board. There was some evidence in the record before the Supreme Court in Wygant that that had been the Jackson school board’s purpose; there is very little evidence that it was the South Bend board’s purpose. The goal advanced by the board in the district court — the goal to which all of the board’s evidence was oriented — was to correct a discrepancy between the percentage of black teachers and the percentage of black students. Such a discrepancy is, in Justice O’Connor’s view, “not probative of employment discrimination,” 106 S.Ct. at 1857 (emphasis added), and therefore cannot, in her view, justify racially discriminatory layoffs. For her the proper comparison in deciding whether black teachers have been discriminated against is not between the percentage of black teachers and the percentage of black students but between the percentage of qualified black teaching applicants who are hired and the percentage of qualified white applicants who are hired; if 10 percent of the qualified blacks are hired but 20 percent of the qualified whites are hired, this would be evidence of racial discrimination in hiring. See id.) J. Edinger & Son, Inc. v. City of Louisville, 802 F.2d 213, 216 (6th Cir.1986). Nowhere in the transcript of the trial or in the trial exhibits do we find evidence that the purpose of the South Bend school board in seeking to equate the fraction of black teachers to the fraction of black students was to remedy employment discrimination. The district court did not overlook this theory of the defense; the theory simply was not presented to the court. Cf. 593 F.Supp. at 1231. The board put all its forensic eggs in the baskets labeled “role models” and “racial imbalance.” The board’s counsel said at trial, “statistical disparity, that’s all that’s necessary.... So our evidence, [771]*771Your Honor, in terms of justifying this provision, is going to be that of showing the statistical disparage [sic] between the proportion of Blacks in the teaching force of the corporation, and the proportion of [black] students in the student body.”
The record contains some evidence bearing on discrimination against blacks, but because discrimination was not the focus of the district court proceedings, the evidence is sparse, and it is also ambivalent. Far from discriminating against black teachers, the South Bend school board had for years been hiring a much higher fraction of black than of white teaching applicants. As early as 1972 — eight years before the collective bargaining provision challenged in this case — 22 percent of all the new hires were black. In 1974 this figure was 30 percent; in 1980, 55 percent. Granted, this is not the complete picture. In 1975, five years before the layoff provision at issue in this case was adopted, HEW wrote a letter to the school board alleging racial discrimination in the South Bend public school system. However, the only concern expressed in the letter with respect to discrimination in hiring involved the discrepancy between the fraction of black students and the fraction of black teachers — the theory of discrimination discredited by Wygant. And the school board’s reply to the letter detailed the board’s vigorous efforts to recruit black teachers, efforts that included not only soliciting teaching applications from black colleges but also hiring a much higher fraction of black than of white applicants. A second letter that HEW wrote in 1975 is silent on discrimination in hiring, and a third is a form letter apparently written to all public school superintendents in the country. The record also contains an unsworn, unsubstantiated, unelaborated charge by a member of the audience at a public meeting unrelated to this case that the board had unjustly refused to hire five (unnamed) black teaching applicants. Even if this accusation were accepted as true, it would imply — in the context of uncontra-dicted evidence that blacks were favored in hiring, consistently with the board’s goal of raising the percentage of black teachers to the percentage of black students — a mistaken personnel decision rather than an act of deliberate discrimination. Finally, Brown v. Weinberger, 417 F.Supp. 1215, 1221 (D.C.Cir.1976), noted that HEW had years ago accused the South Bend board of some unspecified form of racial discrimination, but the opinion does not suggest that the accusation is true, or concerned discrimination in hiring. And HEW never did bring suit.
South Bend may have engaged in a different form of discrimination — assigning black teachers to teach black students — for which the proper remedy would be to enjoin this practice, as a consent order did in 1980. The order said nothing about giving blacks superseniority, for that would not be a logical remedy for discrimination in assigning teachers. That Indiana had a segregated school system almost 40 years ago is another fact that pertains to discrimination in assigning, not in hiring, teachers. Steering black teachers to black schools could actually lead to hiring more black teachers than if there were no steering, by earmarking all teaching slots in black schools for blacks. Granted, in 1964 only 4 percent of the teachers in the South Bend public school system were black, yet there is no evidence that this was due to discrimination in hiring or assigning; the percentage of blacks in South Bend was also lower then.
