diff --git a/src/constraint/alldifferent.rs b/src/constraint/alldifferent.rs
index 98f9627..d9a6871 100644
--- a/src/constraint/alldifferent.rs
+++ b/src/constraint/alldifferent.rs
@@ -1,6 +1,7 @@
 //! All different implementation.
 
 use std::collections::HashMap;
+use std::rc::Rc;
 
 use ::{Constraint,PuzzleSearch,Val,VarToken};
 
@@ -75,6 +76,19 @@ impl Constraint for AllDifferent {
 
         true
     }
+
+    fn substitute(&self, from: VarToken, to: VarToken)
+            -> Option<Rc<Constraint>> {
+        if let Some(idx) = self.vars.iter().position(|&var| var == from) {
+            if !self.vars.contains(&to) {
+                let mut new_vars = self.vars.clone();
+                new_vars[idx] = to;
+                return Some(Rc::new(AllDifferent{ vars: new_vars }));
+            }
+        }
+
+        None
+    }
 }
 
 #[cfg(test)]
diff --git a/src/constraint/equality.rs b/src/constraint/equality.rs
index 24d4f4d..b35aaba 100644
--- a/src/constraint/equality.rs
+++ b/src/constraint/equality.rs
@@ -1,5 +1,7 @@
 //! Equality implementation.
 
+use std::rc::Rc;
+
 use ::{Constraint,LinExpr,PuzzleSearch,Val,VarToken};
 use intdiv::IntDiv;
 
@@ -140,6 +142,16 @@ impl Constraint for Equality {
 
         true
     }
+
+    fn substitute(&self, from: VarToken, to: VarToken)
+            -> Option<Rc<Constraint>> {
+        let mut eqn = self.eqn.clone();
+        if let Some(coef) = eqn.coef.remove(&from) {
+            eqn = eqn + coef * to;
+        }
+
+        Some(Rc::new(Equality{ eqn: eqn }))
+    }
 }
 
 #[cfg(test)]
diff --git a/src/constraint/mod.rs b/src/constraint/mod.rs
index a7ae8f8..3ae214f 100644
--- a/src/constraint/mod.rs
+++ b/src/constraint/mod.rs
@@ -1,9 +1,11 @@
 //! Constraint trait, and some common constraints.
 //!
 //! Note that all puzzle states visited during the solution search
-//! share the same set of constraints.  This means that you cannot
-//! store additional information about the state (e.g. caches) in the
-//! constraint to reuse later.
+//! share the same set of constraint objects.  This means that you
+//! cannot store additional information about the state (e.g. caches)
+//! in the constraint to reuse later.
+
+use std::rc::Rc;
 
 use ::{PuzzleSearch,Val,VarToken};
 
@@ -28,6 +30,14 @@ pub trait Constraint {
     fn on_updated(&self, _search: &mut PuzzleSearch) -> bool {
         true
     }
+
+    /// Substitute the "from" variable with the "to" variable.
+    ///
+    /// Returns a new constraint with all instances of "from" replaced
+    /// with "to", or None if a contradiction was found in the
+    /// process.
+    fn substitute(&self, from: VarToken, to: VarToken)
+            -> Option<Rc<Constraint>>;
 }
 
 pub use self::alldifferent::AllDifferent;
diff --git a/tests/queens.rs b/tests/queens.rs
index 788fda5..1fc60a0 100644
--- a/tests/queens.rs
+++ b/tests/queens.rs
@@ -4,6 +4,7 @@
 
 extern crate puzzle_solver;
 
+use std::rc::Rc;
 use puzzle_solver::{Constraint,Puzzle,PuzzleSearch,Solution,Val,VarToken};
 
 struct NoDiagonal {
@@ -29,6 +30,11 @@ impl Constraint for NoDiagonal {
 
         true
     }
+
+    fn substitute(&self, _from: VarToken, _to: VarToken)
+            -> Option<Rc<Constraint>> {
+        unimplemented!();
+    }
 }
 
 fn make_queens(n: usize) -> (Puzzle, Vec<VarToken>) {