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Although it causes an Equipment to become attached to a creature, reconfigure is not an “equip ability” for the purpose of cards like Fighter Class and Leonin Shikari.

An Equipment creature can never become attached to itself. If an effect tries to do this, nothing happens.

An Equipment creature with reconfigure can be attached to creatures by effects other than its reconfigure ability, such as the activated ability of Brass Squire.

An Equipment doesn't become tapped when the permanent it's attached to becomes tapped. For example, if you attack with a creature that is equipped with Acquisition Octopus, then use reconfigure to unattach Acquisition Octopus after combat, the Octopus will be untapped and could be used to block during your opponent's turn.

As soon as an Equipment creature with reconfigure stops being a creature, any Equipment and Auras with enchant creature abilities become unattached. Auras that can enchant an Equipment that isn't a creature remain attached to it.

Attaching an Equipment with reconfigure to a creature causes that Equipment to stop being a creature until it becomes unattached. It also loses any creature subtypes it had.

If a permanent with reconfigure is somehow still a creature after it becomes attached (perhaps due to an effect like that of March of the Machines), it immediately becomes unattached from the equipped creature.

If an Equipment with reconfigure somehow loses its abilities while it is attached, the effect causing it to not be a creature continues to apply until it becomes unattached.

Reconfigure represents two activated abilities. Reconfigure [cost] means “[Cost]: Attach this permanent to another target creature you control. Activate only as a sorcery,” and “[Cost]: Unattach this permanent. Activate only if this permanent is attached to a creature and only as a sorcery.”

Similarly, if an Equipment is tapped, its reconfigure abilities may still be activated and it may still become attached to creatures. Becoming attached doesn't untap it. In most cases, an attached Equipment being tapped won't affect gameplay, but it will be relevant if it becomes unattached again before it untaps.