# Puzzles

## Reverse bases

Find two digits \(a\) and \(b\) such that \(ab\) in base 10 is equal to \(ba\) in base 4.

Find two digits \(c\) and \(d\) such that \(cd\) in base 10 is equal to \(dc\) in base 7.

Find two digits \(e\) and \(f\) such that \(ef\) in base 9 is equal to \(fe\) in base 5.

#### Show answer & extension

#### Hide answer & extension

If \(ab\) in base 10 is equal to \(ba\) in base 4, then \(10a+b=4b+a\).

So, \(9a=3b\).

\(a\) and \(b\) must both be less than 4, as they are digits used in base 4, so \(a=1\) and \(b=3\).

So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:

- 23 in base 10 is equal to 32 in base 7.
- 46 in base 10 is equal to 64 in base 7.
- 12 in base 9 is equal to 21 in base 5.
- 24 in base 9 is equal to 42 in base 5.

#### Extension

For which pairs of bases \(A\) and \(B\) can you find two digits \(g\) and \(h\) such that \(gh\) in base \(A\) is equal to \(hg\) in base \(B\)?