Four mothers, each accompanied by their daughter, are buying ribbon in a store. Each mother buys twice as many meters as her daughter; each of the 8 individuals buys as many meters as they pay for one meter, and the price per meter is always an integer in forints.

Mrs. Juhász pays 76 forints more than Mrs. Fehér; Nóra buys 3 meters less than Mrs. Barna; Gizi buys 2 meters more than Hilda, whose bill is 48 forints less than Mrs. Kovács's; What is the name of Mária's mother?
Ha Juhászné's daughter bought $x$ meters, and Fehérné's daughter bought $y$ meters, then according to the problem, J.-né paid $4 x^{2}$ forints for $2 x$ meters, and F.-né paid $4 y^{2}$ forints for $2 y$ meters. Therefore,

$$
4 x^{2}-4 y^{2}=76, \quad \text { or } \quad(x+y)(x-y)=19
$$

Here, $x$ and $y$ are integers, so necessarily

$$
x+y=19, \quad \text { and } \quad x-y=1 \text {, }
$$

from which

$$
x=10, \quad y=9 \text {. }
$$

Assuming that Kovácsné bought $2 u$ meters, and Hilda bought $z$ meters, then according to the problem,

$$
4 u^{2}-z^{2}=48
$$

Since $u$ and $z$ are integers, $z$ must be an even number. Let $z=2 v$, then

$$
4 u^{2}-4 v^{2}=48
$$

or

$$
(u-v)(u+v)=12=1 \cdot 12=2 \cdot 6=3 \cdot 4 .
$$

Since the assumption $u-v=1$ or 3 does not lead to integer solutions, only

$$
u-v=2, \quad \text { and } \quad u+v=6
$$

is possible, from which

$$
u=4, \quad v=2
$$

Therefore, Hilda bought $z=2 v=4$ meters, and thus Gizi bought $4+2=6$ meters. Since Nóra necessarily bought an odd number (three less than an even number) of meters, the 10 meters bought by Juhászné's daughter must have been bought by Mária.

Therefore, Mária's mother is Juhászné.

# Verification: 

| Juhász Mária | $(10 \mathrm{~m}$, | $100 \mathrm{Ft})$ | Juhászné | $(20 \mathrm{~m}$, | $400 \mathrm{Ft})$ |
| :--- | ---: | ---: | :--- | ---: | ---: |
| Fehér Nóra | $(9 \mathrm{~m}$, | $81 \mathrm{Ft})$ | Fehérné | $(18 \mathrm{~m}$, | $324 \mathrm{Ft})$ |
| Barna Gizi | $(6 \mathrm{~m}$, | $36 \mathrm{Ft})$ | Barnáné | $(12 \mathrm{~m}$, | $144 \mathrm{Ft})$ |
| Kovács Hilda | $(4 \mathrm{~m}$, | $16 \mathrm{Ft})$ | Kovácsné | $(8 \mathrm{~m}$, | $64 \mathrm{Ft})$ |

Lukács Gábor (Bp., V., Cukor u. g. IV. o. t.)