7.3. It is known that both КРУГ and КУБ are cubes of some numbers. Find all possible values of the numbers КРУГ and КУБ. List them all and explain why there are no others. (Identical digits are replaced by the same letters, different digits - by different letters.)

Note: 8 and 1000 are cubes, since $8=2 \cdot 2 \cdot 2=2^{3}$, and $1000=10 \cdot 10 \cdot 10=10^{3}$.
Answer: 1728 and 125.

Solution: Notice that in the words КУБ (CUBE) and КРУГ (CIRCLE), two digits repeat - К and У.

Let's list all possible three-digit numbers that are cubes of some numbers:

$5^{3}=125, 6^{3}=216, 7^{3}=343, 8^{3}=512, 9^{3}=729$ (4^3=64 - two-digit; 10^3=1000 - four-digit). Notice that 343 does not work (K ≠ B).

Let's list the possible values for letters К and У:

| $\mathbf{y}$ | 1 | 2 |
| :--- | :--- | :--- |
| $\mathbf{K}$ | 2 or 5 | 1 or 7 |

Let's list all possible four-digit numbers that are cubes of some numbers: $10^{3}=1000, 11^{3}=1331, 12^{3}=1728, 13^{3}=2197, 14^{3}=2744, 15^{3}=3375, 16^{3}=4096, 17^{3}=4913, 18^{3}=5832, 19^{3}=6859, 20^{3}=8000, 21^{3}=9261$ (9^3=729 - three-digit; 22^3=10648 - five-digit).

Notice that 1000, 1331, 2744, 3375, and 8000 do not work, as K, Р, У, and Г are different digits. Among the remaining options, let's list all possible values for letters К and У:

| $\mathbf{y}$ | 1 | 2 | 3 | 5 | 6 | 9 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathbf{K}$ | 4 | 1 | 5 | 6 | 9 | 2 or 4 |

Notice that К and У in the numbers КРУГ and КУБ match only when У = 2, К = 1. Therefore, there is a unique solution: КРУГ = 1728, КУБ = 125.

## Criteria:

7 points - correct answer and complete correct justification;

4-5 points - answer found correctly, but the solution is insufficiently justified;

3 points - the idea to consider the same letters У and К, but the solution is incomplete or has errors;

Only answer - 0 points.