diff --git a/docs/paper/reductions.typ b/docs/paper/reductions.typ
index a0dafe9c..63ec58e7 100644
--- a/docs/paper/reductions.typ
+++ b/docs/paper/reductions.typ
@@ -120,6 +120,7 @@
   "StaffScheduling": [Staff Scheduling],
   "MultiprocessorScheduling": [Multiprocessor Scheduling],
   "MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
+  "SumOfSquaresPartition": [Sum of Squares Partition],
   "SequencingWithinIntervals": [Sequencing Within Intervals],
   "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
   "StringToStringCorrection": [String-to-String Correction],
@@ -2301,6 +2302,14 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   *Example.* Let $A = {3, 7, 1, 8, 2, 4}$ ($n = 6$) and target $B = 11$. Selecting $A' = {3, 8}$ gives sum $3 + 8 = 11 = B$. Another solution: $A' = {7, 4}$ with sum $7 + 4 = 11 = B$.
 ]
 
+#problem-def("SumOfSquaresPartition")[
+  Given a finite set $A = {a_0, dots, a_(n-1)}$ with sizes $s(a_i) in ZZ^+$, a positive integer $K lt.eq |A|$ (number of groups), and a positive integer $J$ (bound), determine whether $A$ can be partitioned into $K$ disjoint sets $A_1, dots, A_K$ such that $sum_(i=1)^K (sum_(a in A_i) s(a))^2 lt.eq J$.
+][
+  Problem SP19 in Garey and Johnson @garey1979. NP-complete in the strong sense, so no pseudo-polynomial time algorithm exists unless $P = NP$. For fixed $K$, a dynamic-programming algorithm runs in $O(n S^(K-1))$ pseudo-polynomial time, where $S = sum s(a)$. The problem remains NP-complete when the exponent 2 is replaced by any fixed rational $alpha > 1$. #footnote[No algorithm improving on brute-force $O(K^n)$ enumeration is known for the general case.] The squared objective penalizes imbalanced partitions, connecting it to variance minimization, load balancing, and $k$-means clustering. Sum of Squares Partition generalizes Partition ($K = 2$, $J = S^2 slash 2$).
+
+  *Example.* Let $A = {5, 3, 8, 2, 7, 1}$ ($n = 6$), $K = 3$ groups, and bound $J = 240$. The partition $A_1 = {8, 1}$, $A_2 = {5, 2}$, $A_3 = {3, 7}$ gives group sums $9, 7, 10$ and sum of squares $81 + 49 + 100 = 230 lt.eq 240 = J$. With a tighter bound $J = 225$, the best achievable partition has group sums ${9, 9, 8}$ yielding $81 + 81 + 64 = 226 > 225$, so the answer is NO.
+]
+
 #{
   let x = load-model-example("SequencingWithReleaseTimesAndDeadlines")
   let n = x.instance.lengths.len()
diff --git a/problemreductions-cli/src/cli.rs b/problemreductions-cli/src/cli.rs
index 27e2bbca..c946abe4 100644
--- a/problemreductions-cli/src/cli.rs
+++ b/problemreductions-cli/src/cli.rs
@@ -232,6 +232,7 @@ Flags by problem type:
   Factoring                       --target, --m, --n
   BinPacking                      --sizes, --capacity
   SubsetSum                       --sizes, --target
+  SumOfSquaresPartition           --sizes, --num-groups, --bound
   PaintShop                       --sequence
   MaximumSetPacking               --sets [--weights]
   MinimumSetCovering              --universe, --sets [--weights]
@@ -500,6 +501,9 @@ pub struct CreateArgs {
     /// Alphabet size for LCS, SCS, or StringToStringCorrection (optional; inferred from the input strings if omitted)
     #[arg(long)]
     pub alphabet_size: Option<usize>,
+    /// Number of groups for SumOfSquaresPartition
+    #[arg(long)]
+    pub num_groups: Option<usize>,
     /// Functional dependencies for MinimumCardinalityKey (semicolon-separated "lhs>rhs" pairs, e.g., "0,1>2;0,2>3")
     #[arg(long)]
     pub dependencies: Option<String>,
diff --git a/problemreductions-cli/src/commands/create.rs b/problemreductions-cli/src/commands/create.rs
index e88054a0..56da8a97 100644
--- a/problemreductions-cli/src/commands/create.rs
+++ b/problemreductions-cli/src/commands/create.rs
@@ -16,6 +16,7 @@ use problemreductions::models::misc::{
     BinPacking, FlowShopScheduling, LongestCommonSubsequence, MinimumTardinessSequencing,
     MultiprocessorScheduling, PaintShop, SequencingWithReleaseTimesAndDeadlines,
     SequencingWithinIntervals, ShortestCommonSupersequence, StringToStringCorrection, SubsetSum,
+    SumOfSquaresPartition,
 };
 use problemreductions::models::BiconnectivityAugmentation;
 use problemreductions::prelude::*;
