POSNER, Chief Judge.

A hospital that serves Medicare patients and provides medical education as well is entitled to a medical-education subsidy from the federal government. 42 U.S.C. § 1395ww(d)(5)(B), (h)(3). The plaintiff is such a hospital, and claiming that it had been shortchanged by the government sought judicial review in the district court, as was its right, 42 U.S.C. § 1395oo(f)(1), and having lost there renews its complaint with us.

The amount of the payment is determined by a pair of complicated formulas, one for "indirect medical education" reimbursement, 42 U.S.C. § 1395ww(d)(5)(B); 42 CFR § 412.118 (1987), and the other for "graduate medical education" reimbursement. 42 U.S.C. § 1395ww(h)(3); 42 CFR § 413.86. The differences between them are not relevant to this case, so we'll limit discussion to the first, the IME reimbursement; everything we say about it is equally applicable to the GME reimbursement.

Only one element of the IME formula is at issue in this case, and that is the number of beds the hospital has. The smaller that number, other things being equal, such as the number of patients, the greater the subsidy the hospital is entitled to. E.g., Medicare Program: Fiscal Year 1986 Changes to the Inpatient Hospital Prospective Payment System, 51 Fed. Reg. 16772, 16775 (May 6, 1986). The consequence is that the higher the ratio of patients to beds, the greater the subsidy. From the bed count are excluded "beds assigned to newborns," 42 CFR § 412.118(b) (1987), but there is an exception for "beds in intensive care units," even if the bed is occupied by a newborn. Medicare Provider Reimbursement Manual, § 2202.7(II)(A) and Note, 2 CCH Medicare and Medicaid Guide ¶ 6096, p. 2057-3 (1988). The regulation has since been changed to exclude "beds or bassinets in healthy newborn nursery," 42 CFR § 412.105(b), but the new regulation does not apply to this case.

Why the subsidy is inverse to the ratio of beds to patients, and why beds for newborns in intensive care are counted (newborns are not eligible for Medicare), are mysteries that the briefs pass over in silence, as did the district judge in his opinion, and that neither counsel was able to elucidate at argument. It is very difficult to apply a regulation without a clue to its purpose, so we asked the parties to submit supplemental briefs addressing the issue. The government explains in its supplemental brief, plausibly enough, that the care of healthy newborns is unlikely to involve skills or experience having significant spillover benefits for Medicare patients, but that the care of a sickly newborn in an intensive-care unit may have such benefits since the elderly are frequently in need of intensive care. Hence the beds assigned to newborns who are in intensive care are included in the bed count in the formula.

This produces the paradox that the more newborns a hospital has in intensive care, the lower the Medicare education subsidy, since, other things being equal, the subsidy is smaller the greater the number of beds counted. But the government explains that among the other things held constant is the number of interns and residents. The government has found that the higher the ratio of that number to the number of beds (and the fewer the number of beds, holding number of interns and residents constant, the higher that ratio will be), the more teaching the hospital will be doing. Medicare Program: Fiscal Year 1986 Changes to the Inpatient Hospital Prospective Payment System, supra, 51 Fed. Reg. at 16775. For if the hospital has fewer beds, it probably has a smaller medical staff, and hence a higher ratio of interns and residents to fully trained doctors — the teachers. The higher that ratio, the more training the fully trained doctors must do. Suppose Hospital A has 300 beds, 75 interns and residents, and 25 fully trained doctors, and Hospital B has 600 beds, 75 interns and residents, and 125 fully trained doctors (so that in both hospitals there is one doctor for every three beds). The fully trained doctors in Hospital A will have much heavier teaching loads than the fully trained doctors in B because the ratio of interns and residents to fully trained doctors is so much higher in A (3:1) than in B (3:5).

Because the subsidy thus is greater the fewer the beds that are counted, and beds occupied by newborn infants are counted only if the beds are in an intensive-care unit, it is in the interest of the government, which wants to minimize its expenditures, to classify as many beds for newborns as possible as intensive-care beds, while it is in the hospital's interest to classify as many beds for newborns as possible as non-intensive-care beds. (Remember the theory: the more beds there are, holding number of interns and residents constant, the more fully trained doctors there must be to handle the increased patient load and therefore the less teaching each one must do.) The plaintiff in this case has what it calls an intermediate-care unit for newborns with health problems, and no unit that provides more intensive care for newborns. Illinois law forbids it to provide intensive care for newborns. See 77 Ill. Admin. Code § 250.1820(f). If any of the newborns in its intermediate-care unit require intensive care, the hospital is required to ship them off to the University of Chicago Hospitals, which operate the only intensive-care units in Chicago for newborns that Illinois recognizes. The hospital also points out that its intermediate-care unit for newborn infants provides less care, as measured by such indicators as the ratio of registered nurses to patients, than the typical intensive-care unit.

