Multiple package instances within a single package slot have been pulled

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Problem

Example problem encountered during emerge --update:

!!! Multiple package instances within a single package slot have been pulled
!!! into the dependency graph, resulting in a slot conflict:
dev-java/icedtea-web:0
(dev-java/icedtea-web-1.5.1-r1:0/0::gentoo, installed) pulled in by
dev-java/icedtea-web:0[icedtea7,nsplugin] required by (dev-java/icedtea-7.2.6.3-r2:7/7::gentoo, installed)
^^^^^^^^
dev-java/icedtea-web:0[icedtea7] required by (dev-java/icedtea-7.2.6.3-r2:7/7::gentoo, installed)
^^^^^^^^
(dev-java/icedtea-web-1.6.1-r1:0/0::gentoo, ebuild scheduled for merge) pulled in by
>=dev-java/icedtea-web-1.6.1:0 required by (dev-java/icedtea-bin-3.0.1:8/8::gentoo, ebuild scheduled for merge)
^^ ^^^^^^^
It may be possible to solve this problem by using package.mask to
prevent one of those packages from being selected. However, it is also
possible that conflicting dependencies exist such that they are
impossible to satisfy simultaneously. If such a conflict exists in
the dependencies of two different packages, then those packages can
not be installed simultaneously. You may want to try a larger value of
the --backtrack option, such as --backtrack=30, in order to see if
that will solve this conflict automatically.
For more information, see MASKED PACKAGES section in the emerge man
page or refer to the Gentoo Handbook.

Solution

The solution may differ according to the exact nature of the problem (same symptoms, different causes).
If the solution to your problem is not listed here, open a new ticket, or, if you know the solution, improve this wiki page.

use --backtrack

Try using --backtrack=10 or with an ever higher number.
man emerge:

--backtrack=COUNT
Specifies an integer number of times to backtrack if dependency calculation fails due to a conflict or an unsatisfied dependency (default: ´3´).

Conflict in world file

In the example given above, the user emerged dev-java/icedtea so that it was listed in the /var/lib/portage/world file.
Simply deleting the corresponding line from the world file and emerging again was enough to solve the problem.