@@ -94,6 +95,7 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.requirements.is_none()
         && args.num_workers.is_none()
         && args.alphabet_size.is_none()
+        && args.num_groups.is_none()
         && args.dependencies.is_none()
         && args.num_attributes.is_none()
         && args.source_string.is_none()
@@ -328,6 +330,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         }
         "SubgraphIsomorphism" => "--graph 0-1,1-2,2-0 --pattern 0-1",
         "SubsetSum" => "--sizes 3,7,1,8,2,4 --target 11",
+        "SumOfSquaresPartition" => "--sizes 5,3,8,2,7,1 --num-groups 3 --bound 240",
         "ComparativeContainment" => {
             "--universe 4 --r-sets \"0,1,2,3;0,1\" --s-sets \"0,1,2,3;2,3\" --r-weights 2,5 --s-weights 3,6"
         }
@@ -1044,6 +1047,34 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // SumOfSquaresPartition
+        "SumOfSquaresPartition" => {
+            let sizes_str = args.sizes.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --sizes, --num-groups, and --bound\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let num_groups = args.num_groups.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --num-groups\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let bound = args.bound.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --bound\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let sizes: Vec<i64> = util::parse_comma_list(sizes_str)?;
+            (
+                ser(SumOfSquaresPartition::try_new(sizes, num_groups, bound)
+                    .map_err(anyhow::Error::msg)?)?,
+                resolved_variant.clone(),
+            )
+        }
+
         // PaintShop
         "PaintShop" => {
             let seq_str = args.sequence.as_deref().ok_or_else(|| {
@@ -3451,6 +3482,7 @@ mod tests {
             schedules: None,
             requirements: None,
             num_workers: None,
+            num_groups: None,
         }
     }
 
diff --git a/problemreductions-cli/tests/cli_tests.rs b/problemreductions-cli/tests/cli_tests.rs
index 1a94f07e..d2a6ffa9 100644
--- a/problemreductions-cli/tests/cli_tests.rs
+++ b/problemreductions-cli/tests/cli_tests.rs
@@ -1361,6 +1361,29 @@ fn test_create_set_basis_rejects_out_of_range_elements() {
     assert!(!stderr.contains("panicked at"), "stderr: {stderr}");
 }
 
+#[test]
+fn test_create_sum_of_squares_partition_rejects_negative_bound_without_panicking() {
+    let output = pred()
+        .args([
+            "create",
+            "SumOfSquaresPartition",
+            "--sizes",
+            "1,2,3",
+            "--num-groups",
+            "2",
+            "--bound=-1",
+        ])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(
+        stderr.contains("Bound must be nonnegative"),
+        "stderr: {stderr}"
+    );
+    assert!(!stderr.contains("panicked at"), "stderr: {stderr}");
+}
+
 #[test]
 fn test_create_minimum_cardinality_key_problem_help_uses_supported_flags() {
     let output = pred()
@@ -3851,6 +3874,34 @@ fn test_create_pipe_to_solve() {
     );
 }
 
+#[test]
+fn test_solve_ilp_error_suggests_brute_force_fallback() {
+    let problem_json = r#"{
+        "type": "SumOfSquaresPartition",
+        "data": {
+            "sizes": [5, 3, 8, 2, 7, 1],
+            "num_groups": 3,
+            "bound": 240
+        }
+    }"#;
+    let tmp = std::env::temp_dir().join("pred_test_sum_of_squares_partition.json");
+    std::fs::write(&tmp, problem_json).unwrap();
+
+    let output = pred()
+        .args(["solve", tmp.to_str().unwrap()])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(
+        stderr.contains("--solver brute-force"),
+        "stderr should suggest the brute-force fallback, got: {stderr}"
+    );
+
+    std::fs::remove_file(&tmp).ok();
+}
+
 #[test]
 fn test_create_multiple_choice_branching_pipe_to_solve() {
     let create_out = pred()
diff --git a/src/example_db/fixtures/examples.json b/src/example_db/fixtures/examples.json
index d954a60d..cc0a8e78 100644
--- a/src/example_db/fixtures/examples.json
+++ b/src/example_db/fixtures/examples.json
@@ -49,6 +49,7 @@
