Let r denote number system radix. The only value(s) of r that satisfy the…

Interpret the numerals in their bases and compare their values.

Compute the value of 12 in base r:

• 12 (base r) = 1·r + 2 = r + 2

Compute the value of 11 in base (r + 1):

• 11 (base r+1) = 1·(r + 1) + 1 = r + 2

Since both expressions equal r + 2, the equality holds algebraically for every r for which the digits used are valid.

Digit-validity requirement:

• The digit '2' appears in 12 (base r), so we need r > 2 (radix must be strictly greater than the largest digit).

Conclusion: All integer radices r > 2 satisfy the given equality.