But of course state law cannot preempt federal law. An HHS regulation defines an intensive-care unit by reference to six criteria. 42 CFR § 413.53(d). The first three don't actually distinguish between intensive care and ordinary hospital care. The first criterion is that the unit be in a hospital, the second that it be a physically separate unit from the routine-care areas of the hospital, and the third that there be written criteria for admission to the unit. But the last three criteria do have bite. The fourth is that there be a registered nurse on duty 24 hours a day, the fifth that there be a minimum ratio of one nurse to two patients, and the sixth that the unit have life-saving equipment of the sort required by critically ill patients. It is uncontested that the plaintiff's intermediate-care unit for newborns satisfies the first three criteria for classification as an intensive-care unit, and there is substantial evidence that it satisfies each of the last three as well. And that is the end of judicial review unless the regulation is invalid, which the hospital does not argue. But we do not understand the government's argument that the hospital's intermediate-care unit for newborns must be an intensive-care unit because intermediate care is the highest level of care that the hospital offers for newborns. What that has to do with anything related to the regulation, to medical education, or to Medicare eludes us. We add that under the new regulation, which confines the exclusion from the bed count to "beds or bassinets in the healthy newborn nursery," intermediate-care beds would be automatically included in the bed count, without need to apply the six-factor test, because healthy babies do not require intermediate care.

The appeal presents another issue, this a procedural one. The government hires private firms, called fiscal intermediaries, to help it administer the Medicare program. 42 U.S.C. § 1395h. A hospital that is dissatisfied with the payment authorized by the intermediary on a claim for reimbursement from the government can obtain review from a Provider Reimbursement Review Board in HHS and eventually from the courts. 42 U.S.C. § 1395oo(a). In 1988 the hospital incurred a large loss on the refunding of its debt and a much smaller loss on the sale of some land that it owned. In its request for reimbursement of its Medicare-related expenses for that year, it deducted from its investment income the entire loss on the debt refunding, reducing its investment income to zero. This allowed it to claim a larger reimbursement, since investment income is an offset to interest expense, an allowable cost item. 42 CFR §§ 413.153(a)(1), (b)(2)(iii) (1987). The hospital did not mention the land loss, which it could also have deducted from its investment income if it had had any investment income left on its books to deduct losses from, and it did not.

The fiscal intermediary ruled that in trying to deduct the entire loss against 1988 investment income, the hospital was violating the government's policy of requiring the loss to be amortized over the period until the debt matured by its original terms. Medicare Provider Reimbursement Manual, § 2.33(A), 1 CCH Medicare and Medicaid Guide ¶ 5184, p. 1729-17 (1994). The hospital appealed to the Provider Reimbursement Review Board. While the appeal was pending, the Supreme Court upheld the policy, Shalala v. Guernsey Memorial Hospital, 514 U.S. 87 (1995), dooming the appeal. The hospital then changed its position and argued to the review board that it should be permitted to deduct the land loss from its 1988 investment income, since without the deduction of the full loss from the restructuring of its debt the hospital now had positive investment income to offset by the loss on the sale of the land. The board ruled that the hospital couldn't switch grounds, because this would mean bypassing the fiscal intermediary.

The board is allowed to review a ruling by a fiscal intermediary only if the provider (the hospital in this case) "is dissatisfied with a final determination of the . . . fiscal intermediary . . . as to the amount of total program reimbursement due the provider." 42 U.S.C. § 1395oo(a) (1)(A)(i). The hospital points out that it was dissatisfied, albeit not on the ground that it had presented to the intermediary. But while the statute is curiously worded, the intent is plain that the provider must give the intermediary a first shot at the issue, provided the issue is within the intermediary's competence, as the issue of deducting the land loss against the investment income was because it did not involve an issue of policy. Bethesda Hospital Ass'n v. Bowen, 485 U.S. 399, 404-05 (1988); see also Little Company of Mary Hospital Health Care Centers v. Shalala, 24 F.3d 984, 993 (7th Cir. 1994); cf. Edgewater Hospital, Inc. v. Bowen, 857 F.2d 1123, 1133-34, as modified, 866 F.2d 228 (7th Cir. 1988). The only issue was the size of the loss and when it was incurred, issues securely within the limited competence of the fiscal intermediary.

The hospital argues that it could not have presented the issue to the fiscal intermediary because, having reduced its investment income to zero by deducting in full the loss from the restructuring of its debt, there was nothing to deduct the land loss from and therefore no way to use it to claim a larger reimbursement. But there was nothing to prevent the hospital from explaining to the fiscal intermediary that it had alternative theories for why it was entitled to the larger reimbursement. If the fiscal intermediary had refused to consider this mode of pleading in the alternative — and we have no reason to think it would have — then the hospital could have bypassed the intermediary and presented the issue to the review board. It would have exhausted its remedies by giving the intermediary a shot at considering the hospital's alternative ground for a larger reimbursement than the intermediary wanted to give it; and no more is required.

Affirmed.