     {"problem":"SteinerTree","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[2,5,2,1,5,6,1],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,3,null],[1,2,null],[1,3,null],[2,3,null],[2,4,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}},"terminals":[0,2,4]},"samples":[{"config":[1,0,1,1,0,0,1],"metric":{"Valid":6}}],"optimal":[{"config":[1,0,1,1,0,0,1],"metric":{"Valid":6}}]},
     {"problem":"StringToStringCorrection","variant":{},"instance":{"alphabet_size":4,"bound":2,"source":[0,1,2,3,1,0],"target":[0,1,3,2,1]},"samples":[{"config":[8,5],"metric":true}],"optimal":[{"config":[5,7],"metric":true},{"config":[8,5],"metric":true}]},
     {"problem":"StrongConnectivityAugmentation","variant":{"weight":"i32"},"instance":{"bound":8,"candidate_arcs":[[4,0,10],[4,3,3],[4,2,3],[4,1,3],[3,0,7],[3,1,3],[2,0,7],[2,1,3],[1,0,5]],"graph":{"inner":{"edge_property":"directed","edges":[[0,1,null],[1,2,null],[2,3,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}}},"samples":[{"config":[0,0,0,1,0,0,0,0,1],"metric":true},{"config":[0,0,0,0,0,0,0,0,0],"metric":false}],"optimal":[{"config":[0,0,0,1,0,0,0,0,1],"metric":true}]},
+    {"problem":"SumOfSquaresPartition","variant":{},"instance":{"bound":240,"num_groups":3,"sizes":[5,3,8,2,7,1]},"samples":[{"config":[1,2,0,1,2,0],"metric":true}],"optimal":[{"config":[0,0,1,0,2,0],"metric":true},{"config":[0,0,1,0,2,1],"metric":true},{"config":[0,0,1,0,2,2],"metric":true},{"config":[0,0,1,1,2,0],"metric":true},{"config":[0,0,1,1,2,1],"metric":true},{"config":[0,0,1,1,2,2],"metric":true},{"config":[0,0,1,2,2,0],"metric":true},{"config":[0,0,1,2,2,1],"metric":true},{"config":[0,0,1,2,2,2],"metric":true},{"config":[0,0,2,0,1,0],"metric":true},{"config":[0,0,2,0,1,1],"metric":true},{"config":[0,0,2,0,1,2],"metric":true},{"config":[0,0,2,1,1,0],"metric":true},{"config":[0,0,2,1,1,1],"metric":true},{"config":[0,0,2,1,1,2],"metric":true},{"config":[0,0,2,2,1,0],"metric":true},{"config":[0,0,2,2,1,1],"metric":true},{"config":[0,0,2,2,1,2],"metric":true},{"config":[0,1,1,0,2,0],"metric":true},{"config":[0,1,1,0,2,2],"metric":true},{"config":[0,1,1,2,2,0],"metric":true},{"config":[0,1,2,0,1,0],"metric":true},{"config":[0,1,2,0,1,1],"metric":true},{"config":[0,1,2,0,1,2],"metric":true},{"config":[0,1,2,2,1,0],"metric":true},{"config":[0,2,1,0,2,0],"metric":true},{"config":[0,2,1,0,2,1],"metric":true},{"config":[0,2,1,0,2,2],"metric":true},{"config":[0,2,1,1,2,0],"metric":true},{"config":[0,2,2,0,1,0],"metric":true},{"config":[0,2,2,0,1,1],"metric":true},{"config":[0,2,2,1,1,0],"metric":true},{"config":[1,0,0,1,2,1],"metric":true},{"config":[1,0,0,1,2,2],"metric":true},{"config":[1,0,0,2,2,1],"metric":true},{"config":[1,0,2,1,0,0],"metric":true},{"config":[1,0,2,1,0,1],"metric":true},{"config":[1,0,2,1,0,2],"metric":true},{"config":[1,0,2,2,0,1],"metric":true},{"config":[1,1,0,0,2,0],"metric":true},{"config":[1,1,0,0,2,1],"metric":true},{"config":[1,1,0,0,2,2],"metric":true},{"config":[1,1,0,1,2,0],"metric":true},{"config":[1,1,0,1,2,1],"metric":true},{"config":[1,1,0,1,2,2],"metric":true},{"config":[1,1,0,2,2,0],"metric":true},{"config":[1,1,0,2,2,1],"metric":true},{"config":[1,1,0,2,2,2],"metric":true},{"config":[1,1,2,0,0,0],"metric":true},{"config":[1,1,2,0,0,1],"metric":true},{"config":[1,1,2,0,0,2],"metric":true},{"config":[1,1,2,1,0,0],"metric":true},{"config":[1,1,2,1,0,1],"metric":true},{"config":[1,1,2,1,0,2],"metric":true},{"config":[1,1,2,2,0,0],"metric":true},{"config":[1,1,2,2,0,1],"metric":true},{"config":[1,1,2,2,0,2],"metric":true},{"config":[1,2,0,0,2,1],"metric":true},{"config":[1,2,0,1,2,0],"metric":true},{"config":[1,2,0,1,2,1],"metric":true},{"config":[1,2,0,1,2,2],"metric":true},{"config":[1,2,2,0,0,1],"metric":true},{"config":[1,2,2,1,0,0],"metric":true},{"config":[1,2,2,1,0,1],"metric":true},{"config":[2,0,0,1,1,2],"metric":true},{"config":[2,0,0,2,1,1],"metric":true},{"config":[2,0,0,2,1,2],"metric":true},{"config":[2,0,1,1,0,2],"metric":true},{"config":[2,0,1,2,0,0],"metric":true},{"config":[2,0,1,2,0,1],"metric":true},{"config":[2,0,1,2,0,2],"metric":true},{"config":[2,1,0,0,1,2],"metric":true},{"config":[2,1,0,2,1,0],"metric":true},{"config":[2,1,0,2,1,1],"metric":true},{"config":[2,1,0,2,1,2],"metric":true},{"config":[2,1,1,0,0