Doing a magic trick with limited memory (from a problem solving course)

Question

I got the following question in a problem solving course:

There are four different objects lying on places 1, 2, 3, 4. A magician closes his eyes and someone from the audience comes. He switches pairs of objects 10 times, and each time shouts the places he switched. Then he does a secret switch and doesn't tell the magician the places. Then he switches another 10 times and shouts the places as before. The magician open his eyes, look at the objects and point on one of the objects that participated in the secret switch.
The magician has a bad memory, thus he can only remember one number between 1 to 10. How does he do it?

My direction is not to find the secret switch itself, but 2-3 options with a mutual object and choose that object, but I can't figure out how.

Answer 1

This is more of a summary than an answer, because the answer might be that it is impossible.

We shall first assume that the magician can only remember integers but otherwise has a finely tuned mind.

We can recall that transpositions are generally non-commutative.

To store the information the magician needs, I can see three options, none of which use the full quota of $10$:

• the magician remembers the permutation of $123$
• the magician remembers the position of $4$ and the parity of $123$
• the magician remembers the last digit and the parity of the remaining digits

Depending on how clever the magician is, he/she calculates each of the possible $24$ outcomes at the halfway point. This might not be necessary however, as they only are required to identify one of the two objects affected by the secret switch.

So we need to determine whether or not this is possible.