Spectator Locates Magician’s Card by Unnamed Magician

Imagine:

The magician and the spectator are seated together at a table. The magician hands the spectator a deck of cards and asks them to examine all of the cards (making sure that they're all different, etc.) and to shuffle them thoroughly. After this is done, the spectator is asked to cut the deck into two approximately equal piles and to keep one for themself and to give the other to the magician.

The magician now says that each of them will bring their respective half under the table and spread through the faces of the cards in order to select (and remove) one, which will then be placed face down on the table (with the rest of the half placed next to that). The magician explains, this is to be done under the table so that there is no chance the face of the spectator's selected card can be glimpsed by the magician (and vice versa). Alternatively, the magician can turn his back as the spectator makes their selection. Once both the spectator and the magician have made a selection, the magician says that each of them will lose their selected card back into their respective half, and after this is done, both the magician and the spectator shuffle their respective halves in order to lose their selected cards further.

With that done, a slip of paper is introduced. The magician explains, he is going to write down the name of his selected card on it so that later he can't be accused of changing it / lying about it. After the writing, the slip is folded and handed over to the spectator who is asked to keep it in their pocket until the end.

At this point, the magician tells the spectator that they each will attempt to locate the other's selected card; so the magician and the spectator exchange halves and each of them pulls out a card at random (sight unseen) from the other's half in hopes to locate the other's card. Once each of them has pulled out a card from the other's half, the magician asks the spectator to name their selected card, after which the magician flips over the card he pulled out, proving that he successfully located their card. Next, the magician names his selected card and tells the spectator to verify that he isn't lying by asking them to check the slip of paper in their pocket. After that, the spectator flips over the card they pulled out, realizing that they too successfully located the magician's card.

Some important conditions of note:

At the beginning of the effect, the spectator gets to examine and shuffle all of the cards to their heart's content.

After the shuffling, the spectator themself splits the deck into two halves and decides which half to give to the magician and which half to keep for themself.

When the magician writes down the name of his selected card on a slip of paper, he hands that slip over to the spectator and never touches it again. The spectator themself opens it at the end.

The above is the description and conditions of the first effect / method provided in the e-book. There are two other methods provided as well to achieve a (very similar) effect. The materials required to perform the first and third methods are standard materials found in a magician's toolkit (you will either already own them or can easily obtain them). However, one of the materials required to perform the second method is unorthodox and quite expensive to buy (or build).