,2],"metric":true},{"config":[2,1,1,2,0,0],"metric":true},{"config":[2,1,1,2,0,2],"metric":true},{"config":[2,2,0,0,1,0],"metric":true},{"config":[2,2,0,0,1,1],"metric":true},{"config":[2,2,0,0,1,2],"metric":true},{"config":[2,2,0,1,1,0],"metric":true},{"config":[2,2,0,1,1,1],"metric":true},{"config":[2,2,0,1,1,2],"metric":true},{"config":[2,2,0,2,1,0],"metric":true},{"config":[2,2,0,2,1,1],"metric":true},{"config":[2,2,0,2,1,2],"metric":true},{"config":[2,2,1,0,0,0],"metric":true},{"config":[2,2,1,0,0,1],"metric":true},{"config":[2,2,1,0,0,2],"metric":true},{"config":[2,2,1,1,0,0],"metric":true},{"config":[2,2,1,1,0,1],"metric":true},{"config":[2,2,1,1,0,2],"metric":true},{"config":[2,2,1,2,0,0],"metric":true},{"config":[2,2,1,2,0,1],"metric":true},{"config":[2,2,1,2,0,2],"metric":true}]},
     {"problem":"TravelingSalesman","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[1,3,2,2,3,1],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[0,3,null],[1,2,null],[1,3,null],[2,3,null]],"node_holes":[],"nodes":[null,null,null,null]}}},"samples":[{"config":[1,0,1,1,0,1],"metric":{"Valid":6}}],"optimal":[{"config":[1,0,1,1,0,1],"metric":{"Valid":6}}]},
     {"problem":"UndirectedTwoCommodityIntegralFlow","variant":{},"instance":{"capacities":[1,1,2],"graph":{"inner":{"edge_property":"undirected","edges":[[0,2,null],[1,2,null],[2,3,null]],"node_holes":[],"nodes":[null,null,null,null]}},"requirement_1":1,"requirement_2":1,"sink_1":3,"sink_2":3,"source_1":0,"source_2":1},"samples":[{"config":[1,0,0,0,0,0,1,0,1,0,1,0],"metric":true}],"optimal":[{"config":[0,0,1,0,1,0,0,0,1,0,1,0],"metric":true},{"config":[1,0,0,0,0,0,1,0,1,0,1,0],"metric":true}]}
   ],
diff --git a/src/lib.rs b/src/lib.rs
index edeb203f..8af9c695 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -62,6 +62,7 @@ pub mod prelude {
         MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop,
         SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals,
         ShortestCommonSupersequence, StaffScheduling, StringToStringCorrection, SubsetSum,
+        SumOfSquaresPartition,
     };
     pub use crate::models::set::{
         ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,
diff --git a/src/models/misc/mod.rs b/src/models/misc/mod.rs
index abad5871..d349bec4 100644
--- a/src/models/misc/mod.rs
+++ b/src/models/misc/mod.rs
@@ -14,6 +14,7 @@
 //! - [`ShortestCommonSupersequence`]: Find a common supersequence of bounded length
 //! - [`StringToStringCorrection`]: String-to-String Correction (derive target via deletions and swaps)
 //! - [`SubsetSum`]: Find a subset summing to exactly a target value
+//! - [`SumOfSquaresPartition`]: Partition integers into K groups minimizing sum of squared group sums
 
 mod bin_packing;
 pub(crate) mod factoring;
@@ -29,6 +30,7 @@ pub(crate) mod shortest_common_supersequence;
 mod staff_scheduling;
 pub(crate) mod string_to_string_correction;
 mod subset_sum;
+pub(crate) mod sum_of_squares_partition;
 
 pub use bin_packing::BinPacking;
 pub use factoring::Factoring;
@@ -44,6 +46,7 @@ pub use shortest_common_supersequence::ShortestCommonSupersequence;
 pub use staff_scheduling::StaffScheduling;
 pub use string_to_string_correction::StringToStringCorrection;
 pub use subset_sum::SubsetSum;
+pub use sum_of_squares_partition::SumOfSquaresPartition;
 
 #[cfg(feature = "example-db")]
 pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::ModelExampleSpec> {
@@ -57,6 +60,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(shortest_common_supersequence::canonical_model_example_specs());
     specs.extend(string_to_string_correction::canonical_model_example_specs());
     specs.extend(minimum_tardiness_sequencing::canonical_model_example_specs());
+    specs.extend(sum_of_squares_partition::canonical_model_example_specs());
     specs.extend(sequencing_with_release_times_and_deadlines::canonical_model_example_specs());
     specs
 }
diff --git a/src/models/misc/sum_of_squares_partition.rs b/src/models/misc/sum_of_squares_partition.rs
new file mode 100644
index 00000000..508efa0d
--- /dev/null
+++ b/src/models/misc/sum_of_squares_partition.rs
@@ -0,0 +1,207 @@
+//! Sum of Squares Partition problem implementation.
