Theorem: Any permutation of n cups marked 1 to n can be written as 2-cycles.

Proof:

We show the proof for the permutation 1 → 2 → 3 → ... → n → 1, and note that this generalizes by replacing this image of k with α(k) for any given permutation α :

• Swap the 1st cup with the 2nd.
• Next, swap the current 1st cup (which used to be in the 2nd spot) with the cup in the 3rd spot.
• Continue, swapping the (current) 1st cup with the cup in the n^th spot: (12)(13) ... (1n)

This product of 2-cycles will give the desired permutation ( 6 1 2 3 4 5 for our cups 1 through 6). You can play around with the cups to see for yourself!