Given the long history of discrimination against black people, in Indiana as elsewhere, we cannot exclude the possibility that the South Bend school board, perhaps until fairly recently, discriminated against black teachers in hiring and that the layoff provision challenged in this case was adopted, in part at least, to correct that discrimination by protecting newly hired black teachers against being laid off in the event of an economic downturn. One would think, however, that if this were so, the board would have argued the point in the district court; for while Wygant, decided later, withdrew certain justifications for such provisions, it did not create a new one (correcting previous discrimination). The board had every incentive to assert all its possible defenses in the district court; any [772]*772not asserted would ordinarily be deemed waived. See, e.g., National Fidelity Life Ins. Co. v. Karaganis, 811 F.2d 357, 360-61 (7th Cir.1987); Benzies v. Illinois Dept. of Mental Health & Developmental Disabilities, 810 F.2d 146, 149 (7th Cir.1987). The Supreme Court did not remand Wy-gant, as Justice Marshall had suggested it do, to permit the Jackson board of education to prove that its layoff provision had been designed to rectify previous discrimination in hiring — of which the board had in fact been accused.
Despite all this it might be arguable as an original matter that the evidence of remedial purpose, although weak, is stronger than in Wygant and that the South Bend school board should have a chance to shore up that evidence on remand — were it not for Justice O’Connor’s insistence that even a remedial layoff plan be “narrowly tailored,” a requirement that the plan in this case flunks even more decisively than the plan in Wygant. Recall that Justice O’Connor was willing to accept the possibility that the layoff plan had been adopted in order to correct the Jackson school board’s “apparent prior discrimination.” But that wasn’t good enough; the plan was invalid because tied to an improper hiring goal, that of equating the fraction of black teachers to the fraction of black students. The plan in the present case is tied to the same goal, and really no more need be said to condemn the plan. But there is more: enough more, indeed, that even Justice Marshall and the two Justices who joined him might think South Bend had gone too far, by erecting an absolute racial preference for blacks. That goes further than necessary to preserve blacks’ gains in times of economic downturn, and further than the proportional preference struck down in Wygant.
Between 1979 and 1981 the South Bend school board hired 62 blacks, and it was the 48 most recently hired of these blacks, 41 of whom had been hired since 1980, who would have been laid off under a racially neutral layoff plan. Thus, no matter how recently hired a black was, he was placed on the seniority ladder above every white teacher. In addition to giving every black an absolute preference over every white, the plan ties the percentage of black teachers to such irrelevant and unpredictable circumstances as the economic health and school-age population of South Bend; the plan uses economic downturns and shrink-ages in the student population as fulcrums for arbitrarily increasing the percentage of black teachers in the public school system. A plan with such effects cannot be held to be “narrowly tailored” to the goal of remedying previous discrimination, even if that was the board’s goal, of which there is, as we have said, almost no evidence in the record, and even if such a goal could save a layoff plan tied to a hiring goal of equating the percentage of black teachers to the percentage of black students, which Justice O’Connor (and a fortiori the other four Justices in the majority in Wygant) believed it could not.
The school board has argued (though not until reargument en banc was granted) that it didn’t really lay off these whites, because it offered them substitute positions, though at reduced compensation. But the board’s counsel acknowledged at argument that his client would have violated th|. equal protection clause if it had tried to solve its financial problems by cutting just white teachers’ wages or fringe benefits (estimated to be worth between $2,000 and $4,000 a year), without laying off anybody. Yet that is what he says the board actually did, by offering to hire the laid-off whites as substitute teachers at a reduced level of compensation.
The judgment of the district court is reversed, and the case is remanded for further proceedings consistent with this opinion.