+//!
+//! Given a finite set of positive integers, K groups, and a bound J,
+//! determine whether the set can be partitioned into K groups such that
+//! the sum of squared group sums is at most J.
+//! NP-complete in the strong sense (Garey & Johnson, SP19).
+
+use crate::registry::{FieldInfo, ProblemSchemaEntry};
+use crate::traits::{Problem, SatisfactionProblem};
+use serde::de::Error;
+use serde::{Deserialize, Deserializer, Serialize};
+
+inventory::submit! {
+    ProblemSchemaEntry {
+        name: "SumOfSquaresPartition",
+        display_name: "Sum of Squares Partition",
+        aliases: &[],
+        dimensions: &[],
+        module_path: module_path!(),
+        description: "Partition positive integers into K groups minimizing sum of squared group sums, subject to bound J",
+        fields: &[
+            FieldInfo { name: "sizes", type_name: "Vec<i64>", description: "Positive integer size s(a) for each element a in A" },
+            FieldInfo { name: "num_groups", type_name: "usize", description: "Number of groups K in the partition" },
+            FieldInfo { name: "bound", type_name: "i64", description: "Upper bound J on the sum of squared group sums" },
+        ],
+    }
+}
+
+/// The Sum of Squares Partition problem (Garey & Johnson SP19).
+///
+/// Given a finite set `A` with sizes `s(a) ∈ Z⁺` for each `a ∈ A`,
+/// a positive integer `K ≤ |A|` (number of groups), and a positive
+/// integer `J` (bound), determine whether `A` can be partitioned into
+/// `K` disjoint sets `A_1, ..., A_K` such that:
+///
+/// `∑_{i=1}^{K} (∑_{a ∈ A_i} s(a))² ≤ J`
+///
+/// # Representation
+///
+/// Each element has a variable in `{0, ..., K-1}` representing its
+/// group assignment. A configuration is satisfying if the sum of
+/// squared group sums does not exceed `J`.
+///
+/// # Example
+///
+/// ```
+/// use problemreductions::models::misc::SumOfSquaresPartition;
+/// use problemreductions::{Problem, Solver, BruteForce};
+///
+/// // 6 elements with sizes [5, 3, 8, 2, 7, 1], K=3 groups, bound J=240
+/// let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+/// let solver = BruteForce::new();
+/// let solution = solver.find_satisfying(&problem);
+/// assert!(solution.is_some());
+/// ```
+#[derive(Debug, Clone, Serialize)]
+pub struct SumOfSquaresPartition {
+    /// Positive integer sizes for each element.
+    sizes: Vec<i64>,
+    /// Number of groups K.
+    num_groups: usize,
+    /// Upper bound J on the sum of squared group sums.
+    bound: i64,
+}
+
+impl SumOfSquaresPartition {
+    fn validate_inputs(sizes: &[i64], num_groups: usize, bound: i64) -> Result<(), String> {
+        if sizes.iter().any(|&size| size <= 0) {
+            return Err("All sizes must be positive (> 0)".to_string());
+        }
+        if num_groups == 0 {
+            return Err("Number of groups must be positive".to_string());
+        }
+        if num_groups > sizes.len() {
+            return Err("Number of groups must not exceed number of elements".to_string());
+        }
+        if bound < 0 {
+            return Err("Bound must be nonnegative".to_string());
+        }
+        Ok(())
+    }
+
+    /// Create a new SumOfSquaresPartition instance, returning validation errors.
+    pub fn try_new(sizes: Vec<i64>, num_groups: usize, bound: i64) -> Result<Self, String> {
+        Self::validate_inputs(&sizes, num_groups, bound)?;
+        Ok(Self {
+            sizes,
+            num_groups,
+            bound,
+        })
+    }
+
+    /// Create a new SumOfSquaresPartition instance.
+    ///
+    /// # Panics
+    ///
+    /// Panics if any size is not positive (must be > 0), if `num_groups` is 0,
+    /// if `num_groups` exceeds the number of elements, or if `bound` is negative.
+    pub fn new(sizes: Vec<i64>, num_groups: usize, bound: i64) -> Self {
+        Self::try_new(sizes, num_groups, bound).unwrap_or_else(|message| panic!("{message}"))
+    }
+
+    /// Returns the element sizes.
