Consider a device with two concentric rings, where the outer ring is fixed and the inner ring can be rotated freely $.$ In addition, each ring have twelve numbers $1$ to $12,$ engraved on the surface in an equally spaced fashion. On the outer ring numbers $1$ to $12$ are placed in increasing order similar to a clock, however the number on the inner ring are scrambled $.$ We say two rings are in alignment if the twele numbers on the outer ring.

The device came with an alignment where no two equal numbers match between the outer and the inner rings $.$ Prove that there is a rotation of the inner ring such tha

$$ t the new alignment has at least two pairs of rqual numbers matching each other.

Explain with examples