# Compare k-means fixed points

> ML Prep screening item.

- Stable ID: `ml-screen-unsupervised-advanced-local-optima`
- Collection: screening
- Area: Machine Learning
- Topic: Unsupervised Learning
- Origin: original
- Difficulty: hard
- Expected time: 11 minutes
- Skills: k-means, clustering-objective
- Item JSON: https://mlprep.iwase.dev/item/ml-screen-unsupervised-advanced-local-optima.json

## Problem

For the points $\{0,4,5,9\}$ with $K=2$, evaluate the objective for contiguous splits after $0$, after $4$, and after $5$, using cluster means. Identify the globally best splits and show that the middle split is a Lloyd fixed point despite being worse.


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `ml-screen-unsupervised-advanced-local-optima`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.