+    pub fn sizes(&self) -> &[i64] {
+        &self.sizes
+    }
+
+    /// Returns the number of groups K.
+    pub fn num_groups(&self) -> usize {
+        self.num_groups
+    }
+
+    /// Returns the bound J.
+    pub fn bound(&self) -> i64 {
+        self.bound
+    }
+
+    /// Returns the number of elements |A|.
+    pub fn num_elements(&self) -> usize {
+        self.sizes.len()
+    }
+
+    /// Compute the sum of squared group sums for a given configuration.
+    ///
+    /// Returns `None` if the configuration is invalid (wrong length or
+    /// out-of-range group index), or if arithmetic overflows `i64`.
+    pub fn sum_of_squares(&self, config: &[usize]) -> Option<i64> {
+        if config.len() != self.sizes.len() {
+            return None;
+        }
+        let mut group_sums = vec![0i128; self.num_groups];
+        for (i, &g) in config.iter().enumerate() {
+            if g >= self.num_groups {
+                return None;
+            }
+            group_sums[g] = group_sums[g].checked_add(i128::from(self.sizes[i]))?;
+        }
+        group_sums
+            .into_iter()
+            .try_fold(0i128, |total, group_sum| {
+                let square = group_sum.checked_mul(group_sum)?;
+                total.checked_add(square)
+            })
+            .and_then(|total| i64::try_from(total).ok())
+    }
+}
+
+#[derive(Deserialize)]
+struct SumOfSquaresPartitionData {
+    sizes: Vec<i64>,
+    num_groups: usize,
+    bound: i64,
+}
+
+impl<'de> Deserialize<'de> for SumOfSquaresPartition {
+    fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
+    where
+        D: Deserializer<'de>,
+    {
+        let data = SumOfSquaresPartitionData::deserialize(deserializer)?;
+        Self::try_new(data.sizes, data.num_groups, data.bound).map_err(D::Error::custom)
+    }
+}
+
+impl Problem for SumOfSquaresPartition {
+    const NAME: &'static str = "SumOfSquaresPartition";
+    type Metric = bool;
+
+    fn variant() -> Vec<(&'static str, &'static str)> {
+        crate::variant_params![]
+    }
+
+    fn dims(&self) -> Vec<usize> {
+        vec![self.num_groups; self.sizes.len()]
+    }
+
+    fn evaluate(&self, config: &[usize]) -> bool {
+        match self.sum_of_squares(config) {
+            Some(sos) => sos <= self.bound,
+            None => false,
+        }
+    }
+}
+
+impl SatisfactionProblem for SumOfSquaresPartition {}
+
+crate::declare_variants! {
+    default sat SumOfSquaresPartition => "num_groups^num_elements",
+}
+
+#[cfg(feature = "example-db")]
+pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::ModelExampleSpec> {
+    vec![crate::example_db::specs::ModelExampleSpec {
+        id: "sum_of_squares_partition",
+        build: || {
+            // sizes=[5,3,8,2,7,1], K=3, J=240
+            // Satisfying: groups {8,1},{5,2},{3,7} -> sums 9,7,10 -> 81+49+100=230 <= 240
+            // Config: a0=5->group1, a1=3->group2, a2=8->group0, a3=2->group1, a4=7->group2, a5=1->group0
+            let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+            crate::example_db::specs::satisfaction_example(problem, vec![vec![1, 2, 0, 1, 2, 0]])
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../../unit_tests/models/misc/sum_of_squares_partition.rs"]
+mod tests;
diff --git a/src/models/mod.rs b/src/models/mod.rs
index 7bd03578..50acaa59 100644
--- a/src/models/mod.rs
+++ b/src/models/mod.rs
@@ -25,7 +25,7 @@ pub use misc::{
     BinPacking, Factoring, FlowShopScheduling, Knapsack, LongestCommonSubsequence,
     MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop,
     SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals, ShortestCommonSupersequence,
-    StaffScheduling, StringToStringCorrection, SubsetSum,
+    StaffScheduling, StringToStringCorrection, SubsetSum, SumOfSquaresPartition,
 };
 pub use set::{
     ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,
diff --git a/src/unit_tests/models/misc/sum_of_squares_partition.rs b/src/unit_tests/models/misc/sum_of_squares_partition.rs
new file mode 100644
index 00000000..f0defb70
--- /dev/null
+++ b/src/unit_tests/models/misc/sum_of_squares_partition.rs
@@ -0,0 +1,230 @@
