diff --git a/Cargo.toml b/Cargo.toml
index b7b8d70..8b1c398 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -13,6 +13,8 @@ categories = ["science"]
 
 [dependencies]
 bit-set = "0.4"
+num-rational = { version = "0.1", default-features = false }
+num-traits = "0.1"
 
 [badges]
 travis-ci = { repository = "wangds/puzzle-solver" }
diff --git a/src/lib.rs b/src/lib.rs
index 0218faf..bf8bf66 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -2,6 +2,8 @@
 //! The puzzle rules are expressed as constraints.
 
 extern crate bit_set;
+extern crate num_rational;
+extern crate num_traits;
 
 use std::collections::HashMap;
 use std::ops::Index;