+use super::*;
+use crate::solvers::{BruteForce, Solver};
+use crate::traits::Problem;
+
+#[test]
+fn test_sum_of_squares_partition_basic() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    assert_eq!(problem.num_elements(), 6);
+    assert_eq!(problem.num_groups(), 3);
+    assert_eq!(problem.bound(), 240);
+    assert_eq!(problem.sizes(), &[5, 3, 8, 2, 7, 1]);
+    assert_eq!(problem.dims(), vec![3; 6]);
+    assert_eq!(
+        <SumOfSquaresPartition as Problem>::NAME,
+        "SumOfSquaresPartition"
+    );
+    assert_eq!(<SumOfSquaresPartition as Problem>::variant(), vec![]);
+}
+
+#[test]
+fn test_sum_of_squares_partition_evaluate_satisfying() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    // Groups: {8,1}=9, {5,2}=7, {3,7}=10 -> 81+49+100=230 <= 240
+    assert!(problem.evaluate(&[1, 2, 0, 1, 2, 0]));
+}
+
+#[test]
+fn test_sum_of_squares_partition_evaluate_unsatisfying() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 225);
+    // Best achievable: sums {9,9,8} -> 81+81+64=226 > 225
+    // Groups: {8,1}=9, {5,2}=7, {3,7}=10 -> 81+49+100=230 > 225
+    assert!(!problem.evaluate(&[1, 2, 0, 1, 2, 0]));
+}
+
+#[test]
+fn test_sum_of_squares_partition_all_in_one_group() {
+    // All elements in one group is maximally imbalanced
+    let problem = SumOfSquaresPartition::new(vec![1, 2, 3], 2, 100);
+    // All in group 0: sum=6, group1=0 -> 36+0=36 <= 100
+    assert!(problem.evaluate(&[0, 0, 0]));
+    // Tight bound: all in group 0 gives 36
+    let problem2 = SumOfSquaresPartition::new(vec![1, 2, 3], 2, 35);
+    assert!(!problem2.evaluate(&[0, 0, 0])); // 36 > 35
+}
+
+#[test]
+fn test_sum_of_squares_partition_sum_of_squares_helper() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    // Groups: {8,1}=9, {5,2}=7, {3,7}=10 -> 81+49+100=230
+    assert_eq!(problem.sum_of_squares(&[1, 2, 0, 1, 2, 0]), Some(230));
+}
+
+#[test]
+fn test_sum_of_squares_partition_invalid_config() {
+    let problem = SumOfSquaresPartition::new(vec![1, 2, 3], 2, 100);
+    // Wrong length
+    assert!(!problem.evaluate(&[0, 0]));
+    assert!(!problem.evaluate(&[0, 0, 0, 0]));
+    // Group index out of range
+    assert!(!problem.evaluate(&[0, 2, 0]));
+    // sum_of_squares returns None for invalid configs
+    assert_eq!(problem.sum_of_squares(&[0, 0]), None);
+    assert_eq!(problem.sum_of_squares(&[0, 2, 0]), None);
+}
+
+#[test]
+fn test_sum_of_squares_partition_two_elements() {
+    // Two elements, 2 groups: balanced vs imbalanced
+    let problem = SumOfSquaresPartition::new(vec![3, 5], 2, 34);
+    // {3},{5} -> 9+25=34 <= 34
+    assert!(problem.evaluate(&[0, 1]));
+    // {3,5},{} -> 64+0=64 > 34
+    assert!(!problem.evaluate(&[0, 0]));
+    // {},{3,5} -> 0+64=64 > 34
+    assert!(!problem.evaluate(&[1, 1]));
+}
+
+#[test]
+fn test_sum_of_squares_partition_brute_force() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    let solver = BruteForce::new();
+    let solution = solver
+        .find_satisfying(&problem)
+        .expect("should find a satisfying solution");
+    assert!(problem.evaluate(&solution));
+}
+
+#[test]
+fn test_sum_of_squares_partition_brute_force_all() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    let solver = BruteForce::new();
+    let solutions = solver.find_all_satisfying(&problem);
+    assert!(!solutions.is_empty());
+    for sol in &solutions {
+        assert!(problem.evaluate(sol));
+    }
+}
+
+#[test]
+fn test_sum_of_squares_partition_unsatisfiable() {
+    // Bound too tight: impossible to satisfy
+    // 3 elements [10, 10, 10], 3 groups, bound 299
+    // Best: each element in its own group -> 100+100+100=300 > 299
+    let problem = SumOfSquaresPartition::new(vec![10, 10, 10], 3, 299);
+    let solver = BruteForce::new();
+    let solution = solver.find_satisfying(&problem);
+    assert!(solution.is_none());
+}
+
+#[test]
+fn test_sum_of_squares_partition_serialization() {
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+    let json = serde_json::to_value(&problem).unwrap();
+    assert_eq!(
+        json,
+        serde_json::json!({
+            "sizes": [5, 3, 8, 2, 7, 1],
+            "num_groups": 3,
+            "bound": 240,
+        })
+    );
+    let restored: SumOfSquaresPartition = serde_json::from_value(json).unwrap();
+    assert_eq!(restored.sizes(), problem.sizes());
+    assert_eq!(restored.num_groups(), problem.num_groups());
+    assert_eq!(restored.bound(), problem.bound());
+}
+
+#[test]
+fn test_sum_of_squares_partition_deserialization_rejects_invalid_fields() {
+    let invalid_cases = [
+        serde_json::json!({
+            "sizes": [-1, 2, 3],
+            "num_groups": 2,
+            "bound": 100,
+        }),
+        serde_json::json!({
+            "sizes": [0, 2, 3],
+            "num_groups": 2,
+            "bound": 100,
+        }),
+        serde_json::json!({
+            "sizes": [1, 2, 3],
+            "num_groups": 0,
+            "bound": 100,
+        }),
+        serde_json::json!({
+            "sizes": [1, 2],
+            "num_groups": 3,
+            "bound": 100,
+        }),
+        serde_json::json!({
+            "sizes": [1, 2, 3],
+            "num_groups": 2,
+            "bound": -1,
+        }),
+    ];
+
+    for invalid in invalid_cases {
+        assert!(serde_json::from_value::<SumOfSquaresPartition>(invalid).is_err());
+    }
+}
+
+#[test]
+fn test_sum_of_squares_partition_sum_overflow_returns_none() {
+    let problem = SumOfSquaresPartition::new(vec![i64::MAX, 1], 1, i64::MAX);
+
+    assert_eq!(problem.sum_of_squares(&[0, 0]), None);
+    assert!(!problem.evaluate(&[0, 0]));
+}
+
+#[test]
+fn test_sum_of_squares_partition_square_overflow_returns_none() {
+    let problem = SumOfSquaresPartition::new(vec![3_037_000_500], 1, i64::MAX);
+
+    assert_eq!(problem.sum_of_squares(&[0]), None);
+    assert!(!problem.evaluate(&[0]));
+}
+
+#[test]
+fn test_sum_of_squares_partition_paper_example() {
+    // Instance from the issue: sizes=[5,3,8,2,7,1], K=3, J=240
+    let problem = SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240);
+
+    // Verify the satisfying partition from the issue:
+    // A1={8,1}(sums to 9), A2={5,2}(sums to 7), A3={3,7}(sums to 10)
+    // Config: a0=5->group1, a1=3->group2, a2=8->group0, a3=2->group1, a4=7->group2, a5=1->group0
+    let config = vec![1, 2, 0, 1, 2, 0];
+    assert!(problem.evaluate(&config));
+    assert_eq!(problem.sum_of_squares(&config), Some(230));
+
+    // Brute force finds satisfying solutions
+    let solver = BruteForce::new();
+    let all = solver.find_all_satisfying(&problem);
+    assert!(!all.is_empty());
+    // All solutions must have sum-of-squares <= 240
+    for sol in &all {
+        let sos = problem.sum_of_squares(sol).unwrap();
+        assert!(sos <= 240);
+    }
+}
+
+#[test]
+#[should_panic(expected = "positive")]
+fn test_sum_of_squares_partition_negative_size_panics() {
+    SumOfSquaresPartition::new(vec![-1, 2, 3], 2, 100);
+}
+
+#[test]
+#[should_panic(expected = "positive")]
+fn test_sum_of_squares_partition_zero_size_panics() {
+    SumOfSquaresPartition::new(vec![0, 2, 3], 2, 100);
+}
+
+#[test]
+#[should_panic(expected = "Number of groups must be positive")]
+fn test_sum_of_squares_partition_zero_groups_panics() {
+    SumOfSquaresPartition::new(vec![1, 2, 3], 0, 100);
+}
+
+#[test]
+#[should_panic(expected = "Number of groups must not exceed")]
+fn test_sum_of_squares_partition_too_many_groups_panics() {
+    SumOfSquaresPartition::new(vec![1, 2], 3, 100);
+}
+
+#[test]
+#[should_panic(expected = "Bound must be nonnegative")]
+fn test_sum_of_squares_partition_negative_bound_panics() {
+    SumOfSquaresPartition::new(vec![1, 2, 3], 2, -1);
+}
diff --git a/src/unit_tests/trait_consistency.rs b/src/unit_tests/trait_consistency.rs
index cc81aba4..a142dc4a 100644
--- a/src/unit_tests/trait_consistency.rs
+++ b/src/unit_tests/trait_consistency.rs
@@ -106,6 +106,10 @@ fn test_all_problems_implement_trait_correctly() {
         &SequencingWithReleaseTimesAndDeadlines::new(vec![1, 2, 1], vec![0, 0, 2], vec![3, 3, 4]),
         "SequencingWithReleaseTimesAndDeadlines",
     );
+    check_problem_trait(
+        &SumOfSquaresPartition::new(vec![5, 3, 8, 2, 7, 1], 3, 240),
+        "SumOfSquaresPartition",
+    );
 }
 
 #[test]